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                        "type": "text",
                        "content": "高等代数笔记"
                    }
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                        "type": "text",
                        "content": "作者：Reichtum"
                    }
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                        "type": "text",
                        "content": "组织：ZheJiang University"
                    }
                ]
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "时间：September 20, 2022"
                    }
                ]
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                        "type": "text",
                        "content": "版本：0.1"
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            "content": {
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                    {
                        "type": "text",
                        "content": "There was no royal road to geometry. ——Euclid"
                    }
                ]
            },
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        },
        {
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    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "目录"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "第1章 多项式 1\n\n1.1 一元多项式 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1\n1.1.1 整除理论 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1\n1.1.2 最大公因式 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\n1.1.3 互素 . . . . . . . . . . . . . . . . . . 3\n1.1.4 不可约多项式与因式分解 4\n1.1.5 余数定理及其应用 5\n1.1.6 重因式与重根 6\n1.1.7 整数与多项式 7\n1.2 特殊数域上的一元多项式理论 8\n1.2.1 复数域上的一元多项式 8\n1.2.2 实数域上的一元多项式 8\n1.2.3 有理数域上一元多项式因式分解 8\n1.2.4 一元有理多项式的根 10\n1.3 多元多项式 10\n1.3.1 基本概念 10\n1.3.2 对称多项式 11\n1.3.3 判别式 12\n1.3.4 结式 12\n第2章 矩阵 14\n2.1 矩阵基础 14\n2.1.1 矩阵概念与运算 14\n2.1.2 矩阵的逆 14\n2.1.3 伴随矩阵的性质 15\n2.1.4 广义逆矩阵 16\n2.1.5 Moore-Penrose广义逆 17\n2.2 分块矩阵 18\n2.2.1 常见矩阵拆分技巧 18\n2.2.2 分块初等变换 19\n2.2.3 分块打洞技巧总结 19\n2.2.4 分块矩阵行列式 20\n2.2.5 分块矩阵求逆 20\n2.3 矩阵的秩 21\n2.3.1 秩的概念 21\n2.3.2 初等变换标准型与矩阵分解 21\n2.3.3 利用初等变换标准型进行矩阵分解 22\n2.3.4 秩不等式 22\n2.3.5 空间分解法研究矩阵秩 24\n2.4 矩阵的迹 25\n2.5 几种技巧矩阵 25"
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "2.5.1 初等矩阵 25  \n2.5.2 基本矩阵：仅一个元素为1的矩阵 25  \n2.5.3 置换矩阵 26  \n2.6 特殊矩阵性质总结 27  \n2.6.1 对角矩阵 27  \n2.6.2 上、下三角矩阵 27  \n2.6.3 实对称矩阵 28  \n2.6.4 反对称矩阵 28  \n2.6.5 Hermite矩阵 30  \n2.6.6 正交矩阵 30  \n2.6.7 Unitary矩阵 31  \n2.6.8 幂零矩阵 31  \n2.6.9 幂等矩阵 31  \n2.6.10 对角占优矩阵 33  \n\n第3章 行列式 34  \n3.1 行列式基本概念与性质 34  \n3.2 行列式展开 34  \n3.3 特殊的行列式 34  \n3.3.1 爪型行列式 34  \n3.3.2 大对角型(三斜线) 35  \n3.3.3 行和相同型 35  \n3.3.4 循环型 35  \n3.3.5 行列式加边法 36  \n3.4 Cramer法则 36  \n\n第4章 线性方程组 37  \n4.1 线性方程组解的判定定理 37  \n4.1.1 解存在判定定理 37  \n4.1.2 线性方程组同解 37  \n4.1.3 解个数判定定理 37  \n4.2 线性方程组的求解 37  \n4.2.1 Ax=0求解 37  \n4.2.2 Ax=b求解 38  \n4.2.3 线性方程组公共解的计算 39  \n4.3 最小二乘解 39  \n4.3.1  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T A"
                    },
                    {
                        "type": "text",
                        "content": " 的研究 39  \n4.3.2 正规方程组 39  \n\n第5章 线性空间与线性变换 41  \n5.1 线性空间 41  \n5.1.1 线性空间的基本概念 41  \n5.1.2 线性相关、线性无关与向量组的秩 41  \n5.1.3 线性空间同构 42  \n5.1.4 基、坐标 43  \n5.1.5 坐标变换 44"
                    }
                ]
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "目录"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "ii"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "5.1.6 和与交、维数定理 45  \n5.1.7 直和理论 47  \n5.1.8 商空间 48  \n5.2 线性映射 48  \n5.2.1 线性映射的概念 48  \n5.2.2 核与像 48  \n5.2.3 幂等变换的核与像 50  \n5.2.4 一般线性变换的核像分解 50  \n5.3 线性变换 51  \n5.3.1 线性变换基本概念 51  \n5.3.2 线性变换在基下的矩阵、变换后的坐标 51  \n5.3.3 线性变换在不同基下的矩阵：相似关系 52  \n5.3.4 相似矩阵的性质 53  \n5.4 特征值、相似对角化 53  \n5.4.1 特征值与特征向量 53  \n5.4.2 特征值与特征向量进一步结论 54  \n5.4.3 相似的进一步研究 55  \n5.4.4 相似对角化 56  \n5.4.5 相似上三角型 57  \n5.5 最小多项式 58  \n5.5.1 最小多项式与基本性质 58  \n5.5.2 Hamilton-Cayley 定理 58  \n5.5.3 最小多项式与相似对角化 59  \n\n6章 Jordan 标准型 60  \n6.1 Jordan 标准形：空间分解法 60  \n6.1.1 空间第二分解定理：循环不变子空间分解 60  \n6.1.2 核像升链与根子空间 60  \n6.1.3 空间第一分解定理：根子空间分解 61  \n6.1.4 Jordan 标准形：空间分解法 62  \n6.2 空间分解法：零化多项式角度 62  \n6.2.1 根子空间分解：矩阵角度 62  \n6.2.2 根子空间分解：线性变换角度 63  \n6.2.3 Jordan 阵与循环不变子空间 64  \n6.3 Jordan 标准形： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵法 65  \n6.3.1  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵与相抵 65  \n6.3.2 行列式因子、不变因子 66  \n6.3.3 初等因子与 Jordan 标准型 67  \n6.4  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵的进一步研究 68  \n6.4.1 Jordan 标准型的直接计算 68  \n6.4.2  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k"
                    },
                    {
                        "type": "text",
                        "content": " 阶 Jordan 块的个数 70  \n6.4.3 分块矩阵的 Jordan 阵 71  \n6.4.4 过渡矩阵的计算 71"
                    }
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        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "目录"
                    }
                ]
            },
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        },
        {
            "type": "page_footer",
            "content": {
                "page_footer_content": [
                    {
                        "type": "text",
                        "content": "第7章 双线性函数与二次型"
                    }
                ]
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        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "73"
                    }
                ]
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            "type": "page_number",
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                "page_number_content": [
                    {
                        "type": "text",
                        "content": "iii"
                    }
                ]
            },
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    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "7.1 双线性函数 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73\n7.1.1 双线性函数概念 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73\n7.1.2 双线性函数度量矩阵 . . . . . . . . . . . . . . . . . . . . . . . . . 73\n7.1.3 双线性函数不同基下的矩阵：合同 ..... 73\n7.2 二次型 ..... 74\n7.2.1 二次型概念及其矩阵 ..... 74\n7.2.2 二次型的标准型存在与配方法 ..... 75\n7.2.3 二次型的标准型：初等变换法 ..... 77\n7.2.4 二次型的标准型：正交变换法 ..... 77\n7.2.5 R, C 上的规范型 ..... 79\n7.3 (半) 正定二次型的研究 ..... 79\n7.3.1 正定二次型的概念与判别 ..... 79\n7.3.2 半正定二次型的概念与判别 ..... 81\n7.3.3  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C^T C"
                    },
                    {
                        "type": "text",
                        "content": " 的 (半) 正定性 ..... 82\n7.3.4 正定矩阵同时对角化 ..... 83\n7.3.5 正定矩阵的四则运算性质 ..... 84\n7.3.6 正定矩阵分解进阶 ..... 84\n7.3.7 负定二次型 ..... 85\n8 章 内积空间 ..... 86\n8.1 Euclid 空间 ..... 86\n8.1.1 Euclid 空间与正交 ..... 86\n8.1.2 标准正交基 ..... 87\n8.1.3 正交和与正交补 ..... 88\n8.1.4 正交变换 ..... 89\n8.1.5 镜面反射 ..... 90\n8.2 Unitary 空间 ..... 90\n8.2.1 Unitary 空间概念 ..... 90\n9 章 对偶空间 ..... 91\n9.1 对偶空间与对偶基 ..... 91"
                    }
                ]
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        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "目录"
                    }
                ]
            },
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            ]
        },
        {
            "type": "page_number",
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                "page_number_content": [
                    {
                        "type": "text",
                        "content": "iv"
                    }
                ]
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    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "第1章 多项式"
                    }
                ],
                "level": 2
            },
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        {
            "type": "title",
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                "title_content": [
                    {
                        "type": "text",
                        "content": "1.1 一元多项式"
                    }
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                "level": 2
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        },
        {
            "type": "title",
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                "title_content": [
                    {
                        "type": "text",
                        "content": "1.1.1 整除理论"
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                ],
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            },
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        {
            "type": "title",
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                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 1.1 (整除)"
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                ],
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                    {
                        "type": "equation_inline",
                        "content": "g(x), f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 上的多项式，若  "
                    },
                    {
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                        "content": "\\exists h(x)"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = g(x)h(x)"
                    },
                    {
                        "type": "text",
                        "content": "，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 整除  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": "，记为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)|f(x)"
                    },
                    {
                        "type": "text",
                        "content": "。称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的因式， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的倍式。"
                    }
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        {
            "type": "title",
            "content": {
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                    {
                        "type": "text",
                        "content": "定理 1.1 (整除的数域不变性)"
                    }
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                "level": 2
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            "content": {
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                        "type": "equation_inline",
                        "content": "\\mathbb{P}, \\mathbb{F}"
                    },
                    {
                        "type": "text",
                        "content": " 为两个数域， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathrm{P} \\subseteq \\mathbb{F}"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x) \\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " (在小数域中)，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 中  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)|f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{F}"
                    },
                    {
                        "type": "text",
                        "content": " 中  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)|f(x)"
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记该定理的意义在于可以在最大的数域(方便因式分解)中证明整除，即可证明小数域中整除。"
                    }
                ]
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.2 (带余除法)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall f(x), g(x) \\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": "，存在唯一  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q(x), r(x) \\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = q(x)g(x) + r(x)"
                    },
                    {
                        "type": "text",
                        "content": "。这里  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(x) = 0"
                    },
                    {
                        "type": "text",
                        "content": " 或者  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\partial(r(x)) < \\partial(g(x))"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习1.1带余除法的计算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(1)f(x) = x^{4} + 2x^{3} - 5x + 7, g(x) = x^{2} - 3x + 1"
                    },
                    {
                        "type": "text",
                        "content": "，求  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 除  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的商式与余式解(1)商为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{2} + 5x + 14"
                    },
                    {
                        "type": "text",
                        "content": "，余式为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "32x - 7"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 1.2 带余除法的应用 (1) 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)=ax+b"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)\\in\\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " ，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)|f^{2}(x)"
                    },
                    {
                        "type": "text",
                        "content": "  当且仅当   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)|f(x)"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), h(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 为三个非零多项式，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "h(x)|(f(x)-g(x))"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 除以  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "h(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的余式相同"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "f (x), g (x)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/52296e4b770609a240af12796930431f0ca7a38f0c02f8ecf854c5bf814f4b0e.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "g ^ {m} (x) | f ^ {m} (x)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f94f489fb6d56eaf837bc5fe9366ade852b6ea44eb76dc6a7db46daae09d7724.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "g (x) | f (x)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/07b19a96169411cebf763072450a5fb3ab129c8e5eb83499d14642057ded60b0.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 右推左显然。左推右:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f^2(x) = g(x)h(x)"
                    },
                    {
                        "type": "text",
                        "content": "，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = q(x)g(x) + r(x)"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f^2(x) = q^2(x)g^2(x) + 2q(x)g(x)r(x) + r^2(x)"
                    },
                    {
                        "type": "text",
                        "content": "，根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)|f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)|r^2(x)"
                    },
                    {
                        "type": "text",
                        "content": "，显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(x) = 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=q_{1}(x)h(x)+r_{1}(x), g(x)=q_{1}(x)h(x)+r_{2}(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，相减得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "f (x) - g (x) = h (x) \\left[ q _ {1} (x) - q _ {2} (x) \\right] + r _ {1} (x) - r _ {2} (x)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f6bca2e126e696ab0a2643e9f639725e49e348247bd161c57b912ea20714bf6b.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\partial (r_1(x) - r_2(x)) < \\partial (h(x))"
                    },
                    {
                        "type": "text",
                        "content": " ，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "h(x)|(r_1(x) - r_2(x))"
                    },
                    {
                        "type": "text",
                        "content": " ，因此当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r_1(x) = r_2(x)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 右推左：用   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=h(x)g(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，两侧取 m 次方即可。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "左推右：设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=c_{1}p_{1}^{r_{1}}(x)\\cdots p_{s}^{r_{s}}(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)=c_{2}p_{1}^{t_{1}}(x)\\cdots p_{s}^{t_{s}}(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，其中   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p_{1}(x)\\sim p_{s}(x)"
                    },
                    {
                        "type": "text",
                        "content": "  两两不可约且互素，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  可以为 0 或非负整数。根据条件可知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "mt_{i}\\leq mr_{i}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "t_{i}\\leq r_{i}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)|f(x)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "注意本题用带余除法可能不好说，因此稳妥起见还是用因式分解更好。"
                    }
                ]
            },
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            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.1.2 最大公因式"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 1.2 (最大公因式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x), d(x) \\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": "，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x)"
                    },
                    {
                        "type": "text",
                        "content": " 同时整除  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": "，则称其为公因式。若没有可以整除  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的公因式存在，则称其为最大公因式，记为"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "d (x) = (f (x), g (x))",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1615f23a641ad8d47217b95ecf7333badb7316f6866c76f03a360bbfc8ff3bb2.jpg"
                }
            },
            "bbox": [
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.3 (最大公因式表示定理)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\forall f(x), g(x) \\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": "，均存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x) = (f(x), g(x)) \\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": "，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists u(x), v(x) \\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x) = u(x)f(x) + v(x)g(x)"
                    },
                    {
                        "type": "text",
                        "content": " (反之不可，互素时才可)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 假设通过辗转相除法得到:"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} f (x) = q _ {1} (x) g (x) + r _ {1} (x) \\\\ g (x) = q _ {2} (x) r _ {1} (x) + r _ {2} (x) \\\\ r _ {1} (x) = q _ {3} (x) r _ {2} (x) + r _ {3} (x) \\\\ \\vdots \\\\ r _ {s - 3} (x) = q _ {s - 1} (x) r _ {s - 2} (x) + r _ {s - 1} (x) \\\\ r _ {s - 2} (x) = q _ {s} (x) r _ {s - 1} (x) + r _ {s} (x) \\\\ r _ {s - 1} (x) = q _ {s + 1} (x) r _ {s} (x) + 0 \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/d8c6baeaa5211a659b0f9f815867e8ff6b205e71d58f2e6ff6b1deb3a8104c40.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则此时可以得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r_s(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的最大公因式，并且从倒数第二个等式开始往回推，得到"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\begin{array}{r l} r _ {s} (x) & = r _ {s - 2} (x) - q _ {s} (x) r _ {s - 1} (x) \\\\ & = r _ {s - 2} (x) - q _ {s} (x) (r _ {s - 3} (x) - q _ {s - 1} (x) r _ {s - 2} (x)) \\\\ & = \\dots \\dots \\\\ & = u (x) f (x) + v (x) g (x) \\end{array}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/acdf54e1544e0218c7dfbc37bafbb93d12856e8eafd24757f65cd9995021ba80.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "即逐个将前面的等式代入方程，逐个消去   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r_{s-2}(x), \\cdots, r_{2}(x), r_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，最终只剩下   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x), f(x)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 最大公因式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x) = u(x)f(x) + v(x)g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x)"
                    },
                    {
                        "type": "text",
                        "content": " 和  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "v(x)"
                    },
                    {
                        "type": "text",
                        "content": " 不是唯一的，如果要唯一则需要加上额外的限制条件。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习1.3辗转相除法  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(1)f(x) = x^{4} + 3x^{3} - x^{2} - 4x - 3, g(x) = 3x^{3} + 10x^{2} + 2x - 3"
                    },
                    {
                        "type": "text",
                        "content": "，求  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f(x), g(x))"
                    },
                    {
                        "type": "text",
                        "content": "，并求  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x), v(x)"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x)f(x) + v(x)g(x) = (f(x), g(x))"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 已知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = x^4 + x^3 - 3x^2 - 4x - 1"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x) = x^3 + x^2 - x - 1"
                    },
                    {
                        "type": "text",
                        "content": "，求满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x)f(x) + v(x)g(x) = x + 1"
                    },
                    {
                        "type": "text",
                        "content": " 的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x), v(x)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解 (1) 第一题过程如下图所示。(2) 可先求出最大公因式，再提取"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.4 (保最大公因式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x) \\in P[x]"
                    },
                    {
                        "type": "text",
                        "content": "，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a, b, c, d \\in P"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "ad - bc \\neq 0"
                    },
                    {
                        "type": "text",
                        "content": "，则"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(f (x), g (x)) = (a f (x) + b g (x), c f (x) + d g (x))",
                "math_type": "latex",
                "image_source": {
                    "path": "images/6c8cc5926f0f741071efdb375293bd177baa2d66b71a346f1de1a5b2b454d363.jpg"
                }
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            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 假设左侧为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，右侧为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，下面要证   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x)=d_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，不妨证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x)|d_{1}(x), d_{1}(x)|d(x)"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x)|d_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": " ：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是左侧最大公因式，其是右侧每个部分的最大公因式，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x)|d_{1}(x)"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d_{1}(x)|d(x)"
                    },
                    {
                        "type": "text",
                        "content": " ：由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "ad - bc \\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " ，从方程组角度   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": "  也可以用   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "af(x) + bg(x), cf(x) + dg(x)"
                    },
                    {
                        "type": "text",
                        "content": "  线性表出，"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "1.1 一元多项式"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "2"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\begin{array}{c} \\text {   ①   } \\\\ q _ {1} (x) = x - \\frac {1}{3} \\left| \\begin{array}{l l} f (x) & \\frac {1}{3} g (x) \\\\ x ^ {4} + 3 x ^ {3} - x ^ {2} - 4 x - 3 & x ^ {3} + \\frac {1 0}{3} x ^ {2} + \\frac {2}{3} x - 1 \\\\ x ^ {4} + \\frac {1 0}{3} x ^ {3} + \\frac {2}{3} x ^ {2} - x & x ^ {3} + 5 x ^ {2} + 6 x \\\\ \\hline - \\frac {1}{3} x ^ {3} - \\frac {5}{3} x ^ {2} - 3 x - 3 & - \\frac {5}{3} x ^ {2} - \\frac {1 6}{3} x - 1 \\\\ - \\frac {1}{3} x ^ {3} - \\frac {1 0}{9} x ^ {2} - \\frac {9}{9} x + \\frac {1}{3} & - \\frac {5}{3} x ^ {2} - \\frac {2 5}{3} x - 1 0 \\\\ \\hline r _ {1} (x) = - \\frac {5}{9} x ^ {2} - \\frac {2 5}{9} x - \\frac {1 0}{3} & r _ {2} (x) = 3 x + 9 \\\\ - \\frac {9}{5} r _ {1} (x) = x ^ {2} + 5 x + 6 & \\frac {1}{3} r _ {2} (x) = x + 3. \\end{array} \\right| \\quad \\text {   ②   } \\\\ q _ {2} (x) = x - \\frac {5}{3} \\end{array}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/7a838945b44f980639fcc44c605e93f076a1dfba4333578731d03df61edbd8c1.jpg"
                }
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        {
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            "content": {
                "math_content": "\\begin{array}{r l} & {\\mathrm{因此}: ① f (x) = \\frac {1}{3} g _ {1} (x) g (x) + r _ {1} (x) \\quad ② g (x) = g _ {2} (x) (- \\frac {9}{5} r _ {1} (x)) + r _ {2} (x)} \\\\ & {③ - \\frac {9}{5} r _ {1} (x) = (x + 2) \\frac {1}{3} r _ {2} (x) \\qquad r _ {2} (x) \\mathrm{即大公因式}} \\\\ & {\\Rightarrow ① ^ {\\prime}: r _ {1} (x) = f (x) - \\frac {1}{3} g _ {1} (x) g (x), ② ^ {\\prime} r _ {2} (x) = g (x) + \\frac {9}{5} g _ {2} (x) r _ {1} (x)} \\\\ & {\\mathrm{将} r _ {1} (x) \\mathrm{代入} ② ^ {\\prime} \\mathrm{中即可}} \\end{array}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/62124d637dcf7c0555edf04a94b94565296c85c99c09baacf6428744efddb4dc.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "具体用 cramer 法则求出即可，如   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=\\frac{df_{1}(x)-bg_{1}(x)}{ad-bc}, g(x)=\\frac{ag_{1}(x)-cf_{1}(x)}{ad-bc}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此与 (1) 同理"
                    }
                ]
            },
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "推论 1.1 (加减保最大公因式)"
                    }
                ],
                "level": 2
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "(f (x), f (x) - g (x)) = (f (x), f (x) + g (x)) = (f (x), g (x))",
                "math_type": "latex",
                "image_source": {
                    "path": "images/dc5dfccb9397a2c173be4529d574ea2bb00506db23dab5f7c88b8a74f9ab2698.jpg"
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.5 (最小公倍式的表示)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的首 1 最小公倍式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "[f(x), g(x)] = \\frac{f(x)g(x)}{(f(x), g(x))}"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f(x), g(x)) = d(x), f(x) = d(x)f_1(x), g(x) = d(x)g_1(x)"
                    },
                    {
                        "type": "text",
                        "content": "，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f_1(x), g_1(x)) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，记  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "s(x) = \\frac{f(x)g(x)}{d(x)} = d(x)f_1(x)g_1(x) = f(x)g_1(x) = f_1(x)g(x)"
                    },
                    {
                        "type": "text",
                        "content": "，下面说明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "s(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是最小公倍式：(1)是公倍式：显然(2)最小：设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "h(x)"
                    },
                    {
                        "type": "text",
                        "content": " 也是公倍式，设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "h(x) = f(x)m(x)"
                    },
                    {
                        "type": "text",
                        "content": "，根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)|h(x)"
                    },
                    {
                        "type": "text",
                        "content": " 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x)g_1(x)|d(x)f_1(x)m(x)"
                    },
                    {
                        "type": "text",
                        "content": "，根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g_1(x)"
                    },
                    {
                        "type": "text",
                        "content": " 与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_1(x)"
                    },
                    {
                        "type": "text",
                        "content": " 互素，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g_1(x)|m(x)"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g_1(x)f(x)|f(x)m(x)"
                    },
                    {
                        "type": "text",
                        "content": "，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "s(x)|h(x)"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "s(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是最小的。"
                    }
                ]
            },
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.1.3 互素"
                    }
                ],
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        {
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            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 1.3 (互素)"
                    }
                ],
                "level": 2
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x) \\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " 的最大公因式为1，则称它们互素。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.6 (互素充要条件)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x) \\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " 互素当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists u(x), v(x) \\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x)f(x) + v(x)g(x) = 1"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 左推右显然 (根据最大公因式定理)。右推左：设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f(x), g(x)) = d(x)"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x)|f(x), d(x)|g(x)"
                    },
                    {
                        "type": "text",
                        "content": "，设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = d(x)f_1(x), g(x) = d(x)g_1(x)"
                    },
                    {
                        "type": "text",
                        "content": "，从而"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "d (x) | [ u (x) f _ {1} (x) + v (x) g _ {1} (x) ] = 1",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f16e26ceb62015bae5cbf3789b1c37e9c09a668184b13dba99675dffa4b4a70b.jpg"
                }
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "1.1 一元多项式"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "3"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d(x), u(x)f_1(x) + v(x)g_1(x)"
                    },
                    {
                        "type": "text",
                        "content": " 均为常数，故  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 互素笔记上述  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x), v(x)"
                    },
                    {
                        "type": "text",
                        "content": " 在无条件限制下也是不一定唯一的。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习1.4互素相关应用(1)若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x),g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 互素，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x^{m}),g(x^{m})"
                    },
                    {
                        "type": "text",
                        "content": " 互素"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 重点：若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f(x), g(x)) = d(x)"
                    },
                    {
                        "type": "text",
                        "content": "，证明： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f(x^m), g(x^m)) = d(x^m)"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f(x), g(x)) = 1, (f(x), h(x)) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，证明： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f(x), g(x)h(x)) = 1"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 互素，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x)f(x) + v(x)g(x) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，从而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x^{m})f(x^{m}) + v(x^{m})g(x^{m}) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，由于是互素充要条件，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x^{m}), g(x^{m})"
                    },
                    {
                        "type": "text",
                        "content": " 互素"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 不能直接用最大公因式表示，因为不是充要条件，要转到互素上。设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = d(x)m(x), g(x) = d(x)n(x)"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(m(x), n(x)) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，因此"
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "u (x ^ {k}) m (x ^ {k}) + v (x ^ {k}) n (x ^ {k}) = 1",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f88139cc2094b72c4b43563c18894cfac85c61afc0c2d820de75c6813fae3ea3.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m(x^{k}), n(x^{k})"
                    },
                    {
                        "type": "text",
                        "content": " 互素，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x^{k}) = d(x^{k})m(x^{k}), g(x^{k}) = d(x^{k})n(x^{k})"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f(x^{k}), g(x^{k})) = d(x^{k})"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 根据互素，有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u_{1}(x)f(x) + v_{1}(x)g(x) = 1, u_{2}(x)f(x) + v_{2}(x)h(x) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，两式相乘得到"
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ u _ {1} (x) u _ {2} (x) f (x) + v _ {1} (x) g (x) + v _ {2} (x) h (x) \\right] f (x) + v _ {1} (x) v _ {2} (x) g (x) h (x) = 1",
                "math_type": "latex",
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                    "path": "images/7a1016538406f90afb4bf32ca0cb1fad94b00d7b0c02f2a3bbac7fd85ada78a7.jpg"
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此根据充要条件得到  "
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                        "content": "(f(x), g(x)h(x)) = 1"
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                ]
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 1.5 互素充要条件推广   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
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                        "type": "equation_inline",
                        "content": "g(x)"
                    },
                    {
                        "type": "text",
                        "content": "  是次数大于 0 的互素多项式，则存在唯一的   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x)"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "v(x)"
                    },
                    {
                        "type": "text",
                        "content": "  使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x)f(x) + v(x)g(x) = 1"
                    },
                    {
                        "type": "text",
                        "content": " ，这里   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\partial(u(x)) < \\partial(g(x))"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\partial(v(x)) < \\partial(f(x))"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 存在性：由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 互素，故存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a(x), b(x)"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a(x)f(x) + b(x)g(x) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 也互素，作带余除法  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a(x) = q(x)g(x) + u(x)"
                    },
                    {
                        "type": "text",
                        "content": "，代入公式得到"
                    }
                ]
            },
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            "type": "equation_interline",
            "content": {
                "math_content": "(q (x) g (x) + u (x)) f (x) + b (x) g (x) = 1 \\Rightarrow u (x) f (x) + (b (x) + q (x) f (x)) g (x) = 1",
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        {
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                    {
                        "type": "text",
                        "content": "记  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "v(x) = b(x) + q(x)f(x)"
                    },
                    {
                        "type": "text",
                        "content": "，下面证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\partial(v(x)) < \\partial(f(x))"
                    },
                    {
                        "type": "text",
                        "content": "。反设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\partial(v(x)) \\geq \\partial(f(x))"
                    },
                    {
                        "type": "text",
                        "content": "，根据  "
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                    {
                        "type": "equation_inline",
                        "content": "\\partial(u(x)) < \\partial(g(x))"
                    },
                    {
                        "type": "text",
                        "content": " 可知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\partial(v(x)g(x)) > \\partial(u(x)f(x))"
                    },
                    {
                        "type": "text",
                        "content": "，显然它们的和不为 1，矛盾。"
                    }
                ]
            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 唯一性：假设还有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u_{1}(x)f(x) + v_{1}(x)g(x) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，那么有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(u(x) - u_{1}(x))f(x) + (v(x) - v_{1}(x))g(x) = 0"
                    },
                    {
                        "type": "text",
                        "content": "，即"
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                ]
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            "type": "equation_interline",
            "content": {
                "math_content": "(u (x) - u _ {1} (x)) f (x) = (v _ {1} (x) - v (x)) g (x)",
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                "image_source": {
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                        "content": "由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)"
                    },
                    {
                        "type": "text",
                        "content": "  互素，得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)\\left|(v_{1}(x)-v(x)\\right)"
                    },
                    {
                        "type": "text",
                        "content": " ，而后者次数低于前者，只能   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "v_{1}(x)=v(x)"
                    }
                ]
            },
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.1.4 不可约多项式与因式分解"
                    }
                ],
                "level": 2
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        {
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                        "type": "text",
                        "content": "定义 1.4 (不可约多项式)"
                    }
                ],
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x) \\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " 且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\partial(p(x)) \\geq 1"
                    },
                    {
                        "type": "text",
                        "content": "，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)"
                    },
                    {
                        "type": "text",
                        "content": " 不能表示成  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " 中两个次数小于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的多项式的乘积，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " 中的不可约多项式。"
                    }
                ]
            },
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        {
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            "content": {
                "title_content": [
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                        "type": "text",
                        "content": "定理 1.7 (不可约多项式的性质)"
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                ],
                "level": 2
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)\\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " 是不可约多项式，则"
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                ]
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)"
                    },
                    {
                        "type": "text",
                        "content": " 与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall f(x) \\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " 要么互素，要么  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x) | f(x)"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)|f(x)g(x)"
                    },
                    {
                        "type": "text",
                        "content": "，则要么  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)|f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 要么  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)|g(x)"
                    }
                ]
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        {
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            "content": {
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 不可约多项式经常用于证明乘积形式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(h(x), f(x)g(x)) = 1"
                    },
                    {
                        "type": "text",
                        "content": " 的命题。例如反设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(h(x), f(x)g(x)) = d(x)"
                    },
                    {
                        "type": "text",
                        "content": "，不可约多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)|d(x)"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)"
                    },
                    {
                        "type": "text",
                        "content": " 要么整除  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": "，要么整除  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)"
                    }
                ]
            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 1.6 利用不可约多项式性质 + 反证法 (1) 重点: 已知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f(x), g(x)) = 1"
                    },
                    {
                        "type": "text",
                        "content": " ，证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f(x)g(x), f(x) + g(x)) = 1"
                    }
                ]
            },
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                        "content": "4"
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                ]
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "1.1 一元多项式"
                    }
                ]
            },
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    ],
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 已知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{2}(x)"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\cdots"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{s}(x)"
                    },
                    {
                        "type": "text",
                        "content": "  两两互素，记   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "F_{i}(x) = f_{1}(x) \\cdots f_{i-1}(x) f_{i+1}(x) \\cdots f_{s}(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "F_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "F_{2}(x)"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\cdots"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "F_{s}(x)"
                    },
                    {
                        "type": "text",
                        "content": "  互素"
                    }
                ]
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        {
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 稍加变化: 复数域上的多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 没有重因式, 证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f(x) + 2f'(x), 3f^2(x)) = 1"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 反设有不可约多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q(x)"
                    },
                    {
                        "type": "text",
                        "content": " 作为公因式，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q(x)"
                    },
                    {
                        "type": "text",
                        "content": " 同时是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)g(x), f(x) + g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 因式。根据前者得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q(x)"
                    },
                    {
                        "type": "text",
                        "content": " 至少整除  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 之一，不妨设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q(x)|f(x)"
                    },
                    {
                        "type": "text",
                        "content": "，根据第二个式子得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q(x)"
                    },
                    {
                        "type": "text",
                        "content": " 也一定整除  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)"
                    },
                    {
                        "type": "text",
                        "content": "，否则无法得到多项式。因此矛盾。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 同种方法，随便挑  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "F_{i}(x), F_{j}(x)"
                    },
                    {
                        "type": "text",
                        "content": " 说一说就行"
                    }
                ]
            },
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        {
            "type": "paragraph",
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                    {
                        "type": "text",
                        "content": "(3) 同样设不可约   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，根据   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "3f^{2}(x)"
                    },
                    {
                        "type": "text",
                        "content": "  可知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q(x)|f(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，再根据   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q(x)|f(x)+2f'(x)"
                    },
                    {
                        "type": "text",
                        "content": "  可知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q(x)|f'(x)"
                    },
                    {
                        "type": "text",
                        "content": " （否则不是多项式），根据重因式充要条件可知。"
                    }
                ]
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 1.7 证明不可约多项式 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是次数大于 0 的多项式，若对任意多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": "，根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)|f(x)g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 可以得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)|f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 或者  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)|g(x)"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是不可约多项式。"
                    }
                ]
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 反设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)"
                    },
                    {
                        "type": "text",
                        "content": " 不是不可约多项式，则存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x) = f(x)g(x)"
                    },
                    {
                        "type": "text",
                        "content": "，此时  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)|f(x)g(x)"
                    },
                    {
                        "type": "text",
                        "content": "，但是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x) \\nmid f(x), p(x) \\nmid g(x)"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.8 (因式分解唯一性)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 上次数大于等于1的多项式，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 可唯一分解为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 上有限个不可约多项式乘积："
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "f (x) = c p _ {1} ^ {r _ {1}} (x) p _ {2} ^ {r _ {2}} (x) \\dots p _ {s} ^ {r _ {s}} (x)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/b6456ce00bfaadcda6d777e2aa09c90dffcdcc29cc9430c177cdc822dbac6f1f.jpg"
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.1.5 余数定理及其应用"
                    }
                ],
                "level": 2
            },
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        {
            "type": "title",
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                    {
                        "type": "text",
                        "content": "定理 1.9 (余数定理)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x)\\in \\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " ，对  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in \\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " ，总有"
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "f (x) = (x - \\alpha) q (x) + f (\\alpha)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/3a73b36902be1acc2e5fc799210ba3250b308cc3cb717ab98ffd808362da8602.jpg"
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            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 用带余除法得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=(x-\\alpha)q(x)+c"
                    },
                    {
                        "type": "text",
                        "content": " ，而代入   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x=\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "  得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(\\alpha)=0+c"
                    },
                    {
                        "type": "text",
                        "content": " ，即   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "c=f(\\alpha)"
                    },
                    {
                        "type": "text",
                        "content": " ，因此"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "f (x) = (x - \\alpha) q (x) + f (\\alpha)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/ceba30f1478cab7c6e872a3e4d3e7b0df81c2973d30b29599c39ffb393d7d5f0.jpg"
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "推论 1.2"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\in \\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 是多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的根当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x - \\alpha) \\mid f(x)"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 余数定理的主要价值在于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 上的因式等价于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 上的根，两者可以灵活互换。判断重根可以转换为判断重因式，判断整除可以转换为根。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 1.8 余数定理的应用 (1) 若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x-1)\\mid f(x^{m})"
                    },
                    {
                        "type": "text",
                        "content": " ，证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x^{m}-1)\\mid f(x^{m})"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{2} + x + 1 \\mid x^{3m} + x^{3n + 1} + x^{3p + 2}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 已知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{2} + x + 1 \\mid f_{1}(x^{3}) + xf_{2}(x^{3})"
                    },
                    {
                        "type": "text",
                        "content": "，证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x - 1) \\mid f_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": " 且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x - 1) \\mid f_{2}(x)"
                    }
                ]
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(4) 已知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{2} + 1 \\mid f_{1}(x^{4}) + xf_{2}(x^{4})"
                    },
                    {
                        "type": "text",
                        "content": "，证明 1 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{1}(x), f_{2}(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的公共根。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x - 1) \\mid f(x^m)"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = 1"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x^m)"
                    },
                    {
                        "type": "text",
                        "content": " 的根，故  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = 1"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的根，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x - 1) \\mid f(x)"
                    },
                    {
                        "type": "text",
                        "content": "，故  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x^m - 1) \\mid f(x^m)"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 放到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上考虑:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\omega = e^{i\\frac{2\\pi}{3}}"
                    },
                    {
                        "type": "text",
                        "content": ", 则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{2} + x + 1 = (x - \\omega)(x - \\omega^{2})"
                    },
                    {
                        "type": "text",
                        "content": ", 代入  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\omega"
                    },
                    {
                        "type": "text",
                        "content": " 则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\omega^{3m} + \\omega^{3n+1} + \\omega^{3p+2} = 1 + \\omega + \\omega^{2} = 0"
                    },
                    {
                        "type": "text",
                        "content": ", 代入  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\omega^{2}"
                    },
                    {
                        "type": "text",
                        "content": " 则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\omega^{6m} + \\omega^{6n+2} + \\omega^{6p+4} = 1 + \\omega^{2} + \\omega^{4} = 1 + \\omega^{2} + \\omega = 0"
                    },
                    {
                        "type": "text",
                        "content": ", 根据余数定理可知成立"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(4)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{2}+1"
                    },
                    {
                        "type": "text",
                        "content": "  复数根为 x=i，根据   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x^{2}+1)\\mid f_{1}(x^{4})+xf_{2}(x^{4})"
                    },
                    {
                        "type": "text",
                        "content": "  可知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{1}(1)+if_{2}(1)=0"
                    },
                    {
                        "type": "text",
                        "content": " ，从而为公共根。"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "5"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "1.1 一元多项式"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.1.6 重因式与重根"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.10 (重因式性质汇总)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是次数大于0的多项式，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是不可约多项式，则"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 核心：若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k"
                    },
                    {
                        "type": "text",
                        "content": " 重因式，则其是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f'(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k - 1"
                    },
                    {
                        "type": "text",
                        "content": " 重因式。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 无重因式充要： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 没有重因式当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), f'(x)"
                    },
                    {
                        "type": "text",
                        "content": " 互素。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 不可约因式：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\frac{f(x)}{(f(x), f'(x))}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 所有不可约因式 (重数均为 1) 乘积的常数倍"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 重因式与重根的关系：重因式不随数域而改变，而重根会随着数域而改变。例如  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x^{2} + 1)^{2}"
                    },
                    {
                        "type": "text",
                        "content": "，其在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C},\\mathbb{R},\\mathbb{Q}"
                    },
                    {
                        "type": "text",
                        "content": " 上都有重因式，但是只在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上有重根。因此有重根一定有重因式，有重因式不一定有重根。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习1.9用根判断因式(1)重点：设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "h(x), f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 为实系数多项式满足以下等式，证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{2} + 1|f(x), x^{2} + 1|g(x)"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} (x ^ {2} + 1) h (x) + (x + 1) f (x) + (x - 2) g (x) = 0 \\\\ (x ^ {2} + 1) k (x) + (x - 1) f (x) + (x + 2) g (x) = 0 \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/11fc655f774a6dfcf7d763096fe4d6013c65a6fdc3ff3af635dd83b095db9e2c.jpg"
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            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 要证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{R}"
                    },
                    {
                        "type": "text",
                        "content": " 上  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x^2 + 1)|f(x)"
                    },
                    {
                        "type": "text",
                        "content": "，只需要证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x - i)|f(x), (x + i)|f(x)"
                    },
                    {
                        "type": "text",
                        "content": "，等价于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = \\pm i"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的根。代入  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = i"
                    },
                    {
                        "type": "text",
                        "content": " 得到："
                    }
                ]
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            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} (i + 1) f (i) + (i - 2) g (i) = 0 \\\\ (i - 1) f (i) + (i + 2) g (i) = 0 \\end{array} \\right.",
                "math_type": "latex",
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                    "path": "images/6bb4faaa85fd671a79aa324589ed6c07480f713cfd5fda903b30d3851cd7ae37.jpg"
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            "content": {
                "paragraph_content": [
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                        "type": "text",
                        "content": "解得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(i) = g(i) = 0"
                    },
                    {
                        "type": "text",
                        "content": " 。同理可证  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(-i) = g(-i) = 0"
                    }
                ]
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.11 (重根性质汇总)"
                    }
                ],
                "level": 2
            },
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}[x]"
                    },
                    {
                        "type": "text",
                        "content": " 上次数大于0的多项式，则"
                    }
                ]
            },
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 重根： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_0"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k"
                    },
                    {
                        "type": "text",
                        "content": " 重根当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_0) = f'(x_0) = \\cdots = f^{(k-1)}(x_0) = 0"
                    },
                    {
                        "type": "text",
                        "content": "，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f^{(k)}(x_0) \\neq 0"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 重根判定定理： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 重因式当且仅当其是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), f'(x)"
                    },
                    {
                        "type": "text",
                        "content": " 公因式。"
                    }
                ]
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上的重根问题等价于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上的重因式问题，因此判断多项式是否有重根时，可判断其在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上是否有重因式。重因式的好处是可以用多项式理论，重根的好处是可以带进去算，要相互补充。"
                    }
                ]
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
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                        "type": "text",
                        "content": "练习 1.10 判定重根存在性 (1) 证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=1+x+\\frac{x^{2}}{2!}+\\cdots+\\frac{x^{n}}{n!}"
                    },
                    {
                        "type": "text",
                        "content": "  没有重根"
                    }
                ]
            },
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=x^{n}+nx^{n-1}+n(n-1)x^{n-2}+\\cdots+n!"
                    },
                    {
                        "type": "text",
                        "content": "  无重根证明 (1) 直接证重根不好证，转换为重因式，若在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上无重因式则肯定无重根。根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f'(x) = 1 + x + \\frac{x^2}{2!} + \\cdots + \\frac{x^{n-1}}{(n-1)!}"
                    },
                    {
                        "type": "text",
                        "content": " 得到"
                    }
                ]
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "(f (x), f ^ {\\prime} (x)) = (f (x), f (x) - f ^ {\\prime} (x)) = (f (x), \\frac {x ^ {n}}{n !}) = (f (x), x ^ {n})",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1d818e76934d1506db4f41af1d29926627d59abde1ba1704bb08a716aa8a949e.jpg"
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^n"
                    },
                    {
                        "type": "text",
                        "content": " 项的原因， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f(x), f'(x)) = x^k, k \\geq 0"
                    },
                    {
                        "type": "text",
                        "content": "。而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(0) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，这说明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x"
                    },
                    {
                        "type": "text",
                        "content": " 不是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的因式，从而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k = 0"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), f'(x)"
                    },
                    {
                        "type": "text",
                        "content": " 互素，无重根。"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 也用无重因式证明"
                    }
                ]
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 1.11 重根、重因式相互转化 (1) 求   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{5} + x^{4} - 6x^{3} - 14x^{2} - 11x - 3"
                    },
                    {
                        "type": "text",
                        "content": "  的重根和重数"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
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                        "type": "text",
                        "content": "(2) 重点: 求  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x - 1)^{2} | Ax^{4} + Bx^{2} + 1"
                    }
                ]
            },
            "bbox": [
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        {
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            "content": {
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                    {
                        "type": "text",
                        "content": "(3) 重点：求  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "t"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = x^{3} - 3x^{2} + tx - 1"
                    },
                    {
                        "type": "text",
                        "content": " 有重根"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(4) 重点 ZJU2018.2: 证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{3} + px + q"
                    },
                    {
                        "type": "text",
                        "content": " 有重根的充要条件  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "4p^{3} + 27q^{2} = 0"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "1.1 一元多项式"
                    }
                ]
            },
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        },
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            "type": "page_number",
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                "page_number_content": [
                    {
                        "type": "text",
                        "content": "6"
                    }
                ]
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        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(5)ZJU2021: 求  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "t"
                    },
                    {
                        "type": "text",
                        "content": " 为何值时,  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = x^3 + 6x^2 + tx + 8"
                    },
                    {
                        "type": "text",
                        "content": " 有重根, 并求重根解 (1) 可以尝试猜。一般用辗转相除法算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), f'(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的最大公因式, 最终答案为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x + 1)^4(x - 3)"
                    }
                ]
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 根据重根与重因式关系，在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上必有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(1) = 0, f'(1) = 0"
                    },
                    {
                        "type": "text",
                        "content": "，因此得出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A + B + 1 = 0, 4A + 2B = 0"
                    },
                    {
                        "type": "text",
                        "content": "，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = -2, B = 1"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 同理考虑  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上的因式分解， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f'(x) = 3x^2 - 6x + t"
                    },
                    {
                        "type": "text",
                        "content": "，因此考虑  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\frac{1}{3} f'(x) = x^2 - 2x + \\frac{t}{3}"
                    },
                    {
                        "type": "text",
                        "content": " 辗转相除法，第一步得出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = \\frac{1}{3} f'(x)(x - 1) + (\\frac{2}{3} t - 2)x + (\\frac{t}{3} - 1)"
                    },
                    {
                        "type": "text",
                        "content": "，如果  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "t = 3"
                    },
                    {
                        "type": "text",
                        "content": " 则直接整除有重根，如果  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "t \\neq 3"
                    },
                    {
                        "type": "text",
                        "content": "，则接着算， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\frac{1}{3} f' = (\\frac{1}{\\frac{2}{3} t - 2})r_1(x)(x - \\frac{5}{2}) + \\frac{t}{3} + \\frac{5}{4}"
                    },
                    {
                        "type": "text",
                        "content": "，则需要  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\frac{t}{3} + \\frac{5}{4} = 0"
                    },
                    {
                        "type": "text",
                        "content": "，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "t = -\\frac{15}{4}"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(4) 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上考虑，有重根即有重因式， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f'(x) = 3x^2 + p"
                    },
                    {
                        "type": "text",
                        "content": "，先有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = \\frac{1}{3} f'(x)x + \\frac{2}{3} px + q"
                    },
                    {
                        "type": "text",
                        "content": "，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p = q = 0"
                    },
                    {
                        "type": "text",
                        "content": "，则成立，下面考虑  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p \\neq 0"
                    },
                    {
                        "type": "text",
                        "content": "。得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r_1(x) = \\frac{3}{2p} r_1(x)(x - \\frac{3q}{2p}) + \\frac{p}{3} + \\frac{9q^2}{4p^2}"
                    },
                    {
                        "type": "text",
                        "content": "，因此得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "4p^3 + 27q^2 = 0"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(5)t=12 时有 3 重 x=-2 根，t=-15 时有 2 重 x=1 根。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 1.12 导数重根反推函数重根？  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{0}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f'(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的 k 重根，举例说明其不能推出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{0}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k+1"
                    },
                    {
                        "type": "text",
                        "content": " 重根，该命题需要  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{0}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的根。"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解例如  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = x^{k + 1} + 1"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_0 = 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记上面的结论非常重要，在利用导数验证根的重数时，一定要验证其是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 本身的根。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.1.7 整数与多项式"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.12 (整数与多项式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定理1.12（整数与多项式） "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m,n"
                    },
                    {
                        "type": "text",
                        "content": " 为正整数， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(m,n) = d"
                    },
                    {
                        "type": "text",
                        "content": " 的充要条件是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x^{m} - 1, x^{n} - 1) = x^{d} - 1"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 左推右是公因式：设 m = kd，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{m} - 1 = (x^{d})^{k} - 1 = (x^{d} - 1)(1 + x^{d} + \\cdots + x^{(k-1)d})"
                    },
                    {
                        "type": "text",
                        "content": " ，这说明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{d} - 1|x^{m} - 1"
                    },
                    {
                        "type": "text",
                        "content": " ，另一侧同理，因此是公因式"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 最大: 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^n - 1, x^m - 1"
                    },
                    {
                        "type": "text",
                        "content": " 的公因式, 下面证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)|x^d - 1"
                    },
                    {
                        "type": "text",
                        "content": " 。由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^n - 1"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上无重根 (因为和导数互素), 因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)"
                    },
                    {
                        "type": "text",
                        "content": " 也没有重根, 从而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x) = a(x - x_1) \\cdots (x - x_s)"
                    },
                    {
                        "type": "text",
                        "content": ", 这里  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_1 \\sim x_s"
                    },
                    {
                        "type": "text",
                        "content": " 是互异复数, 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_i^n = x_i^m = 1"
                    },
                    {
                        "type": "text",
                        "content": ", 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_i^d = x_i^{um + vn} = (x_i^m)^u(x_i^n)^v = 1"
                    },
                    {
                        "type": "text",
                        "content": ", 因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x - x_i|x^d - 1"
                    },
                    {
                        "type": "text",
                        "content": ",  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x - x_1) \\cdots (x - x_s)|x^d - 1"
                    },
                    {
                        "type": "text",
                        "content": ", 因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x)|x^d - 1"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 右推左：若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(m, n) = d_{1}"
                    },
                    {
                        "type": "text",
                        "content": "，则根据前面的结论  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x^{m} - 1, x^{n} - 1) = x^{d_{1}} - 1 = x^{d} - 1"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d_{1} = d"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "练习1.13整除与多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(1)d, n \\in \\mathbb{N}^{+}"
                    },
                    {
                        "type": "text",
                        "content": "，证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{d} - 1|x^{n} - 1"
                    },
                    {
                        "type": "text",
                        "content": " 的充要条件为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d|n"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)m,n 是大于 1 的自然数，证明：m,n 互素的充要条件是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1 + x + x^{2} + \\cdots + x^{m-1}"
                    },
                    {
                        "type": "text",
                        "content": "  与   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1 + x + x^{2} + \\cdots + x^{n-1}"
                    },
                    {
                        "type": "text",
                        "content": "  互素"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)=1+x+x^{2}+\\cdots+x^{n},f(x)=g(x^{k}),(k,n+1)=1"
                    },
                    {
                        "type": "text",
                        "content": " ，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)|f(x)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 本题即前面题目的特例。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "右推左：设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{d}-1=(x-x_{1})\\cdots(x-x_{d})(\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": "  上)，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{i}^{d}=1"
                    },
                    {
                        "type": "text",
                        "content": " ，由于 n=dk，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{i}^{n}=(x_{i}^{d})^{k}=1"
                    },
                    {
                        "type": "text",
                        "content": " ，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{d}-1|x^{n}-1"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "左推右：设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{d}-1=(x-x_{1})\\cdots(x-x_{d})"
                    },
                    {
                        "type": "text",
                        "content": " ，由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{d}-1|x^{n}-1,\\quad x_{i}^{d}=x_{i}^{n}=1"
                    },
                    {
                        "type": "text",
                        "content": " ，假设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n=kd+r, r\\neq0"
                    },
                    {
                        "type": "text",
                        "content": " ，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{i}^{n}=x_{i}^{kd+r}=x_{i}^{r}"
                    },
                    {
                        "type": "text",
                        "content": " ，由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{i}=1"
                    },
                    {
                        "type": "text",
                        "content": "  只有一重，其他   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{i}\\neq1"
                    },
                    {
                        "type": "text",
                        "content": " ，对于这些   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{i}^{r}\\neq1"
                    },
                    {
                        "type": "text",
                        "content": " ，因此矛盾。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)m,n 互素当且仅当   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x^{m}-1,x^{n}-1)=x-1"
                    },
                    {
                        "type": "text",
                        "content": " ，等价于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(1+x+x^{2}+\\cdots+x^{m-1},1+x+\\cdots+x^{n-1})=1"
                    },
                    {
                        "type": "text",
                        "content": " \n\n(3) 根据   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)=\\frac{x^{n+1}-1}{x-1}"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=\\frac{x^{k(n+1)}-1}{x^{k}-1}"
                    },
                    {
                        "type": "text",
                        "content": " ，此时   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)|f(x)"
                    },
                    {
                        "type": "text",
                        "content": "  等价于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x^{k}-1)(x^{n+1}-1)|(x-1)(x^{k(n+1)}-1)"
                    },
                    {
                        "type": "text",
                        "content": " ，由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(k,n+1)=1"
                    },
                    {
                        "type": "text",
                        "content": " ，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{k}-1"
                    },
                    {
                        "type": "text",
                        "content": "  与   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{n+1}"
                    },
                    {
                        "type": "text",
                        "content": "  的最大公因为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x-1)"
                    },
                    {
                        "type": "text",
                        "content": " ，而   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x^{k(n+1)}-1)"
                    },
                    {
                        "type": "text",
                        "content": "  是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x^{k}-1)(x^{n+1}-1)"
                    },
                    {
                        "type": "text",
                        "content": " ，因此根据最小公倍式的性质显然可知。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "7"
                    }
                ]
            },
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "1.2 特殊数域上的一元多项式理论"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.2 特殊数域上的一元多项式理论"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.2.1 复数域上的一元多项式"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定理 1.13 (代数学基本定理)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "任意次数大于等于1的多项式在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上至少有一个根。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.14 (复数域因式分解)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上任意次数大于等于1的多项式可唯一地分解为一次因式的乘积，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 次多项式在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上恰有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 个复根。"
                    }
                ]
            },
            "bbox": [
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                272
            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.2.2 实数域上的一元多项式"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定理 1.15 (实数域因式分解)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{R}"
                    },
                    {
                        "type": "text",
                        "content": " 上的多项式总能唯一地分解为一次因式和不可约二次因式的乘积。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.2.3 有理数域上一元多项式因式分解"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "复不可约多项式是一次的，实不可约多项式是一次或二次的，下面将说明有理不可约多项式的次数可以是任意的。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 1.5 (本原多项式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若整系数多项式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)=b_{n}x^{n}+\\cdots+b_{1}x+b_{0}"
                    },
                    {
                        "type": "text",
                        "content": "  的系数   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "b_{0}\\sim b_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  是互素的 (整体互素而非两两互素)，即"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(b _ {1}, \\dots , b _ {n}) = 1",
                "math_type": "latex",
                "image_source": {
                    "path": "images/006b478a0c014f9aae5a336b7761d6b37f65279b48f954309ddb5ba044f9ad60.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是一个本原多项式"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.16 (Gauss 引理)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "两个本原多项式的乘积仍为本原多项式"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = \\sum_{k=0}^{n} a_k x^k, g(x) = \\sum_{k=0}^{m} b_k x^k"
                    },
                    {
                        "type": "text",
                        "content": "，设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "h(x) = f(x)g(x) = \\sum_{k=0}^{n+m} d_k x^k"
                    },
                    {
                        "type": "text",
                        "content": "。假设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "h(x)"
                    },
                    {
                        "type": "text",
                        "content": " 不是本原多项式，则存在素数  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p"
                    },
                    {
                        "type": "text",
                        "content": " 整除所有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d_k"
                    },
                    {
                        "type": "text",
                        "content": "，由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 本原，因此从常数项开始搜索，存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i, j"
                    },
                    {
                        "type": "text",
                        "content": "，使得"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} p \\mid a _ {0}, \\dots , p \\mid a _ {i - 1}, p \\nmid a _ {i} \\\\ p \\mid b _ {0}, \\dots , p \\mid b _ {j - 1}, p \\nmid b _ {j} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/425ac0351266222a97cc1218a0b2b3de186c6332d8ef91a14d453bf7b6f50356.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d_{i + j} = a_i b_j + \\sum_{k=0, k \\neq i}^{i+j} a_k b_{i+j-k}"
                    },
                    {
                        "type": "text",
                        "content": "，除了第一项外均至少有一项被  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p"
                    },
                    {
                        "type": "text",
                        "content": " 整除，第一项不被  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p"
                    },
                    {
                        "type": "text",
                        "content": " 整除，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p \\nmid d_{i+j}"
                    },
                    {
                        "type": "text",
                        "content": "，这与假设矛盾。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.17 (有理多项式变本原多项式)"
                    }
                ],
                "level": 2
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是有理多项式， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 为本原多项式，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 可表示为有理数  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r"
                    },
                    {
                        "type": "text",
                        "content": " 与本原多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的乘积"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 有理多项式分解说明了有理多项式的因式分解问题完全可以转换成本原多项式的因式分解，而下面的定理说明了本原多项式能否分解为两个有理多项式之积与能否分解为整系数多项式之积是等价的。"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "1.2 特殊数域上的一元多项式理论"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "8"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习1.14 本原多项式 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是本原多项式， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x) \\mid f(x)"
                    },
                    {
                        "type": "text",
                        "content": "，证明： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\frac{f(x)}{g(x)}"
                    },
                    {
                        "type": "text",
                        "content": " 也是本原多项式 (2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是本原多项式。证明： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x+1)"
                    },
                    {
                        "type": "text",
                        "content": " 也是本原多项式"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是本原多项式，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x+1)"
                    },
                    {
                        "type": "text",
                        "content": " 也是本原多项式。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = g(x)q(x)"
                    },
                    {
                        "type": "text",
                        "content": "，由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 为本原多项式， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q(x)"
                    },
                    {
                        "type": "text",
                        "content": " 必然是有理系数的，设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q(x) = rq_1(x)"
                    },
                    {
                        "type": "text",
                        "content": "，这里  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q_1(x)"
                    },
                    {
                        "type": "text",
                        "content": " 为本原多项式，因此"
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "f (x) = r g (x) q _ {1} (x)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/cba35858036b26a588d92f020d115f23d9debde0fb2d2813b73e7fc844bdd48b.jpg"
                }
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "根据 Gauss 引理可知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)q_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": " 仍为本原多项式，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r = \\pm 1"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q(x) = \\pm q_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是本原多项式。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 直接展开得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x+1)=b_{n}x^{n}+\\cdots+b_{1}x+b_{0}"
                    },
                    {
                        "type": "text",
                        "content": " ，其中   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "b_{i}=\\sum_{k=i}^{n}\\binom{k}{i}a_{k}"
                    },
                    {
                        "type": "text",
                        "content": " 。下面证明互素，假设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(b_{0},b_{1},\\cdots,b_{n})=d"
                    },
                    {
                        "type": "text",
                        "content": " ，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} d \\mid b _ {n} \\Rightarrow d \\mid a _ {n} \\\\ d \\mid b _ {n - 1} \\Rightarrow d \\mid \\binom {n} {1} a _ {n} + a _ {n - 1} \\Rightarrow d \\mid a _ {n - 1} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/34d91746148319a83fc575f4d88ba5e20722b5850cedbab63bc3949dfad944f5.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "同理可以得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d \\mid a_{n-2}, \\cdots, d \\mid a_{1}, d \\mid a_{0}"
                    },
                    {
                        "type": "text",
                        "content": " ，这与   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": "  是本原多项式矛盾。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.18 (整系数多项式因式分解)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "整系数多项式若能分解为两个较低次有理多项式乘积，则一定也能分解为两个较低次整系数多项式乘积"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = g(x)h(x)"
                    },
                    {
                        "type": "text",
                        "content": "，这里  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x), h(x) \\in \\mathbb{Q}[x]"
                    },
                    {
                        "type": "text",
                        "content": "，则存在本原多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{1}(x), h_{1}(x), g_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": " 使得"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "f (x) = a f _ {1} (x), g (x) = r g _ {1} (x), h (x) = s h _ {1} (x), a \\in \\mathbb {Z}, r, s \\in \\mathbb {Q}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/725d7d80c1a28e9a369e188e4ac0cc0b5b267e5085e96491b3d1cbd80feb4aed.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "从而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "af_{1}(x) = rsg_{1}(x)h_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": "，根据Gauss引理可知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g_{1}(x)h_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是本原多项式，不妨设最高项系数均正，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{1}(x) = g_{1}(x)h_{1}(x), a = rs"
                    },
                    {
                        "type": "text",
                        "content": "，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "rs \\in \\mathbb{Z}"
                    },
                    {
                        "type": "text",
                        "content": "，于是"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "f (x) = (r s g _ {1} (x)) h _ {1} (x)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/cf886079f5dd737b0e0b5477af7d30ef7c3139d31bf8d6b3c29c74db0e7ec710.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "其中  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "rsg_{1}(x), h_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": " 都是整系数多项式。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 1.15 整系数多项式整数根 (1) 若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\neq \\pm1"
                    },
                    {
                        "type": "text",
                        "content": "  为整系数多项式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": "  的整数根，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\frac{f(1)}{\\alpha-1}, \\frac{f(-1)}{\\alpha+1}"
                    },
                    {
                        "type": "text",
                        "content": "  都是整数 (2) ZJU2017.1:   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": "  为整系数多项式，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(0), f(1)"
                    },
                    {
                        "type": "text",
                        "content": "  为奇数，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": "  无整数根"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3)ZJU2019.4:   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{1}, \\cdots, a_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  为互不相同的整数,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{1} \\cdots a_{n} + 1"
                    },
                    {
                        "type": "text",
                        "content": "  不是某个整数的平方, 证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = (x + a_{1}) \\cdots (x + a_{n}) + 1"
                    },
                    {
                        "type": "text",
                        "content": "  不能表示为 Q 上两个次数大于等于 1 的多项式的乘积。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的根，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = (x - \\alpha)(b_{n-1}x^{n-1} + \\cdots + b_1x + b_0)"
                    },
                    {
                        "type": "text",
                        "content": "，这里  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "b_0, \\cdots, b_{n-1}"
                    },
                    {
                        "type": "text",
                        "content": " 也都是整数，因此"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "f (1) = (1 - \\alpha) (b _ {n - 1} + \\dots + b _ {1} + b _ {0}) \\Rightarrow \\frac {f (1)}{\\alpha - 1} = - b _ {0} - \\dots - b _ {n - 1} \\in \\mathbb {Z}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/c54becd27646351fe8959ec9c49125d4e45e4b028f9a299c8e31564e8f6fee04.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f"
                    },
                    {
                        "type": "text",
                        "content": " 有整数根，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = (x - a)f_0(x)"
                    },
                    {
                        "type": "text",
                        "content": "，这里  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_0(x)"
                    },
                    {
                        "type": "text",
                        "content": " 为整系数多项式，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(0) = (-a)f_0(0), f(1) = (1 - a)f_0(1)"
                    },
                    {
                        "type": "text",
                        "content": "，而根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(0), f(1)"
                    },
                    {
                        "type": "text",
                        "content": " 为奇数得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "-a, 1 - a"
                    },
                    {
                        "type": "text",
                        "content": " 同时为奇数，矛盾。"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q[x]"
                    },
                    {
                        "type": "text",
                        "content": " 上可约，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{Z}[x]"
                    },
                    {
                        "type": "text",
                        "content": " 也可约，不妨设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = g(x)h(x), g(x), h(x) \\in \\mathbb{Z}[x]"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(-a_i) = g(-a_i)h(-a_i) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，这说明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(-a_i) = h(-a_i) = \\pm 1"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x) - h(x)"
                    },
                    {
                        "type": "text",
                        "content": " 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 个不同的根，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\partial(g(x) - h(x)) < n"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x) \\equiv h(x)"
                    },
                    {
                        "type": "text",
                        "content": "，根据条件  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = g^2(x)"
                    },
                    {
                        "type": "text",
                        "content": "，常数项为整数的平方，这与条件不是平方矛盾。"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "9"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "1.2 特殊数域上的一元多项式理论"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.2.4 一元有理多项式的根"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.19 (有理多项式的根)"
                    }
                ],
                "level": 2
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x)=a_{n}x^{n}+\\cdots+a_{1}x+a_{0}"
                    },
                    {
                        "type": "text",
                        "content": " 是整系数多项式，若既约分数  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha=\\frac{q}{p}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 的有理根，则必有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "q\\mid a_{0},p\\mid a_{n}"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 假设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\frac{q}{p}"
                    },
                    {
                        "type": "text",
                        "content": " 是解，则代入多项式得到："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "a _ {n} \\frac {q ^ {n}}{p ^ {n}} + \\dots + a _ {1} \\frac {q}{p} + a _ {0} = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/cf91596a036f32edf160602c04a85add3861472921cf70a5b0e13882a92b46a9.jpg"
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            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "两边乘上  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p^n"
                    },
                    {
                        "type": "text",
                        "content": " 并移项得到："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} a _ {n} q ^ {n} = p (- a _ {n - 1} q ^ {n - 1} - a _ {n - 2} p q ^ {n - 2} - \\dots - a _ {0} p ^ {n - 1}) \\\\ a _ {0} p ^ {n} = q (- a _ {n} q ^ {n - 1} - a _ {n - 1} p q ^ {n - 2} - \\dots - a _ {n - 1} p ^ {n - 1}) \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/6fa23a3e7b34bcbd7140a9bd1b89cf154b10ddfd5c9ed0d5422a154d63f38161.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(p,q) = 1"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(p,q^n) = 1,(q,p^n) = 1"
                    },
                    {
                        "type": "text",
                        "content": " ，因此根据上面两个式子得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "p \\mid a _ {0}, q \\mid a _ {n}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/e103f0ffa255511aebc818a3917f1ce8be7c379fb3f8c6663b84c8c46785331e.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 1.16 计算有理根 (1) 证明:   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = x^{3} + 2x^{2} + x + 1"
                    },
                    {
                        "type": "text",
                        "content": "  在有理数域上不可约"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 求  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = 3x^{4} + 5x^{3} + x^{2} + 5x - 2"
                    },
                    {
                        "type": "text",
                        "content": " 的有理根"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 有理数域不可约即无有理根。而有理根只可能是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\pm 1, f(\\pm 1) \\neq 0"
                    },
                    {
                        "type": "text",
                        "content": "，因此不可约。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 全部列出来验证，结果为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\frac{1}{3}, -2"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.20 (Eisenstein 判别法)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x)=a_{n}x^{n}+\\cdots+a_{1}x+a_{0}"
                    },
                    {
                        "type": "text",
                        "content": " 是整系数多项式，若存在素数 p 使得：(1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p\\nmid a_{n}(2)p\\mid a_{n-1},\\cdots,a_{0}(3)p^{2}\\nmid a_{0}"
                    },
                    {
                        "type": "text",
                        "content": " ，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 在有理数域上是不可约多项式。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 1.17 使用 Eisenstein 判别法 证明在 Q 不可约"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=2x^{4}+3x^{3}-9x^{2}-3x+6"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=x^{n}+2x+2"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p = 3"
                    },
                    {
                        "type": "text",
                        "content": " 即可"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p = 2"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.3 多元多项式"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.3.1 基本概念"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 1.6 (字典排序)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于多元多项式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p(x_{1}, x_{2}, \\cdots, x_{n})"
                    },
                    {
                        "type": "text",
                        "content": "  的乘积项，先按   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1}"
                    },
                    {
                        "type": "text",
                        "content": "  次数由高到低排序，再根据   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{2}, x_{3}, \\cdots"
                    },
                    {
                        "type": "text",
                        "content": "  次数排序，将字典排序得到的第一项称为首项。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.21 (乘积保首项)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n}),g(x_{1},\\cdots,x_{n})"
                    },
                    {
                        "type": "text",
                        "content": "  为多元多项式，则它们乘积的首项等于首项的乘积。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "1.3 多元多项式"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "10"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.3.2 对称多项式"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 1.7 (对称多项式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若多元多项式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n})"
                    },
                    {
                        "type": "text",
                        "content": "  互换任意   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{i},x_{j}"
                    },
                    {
                        "type": "text",
                        "content": "  后多项式保持不变，则称其为对称多项式"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.22 (对称多项式的性质)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对称多项式的和、差、乘积、多项式复合仍为对称多项式。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 1.8 (初等对称多项式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "将形如   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\sigma_{i}=\\sum_{k_{1}<k_{2}<\\cdots<k_{i}}x_{k_{1}}x_{k_{2}}\\cdots x_{k_{i}}"
                    },
                    {
                        "type": "text",
                        "content": "  的对称多项式称为初等对称多项式："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} \\sigma_ {1} = x _ {1} + x _ {2} + \\dots + x _ {n} \\\\ \\sigma_ {2} = x _ {1} x _ {2} + x _ {1} x _ {3} + \\dots + x _ {n - 1} x _ {n} \\\\ \\quad \\vdots \\\\ \\sigma_ {n} = x _ {1} x _ {2} \\dots x _ {n} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1d88d6beeef9f8c98aa767a8c621688e40f422385a4c6c4bb79187d0f323239c.jpg"
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            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.23 (Vieta 定理)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=x^{n}+a_{n-1}x^{n-1}+\\cdots+a_{0}"
                    },
                    {
                        "type": "text",
                        "content": " ，假设其解为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1},\\cdots,x_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，则"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} a _ {n - 1} = - \\sigma_ {1} \\\\ a _ {n - 2} = \\sigma_ {2} \\\\ \\vdots \\\\ a _ {n - i} = (- 1) ^ {i} \\sigma_ {i} \\\\ \\vdots \\\\ a _ {0} = (- 1) ^ {n} \\sigma_ {n} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/005ace7ada8376bf714e0947835195f27dbd8245a4fdce4de2bacb30244bba52.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.24 (复合定理)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "任意对称多项式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n})"
                    },
                    {
                        "type": "text",
                        "content": "  可唯一地写成初等多项式的多项式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\varphi(\\sigma_{1},\\cdots,\\sigma_{n})"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n})"
                    },
                    {
                        "type": "text",
                        "content": "  的首项为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "ax_{1}^{l_{1}}x_{2}^{l_{2}}\\cdots x_{n}^{l_{n}}"
                    },
                    {
                        "type": "text",
                        "content": " ，且   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "l_{1}\\geq l_{2}\\geq\\cdots l_{n}"
                    },
                    {
                        "type": "text",
                        "content": " （根据对称多项式可交换性可知该性质），做多项式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\varphi_{1}=a\\sigma_{1}^{l_{1}-l_{2}}\\sigma_{2}^{l_{2}-l_{3}}\\cdots\\sigma_{n}^{l_{n}}"
                    },
                    {
                        "type": "text",
                        "content": " ，得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\varphi_{1}"
                    },
                    {
                        "type": "text",
                        "content": "  的首项为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "ax_{1}^{l_{1}-l_{2}}(x_{1}x_{2})^{l_{2}-l_{3}}\\cdots(x_{1}\\cdots x_{n})^{l_{n}}=ax_{1}^{l_{1}}\\cdots x_{n}^{l_{n}}"
                    },
                    {
                        "type": "text",
                        "content": " ，取   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{1}(x)=f(x)-\\varphi_{1}"
                    },
                    {
                        "type": "text",
                        "content": " ，以此类推可以得到结果。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 1.18 (1) 将   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1}, x_{2}, x_{3}) = x_{1}^{3} + x_{2}^{3} + x_{3}^{3}"
                    },
                    {
                        "type": "text",
                        "content": "  表达为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\sigma_{1}, \\sigma_{2}, \\sigma_{3}"
                    },
                    {
                        "type": "text",
                        "content": "  的多项式"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解(1)根据首项可知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "l_{1} = 3, l_{2} = l_{3} = 0"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\varphi_{1} = \\sigma_{1}^{l_{1}}\\sigma_{2}^{l_{2}}\\sigma_{3}^{l_{3}} = (x_{1} + x_{2} + x_{3})^{3}"
                    },
                    {
                        "type": "text",
                        "content": "，得到"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "f _ {1} (x) = f (x) - \\varphi_ {1} = - 3 (x _ {1} ^ {2} x _ {2} + x _ {1} ^ {2} x _ {3} + \\dots) - 6 x _ {1} x _ {2} x _ {3}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/d63a1cf451e904d032079bf515824cc18e001abde1bec4ebc2d514662330025b.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f_{1}(x)"
                    },
                    {
                        "type": "text",
                        "content": "  首项为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "-3x_{1}^{2}x_{2}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\varphi_{2}=-3\\sigma_{1}^{2-1}\\sigma_{2}=-3(x_{1}^{2}x_{2}+x_{2}^{2}x_{1}+\\cdots)-9x_{1}x_{2}x_{3}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{2}(x_{1},x_{2},x_{3})=3\\sigma_{3}"
                    },
                    {
                        "type": "text",
                        "content": " 。综上   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},x_{2},x_{3})=3\\sigma_{3}-3\\sigma_{1}\\sigma_{2}+\\sigma_{1}^{3}"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "1.3 多元多项式"
                    }
                ]
            },
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "11"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.3.3 判别式"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对称多项式理论可应用于判断一元高次多项式重根的存在性，这就导出了判别式的概念。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 1.9 (判别式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "给定多项式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=x^{n}+a_{n-1}x^{n-1}+\\cdots+a_{0}\\in\\mathbb{C}[x]"
                    },
                    {
                        "type": "text",
                        "content": " ，设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1},\\cdots,x_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": "  的根，则定义判别式为"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "D (f) := \\prod_ {1 \\leq i <   j \\leq n} (x _ {i} - x _ {j}) ^ {n}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/4ecb7bedf9c618cf9193e2727b75838f26eb26343ea7a86ec66fc7a36a43a2c6.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 有重根当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D(f) = 0"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.25 (判别式的计算)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D(f)"
                    },
                    {
                        "type": "text",
                        "content": "  是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1},\\cdots,x_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  的对称多项式，因此可以表示为初等对称多项式的多项式，再根据 Vieta 定理即可表达为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{0},\\cdots,a_{n-1}"
                    },
                    {
                        "type": "text",
                        "content": "  的多项式"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 1.19 判别式的计算 (1) 计算   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=x^{2}+px+q"
                    },
                    {
                        "type": "text",
                        "content": "  的判别式"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解(1)设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x_{1},x_{2}) = (x_{1} - x_{2})^{2}"
                    },
                    {
                        "type": "text",
                        "content": " ，可以将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x_{1},x_{2})"
                    },
                    {
                        "type": "text",
                        "content": " 表达为初等对称多项式的多项式："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "g (x _ {1}, x _ {2}) = \\sigma_ {1} ^ {2} - 4 \\sigma_ {2}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1596b2df484788854bbfaf78a700b6b8459f9707934e0d7831039404ca285512.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "而根据 Vieta 定理，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p = -\\sigma_{1}, q = \\sigma_{2}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D(f) = p^{2} - 4q"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "1.3.4 结式"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "给定多项式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\\cdots+a_{0}, g(x)=b_{m}x^{m}+\\cdots+b_{0}"
                    },
                    {
                        "type": "text",
                        "content": " ，虽然可以用辗转相除法计算它们的最大公因式，但判断它们是否互素有更好的解决办法，这就是结式"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 1.10 (结式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "考虑两个多项式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\\cdots+a_{0}, g(x)=b_{m}x^{m}+b_{m-1}x^{m-1}+\\cdots+b_{0}"
                    },
                    {
                        "type": "text",
                        "content": " ，则定义两个矩阵："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c c c c c c c} a _ {n} & a _ {n - 1} & a _ {n - 2} & \\dots & a _ {0} & 0 & \\dots & 0 \\\\ 0 & a _ {n} & a _ {n - 1} & a _ {n - 2} & \\dots & a _ {0} & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\ddots & \\ddots & & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & a _ {n} & a _ {n - 1} & a _ {n - 2} & \\dots & a _ {0} \\end{array} \\right] _ {m \\times (n + m)}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f9cbf7350d6b306f4a1b1c82b4d0e4d3f787be4bb20291c1b1833e0b1bfeee24.jpg"
                }
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "B = \\left[ \\begin{array}{c c c c c c c c} b _ {m} & b _ {m - 1} & b _ {m - 2} & \\dots & b _ {0} & 0 & \\dots & 0 \\\\ 0 & b _ {m} & b _ {m - 1} & b _ {m - 2} & \\dots & b _ {0} & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\ddots & \\ddots & & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & b _ {m} & b _ {m - 1} & b _ {m - 2} & \\dots & b _ {0} \\end{array} \\right] _ {n \\times (n + m)}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f4178a15788525b380c63d4a8f791f7275116cada1f53e78ffea04a6062fa12c.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则定义它们的结式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "R(f,g) = \\left| \\begin{array}{l}A\\\\ B \\end{array} \\right|"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 1.26 (结式判定定理)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "给定   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=a_{n}x^{n}+\\cdots+a_{0}, g(x)=b_{m}x^{m}+\\cdots+b_{0}, a_{n}, b_{m}\\neq0"
                    },
                    {
                        "type": "text",
                        "content": " ，则 f, g 不互素的充要条件为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "R(f,g)=0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\partial (f) = n, \\partial (g) = m"
                    },
                    {
                        "type": "text",
                        "content": "，首先  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 不互素当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists u(x), v(x)"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\partial (u) < \\partial (g), \\partial (v) < \\partial (f)"
                    },
                    {
                        "type": "text",
                        "content": "，使得"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "12"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "1.3 多元多项式"
                    }
                ]
            },
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            ]
        }
    ],
    [
        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "#"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "u(x)f(x) = v(x)g(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，不妨设"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} u (x) = u _ {m - 1} x ^ {m - 1} + \\dots + u _ {0} \\\\ v (x) = v _ {n - 1} x ^ {n - 1} + \\dots + v _ {0} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/a1f4b2c78dbcb4ade5d4074f8c74ef6bc28789ad1c0cfb484413925705578190.jpg"
                }
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x)f(x) = v(x)g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 展开对比系数得到"
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                ]
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            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} a _ {n} u _ {m - 1} = b _ {m} v _ {n - 1} \\\\ a _ {n - 1} u _ {m - 1} + a _ {n} u _ {m - 2} = b _ {m - 1} v _ {n - 1} + b _ {m} v _ {n - 2} \\\\ \\vdots \\\\ a _ {0} u _ {0} = b _ {0} v _ {0} \\end{array} \\right.",
                "math_type": "latex",
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                        "type": "text",
                        "content": "将上述线性方程组看成   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u_{0}, \\cdots, u_{m-1}, v_{0}, \\cdots, v_{n-1}"
                    },
                    {
                        "type": "text",
                        "content": "  的线性方程组，则系数矩阵的转置   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C^{T}"
                    },
                    {
                        "type": "text",
                        "content": "  为结式定义中的矩阵："
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                ]
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            "content": {
                "math_content": "C ^ {T} = \\binom{A}{- B}",
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此有解当且仅当  "
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                    {
                        "type": "equation_inline",
                        "content": "|C^T| = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，也等价于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "R(f,g) = 0"
                    }
                ]
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                        "content": "定理 1.27 (结式计算公式)"
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                    {
                        "type": "text",
                        "content": "给定前面的   "
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                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}, \\cdots, \\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  和   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{1}, \\cdots, \\beta_{m}"
                    },
                    {
                        "type": "text",
                        "content": "  分别是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), g(x)"
                    },
                    {
                        "type": "text",
                        "content": "  的复根，则"
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            "content": {
                "math_content": "R (f, g) = a _ {n} ^ {m} b _ {m} ^ {n} \\prod_ {i = 1} ^ {n} \\prod_ {j = 1} ^ {m} (\\alpha_ {i} - \\beta_ {i})",
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                        "content": "定理 1.28 (判别式计算公式)"
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                "level": 2
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                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}[x]"
                    },
                    {
                        "type": "text",
                        "content": " 中多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = a_{n}x^{n} + a_{n - 1}x^{n - 1} + \\dots +a_{0}, a_{n}\\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " ，则  "
                    },
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                        "type": "equation_inline",
                        "content": "f(x)"
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                    {
                        "type": "text",
                        "content": " 的判别式"
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            "content": {
                "math_content": "D (f) = (- 1) ^ {\\frac {n (n - 1)}{2}} a _ {0} ^ {- (2 n - 1)} R (f, f ^ {\\prime})",
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                    "path": "images/c3694a89fa415bbeb2e1b6a0018e4228b13aecfe4afc344ab330609e53eff7b6.jpg"
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                "page_header_content": [
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                        "type": "text",
                        "content": "1.3 多元多项式"
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            "type": "page_number",
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                        "content": "13"
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                        "content": "第2章 矩阵"
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                        "type": "text",
                        "content": "若  "
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                    {
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                        "content": "AB = BA = E"
                    },
                    {
                        "type": "text",
                        "content": " ，则称  "
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                        "content": "B"
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                        "content": " 为  "
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                        "content": "A"
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                    {
                        "type": "text",
                        "content": " 的逆，记  "
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                        "content": "A^{-1}"
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                        "content": "定理 2.1 (矩阵逆的基本性质)"
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            "content": {
                "math_content": "(1) (k A) ^ {- 1} = k ^ {- 1} A ^ {- 1} (2) (A B) ^ {- 1} = B ^ {- 1} A ^ {- 1} (3) \\left(A ^ {T}\\right) ^ {- 1} = \\left(A ^ {- 1}\\right) ^ {T} (4) | A ^ {- 1} | = | A | ^ {- 1}",
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                        "content": "证明 (2)  "
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                        "content": "(AB)^{-1}AB = E"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
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                        "content": "(AB)^{-1} = B^{-1}A^{-1}"
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                        "content": "(3)  "
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                        "content": "(A^{-1})^T A^T = (AA^{-1})^T = E"
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                        "content": "(4)  "
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                        "content": "|A^{-1}A|=|A^{-1}|\\cdot|A|=1"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
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                        "content": "|A^{-1}|=|A|^{-1}"
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                        "content": "定理 2.2 (可逆充要条件)"
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                        "content": "矩阵  "
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                        "content": "A"
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                        "content": " 可以的充要条件如下："
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                        "type": "text",
                        "content": "-  "
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                    {
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                        "content": "|A| \\neq 0"
                    }
                ]
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            "content": {
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                        "content": "-  "
                    },
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                        "content": "A"
                    },
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                        "content": " 可表示为初等矩阵的积"
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                        "content": "定理 2.3 (利用初等行变换求逆)"
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                        "content": "若  "
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                        "content": "A"
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                    {
                        "type": "text",
                        "content": " 可逆，则利用初等行变换可以得到  "
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                    {
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                        "content": "(A,E)\\rightarrow (E,A^{-1})"
                    },
                    {
                        "type": "text",
                        "content": " 。特别地，  "
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                        "content": "(A,B)\\rightarrow (E,A^{-1}B)"
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                        "content": "笔记 同样可以通过初等列变换得到：  "
                    },
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                        "content": "\\begin{pmatrix} A \\\\ B \\end{pmatrix} \\rightarrow \\begin{pmatrix} E \\\\ BA^{-1} \\end{pmatrix}"
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                        "content": "练习 2.1 利用初等变换求逆 (1) 计算下面矩阵的逆:"
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            "content": {
                "math_content": "\\left[ \\begin{array}{c c c} 1 & 1 & - 1 \\\\ 2 & 1 & 0 \\\\ 1 & - 1 & 0 \\end{array} \\right]",
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                        "content": "解结果为  "
                    },
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                        "content": "\\frac{1}{3}\\left[ \\begin{array}{ccc}0 & 1 & 1\\\\ 0 & 1 & -2\\\\ -3 & 2 & -1 \\end{array} \\right]"
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                ]
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                        "type": "text",
                        "content": "练习 2.2 逆矩阵理论练习 (1) A 为方阵，若   "
                    },
                    {
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                        "content": "A^{k}=O"
                    },
                    {
                        "type": "text",
                        "content": " ，证明 E-A 可逆，且   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(E-A)^{-1}=E+A+A^{2}+\\cdots+A^{k-1}"
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                        "content": "(2)  "
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                    {
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                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵，若  "
                    },
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                        "content": "A + B = AB"
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                    {
                        "type": "text",
                        "content": " ，证明： "
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                        "content": "(3) 若  "
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                        "content": "A + B"
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                    {
                        "type": "text",
                        "content": " 可逆，证明  "
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                        "content": "A^{-1} + B^{-1}"
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                        "content": " 也可逆，并求其逆"
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                        "content": "解(1)直接做乘法验证:"
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                "math_content": "(E - A) \\left(E + A + A ^ {2} + \\dots + A ^ {k - 1}\\right) = E + A + A ^ {2} + \\dots + A ^ {k - 1} - A - A ^ {2} - \\dots - A ^ {k} = E - A ^ {k} = E",
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            "content": {
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                        "type": "text",
                        "content": "(2) 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A + B = AB"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
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                        "type": "equation_inline",
                        "content": "(A - E)(B - E) = AB - (A + B) + E = E"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
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                        "content": "(A - E)^{-1} = B - E"
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                        "content": "(3) 根据  "
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                        "content": "B(A^{-1} + B^{-1})A = (A + B)"
                    },
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                        "content": " 得到"
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            "content": {
                "math_content": "(A ^ {- 1} + B ^ {- 1}) B = A ^ {- 1} (A + B) \\Rightarrow (A ^ {- 1} + B ^ {- 1}) B (A + B) ^ {- 1} = A ^ {- 1}",
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A^{-1} + B^{-1})^{-1} = B(A + B)^{-1}A"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 2.2 ((代数) 余子式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A = (a_{ij})"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n\\times n"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵，划去  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i"
                    },
                    {
                        "type": "text",
                        "content": " 行  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "j"
                    },
                    {
                        "type": "text",
                        "content": " 列的行列式称为余子式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{ij}"
                    },
                    {
                        "type": "text",
                        "content": " ，代数余子式为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(-1)^{i + j}M_{ij}"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 2.3 (伴随矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A = (a_{ij})_n"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{ij}"
                    },
                    {
                        "type": "text",
                        "content": "  是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{ij}"
                    },
                    {
                        "type": "text",
                        "content": "  的代数余子式，则其伴随为 (注意行列颠倒!!):"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A ^ {*} = \\left[ \\begin{array}{c c c c} A _ {1 1} & A _ {2 1} & \\dots & A _ {n 1} \\\\ A _ {1 2} & A _ {2 2} & \\dots & A _ {n 2} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ A _ {1 n} & A _ {2 n} & \\dots & A _ {n n} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/874bc68cecf0f87576dd54bb8a4c27d9adfbcb305334509a09f502c22eab1116.jpg"
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.4 (利用伴随求逆)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 及其伴随矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{*}"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AA^{*} = |A|E"
                    },
                    {
                        "type": "text",
                        "content": " ，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 可逆，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1} = \\frac{1}{|A|} A^{*}"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 考虑  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B = AA^{*}"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B_{ij} = \\sum_{k=1}^{n} a_{ik} A_{kj}"
                    },
                    {
                        "type": "text",
                        "content": "，根据行列式的展开式可知"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "B _ {i j} = \\left\\{ \\begin{array}{l l} | A |, & i = j \\\\ 0, & i \\neq j \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/bc9138e59e5c22e1c84993f9c905b1566236d2d8f2be39421dadf69844a32b19.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AA^{*} = |A|E"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.1.3 伴随矩阵的性质"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.5 (伴随运算的性质)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵，则"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "1.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A^{*})^{T} = (A^{T})^{*}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "2.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A^{*})^{-1} = (A^{-1})^{*}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "3.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(AB)^{*} = B^{*}A^{*}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = (a_{ij})"
                    },
                    {
                        "type": "text",
                        "content": "，则"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(A ^ {*}) ^ {T} = \\left[ \\begin{array}{c c c c} A _ {1 1} & A _ {1 2} & \\dots & A _ {1 n} \\\\ A _ {2 1} & A _ {2 2} & \\dots & A _ {2 n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ A _ {n 1} & A _ {n 2} & \\dots & A _ {n n} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/c86118239fc325499c3bd7e7a8a7ec74fe85b407a9d7e3c10f28e78e2fa54d16.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "同理设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A^T)^* = (B_{ij})"
                    },
                    {
                        "type": "text",
                        "content": " ，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B_{ij} = A_{ji}^T"
                    },
                    {
                        "type": "text",
                        "content": " ，因此行列式大小相同。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{*} = |A|A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " ，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A^{*})^{-1} = \\frac{1}{|A|} A"
                    },
                    {
                        "type": "text",
                        "content": " 因此"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(A ^ {- 1}) ^ {*} = | A ^ {- 1} | A = \\frac {1}{| A |} A",
                "math_type": "latex",
                "image_source": {
                    "path": "images/da2261677841eb4d9b86bf1de71c56615d0a62ee142ce97ee2c854430cf03761.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此结论成立"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "15"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.1 矩阵基础"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(3) (A B) ^ {*} = | A B | (A B) ^ {- 1} = | B | (B) ^ {- 1} | A | A ^ {- 1} = B ^ {*} A ^ {*}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/3ce280ccf5acbbadb7106a1d396d4022cf7dabc4572a6b442c5326a3f24bb2f2.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定理 2.6 (伴随矩阵的行列式)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "| A ^ {*} | = | A | ^ {n - 1}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/c1dfdd1ff136234bbb3ebfffd2c30b67b5524a41d45f06608cd55f9bb48f1d8c.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A^{*}| = \\left| |A| A^{-1} \\right| = |A|^{n} \\cdot |A^{-1}| = |A|^{n - 1}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.7 (伴随矩阵的秩)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A^{*}"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的伴随矩阵，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "r (A ^ {*}) = \\left\\{ \\begin{array}{l l} n, & r (A) = n \\\\ 1, & r (A) = n - 1 \\\\ 0, & r (A) <   n - 1 \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/7f42785d741889f71fb44f29c071abefe0160981052c028162c32e2ccb9577e1.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = n"
                    },
                    {
                        "type": "text",
                        "content": " 时， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A^{*}| = |A|^{n - 1} \\neq 0"
                    },
                    {
                        "type": "text",
                        "content": "，因此满秩。 "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = n - 1"
                    },
                    {
                        "type": "text",
                        "content": " 时，由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AA^{*} = |A|E = O"
                    },
                    {
                        "type": "text",
                        "content": "，根据秩不等式："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "r (A A ^ {*}) \\geq r (A) + r (A ^ {*}) - n = r (A ^ {*}) - 1 \\Rightarrow r (A ^ {*}) \\leq 1",
                "math_type": "latex",
                "image_source": {
                    "path": "images/7198cea4267323b64b5255082e5145d28de70e0e5aeb903b6ea0fc44636c17c0.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "而由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = n - 1"
                    },
                    {
                        "type": "text",
                        "content": " ，其至少有一个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n - 1"
                    },
                    {
                        "type": "text",
                        "content": " 阶代数余子式不为0，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A^{*})\\geq 1"
                    },
                    {
                        "type": "text",
                        "content": " ，综上  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A^{*}) = 1"
                    },
                    {
                        "type": "text",
                        "content": " 。其余显然"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.8 (伴随矩阵的特征值)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "矩阵 A 的特征值为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1},\\cdots,\\lambda_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{*}"
                    },
                    {
                        "type": "text",
                        "content": "  的特征值   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{j}^{*}=\\prod_{i\\neq j}\\lambda_{i}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 考虑  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = P^{-1}JP"
                    },
                    {
                        "type": "text",
                        "content": "，由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J"
                    },
                    {
                        "type": "text",
                        "content": " 是上三角阵，根据初等变换法， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 是上三角阵， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J^{*} = |J|J^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 也是上三角阵，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J"
                    },
                    {
                        "type": "text",
                        "content": " 对角元素的代数余子式为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\prod_{i\\neq j}\\lambda_i"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J^{*}"
                    },
                    {
                        "type": "text",
                        "content": " 的对角元为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\prod_{i\\neq j}\\lambda_i"
                    },
                    {
                        "type": "text",
                        "content": "，代数余子式在初等变换下不变 (位置可能变化，"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "但值整体相同)，因此结论成立。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.1.4 广义逆矩阵"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 2.4 (广义逆)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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                592
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A_{m\\times n}"
                    },
                    {
                        "type": "text",
                        "content": " 是数域  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 上的矩阵，若存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G_{n\\times m}"
                    },
                    {
                        "type": "text",
                        "content": " 使得"
                    }
                ]
            },
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            "type": "equation_interline",
            "content": {
                "math_content": "A G A = A",
                "math_type": "latex",
                "image_source": {
                    "path": "images/0bca354b36bc9f30727a66df36218180a039ed7769113934f3d57881c0317cb3.jpg"
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的广义逆矩阵。"
                    }
                ]
            },
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        },
        {
            "type": "image",
            "content": {
                "image_source": {
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                "image_caption": [],
                "image_footnote": []
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "笔记 广义逆矩阵一般不唯一。"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.9 (广义逆的表达式)"
                    }
                ],
                "level": 2
            },
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        },
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                    {
                        "type": "text",
                        "content": "设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{m\\times n}"
                    },
                    {
                        "type": "text",
                        "content": " 的秩为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r"
                    },
                    {
                        "type": "text",
                        "content": " ，设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = P\\left( \\begin{array}{cc}E_r & O\\\\ O & O \\end{array} \\right)Q"
                    },
                    {
                        "type": "text",
                        "content": " ，其中  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P_{m\\times m},Q_{n\\times n}"
                    },
                    {
                        "type": "text",
                        "content": " 为可逆矩阵，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 全部的广义逆矩阵为"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "G = Q ^ {- 1} \\left( \\begin{array}{c c} E _ {r} & C \\\\ D & F \\end{array} \\right) P ^ {- 1}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/629ed2d1e7d7b4e883abd4650383812f8a15d782a82fe0f0b837755d6b857e5a.jpg"
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            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "这里 C, D, F 为任意   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r \\times (m - r)"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(n - r) \\times r"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(m - r) \\times (n - r)"
                    },
                    {
                        "type": "text",
                        "content": "  阶矩阵。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 验证  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G"
                    },
                    {
                        "type": "text",
                        "content": " 是广义逆: 直接乘起来就行, 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AGA = A"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.1 矩阵基础"
                    }
                ]
            },
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        },
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            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "16"
                    }
                ]
            },
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        }
    ],
    [
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 验证广义逆都是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G"
                    },
                    {
                        "type": "text",
                        "content": "：设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G = Q^{-1}\\left( \\begin{array}{cc}B & C\\\\ D & F \\end{array} \\right)P^{-1}"
                    },
                    {
                        "type": "text",
                        "content": "，乘法得到"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A G A = P \\left( \\begin{array}{c c} B & O \\\\ O & O \\end{array} \\right) Q",
                "math_type": "latex",
                "image_source": {
                    "path": "images/773a275e69d0276bf8fa1d2a57e72101d8de2dc8bc9b7455469dea524aaf06af.jpg"
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            },
            "bbox": [
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B = E_{r}"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.10 (非方阵线性方程组解的表达式)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A_{m\\times n}"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的广义逆，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX = b"
                    },
                    {
                        "type": "text",
                        "content": " 的通解可表示为"
                    }
                ]
            },
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        {
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            "content": {
                "math_content": "X = G b + (E _ {n} - G A) Y",
                "math_type": "latex",
                "image_source": {
                    "path": "images/2c226093dd5a1bc4b817bb2f47d8b2053a37015cd71e6fbb6dd6244e1d5c98c4.jpg"
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            },
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                        "type": "text",
                        "content": "这里  "
                    },
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                        "type": "equation_inline",
                        "content": "Y"
                    },
                    {
                        "type": "text",
                        "content": " 是任一  "
                    },
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                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 维向量"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 证明是解：设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Z"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AZ = b"
                    },
                    {
                        "type": "text",
                        "content": " 的解，则此时有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "b = AZ = AGAZ = AGb"
                    },
                    {
                        "type": "text",
                        "content": "。将  "
                    },
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                        "type": "equation_inline",
                        "content": "X"
                    },
                    {
                        "type": "text",
                        "content": " 表达式代入  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX"
                    },
                    {
                        "type": "text",
                        "content": " 验证："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A X = A G b + (A - A G A) Y = b + O = b",
                "math_type": "latex",
                "image_source": {
                    "path": "images/fcfd48515f1a3af97ebc436791c6c0d0b25f137ecea21d3fdcfdaa88ac42ea73.jpg"
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 证明解都有该形式: 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{0}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX = b"
                    },
                    {
                        "type": "text",
                        "content": " 的解, 则"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "X _ {0} = G b + X _ {0} - G b = G b + X _ {0} - G A X _ {0} = G b + (E _ {n} - G A) X _ {0}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/bda651acdeed4388d5fcff1fefb79467b0dd9e7b3607efb26503b0b33ce196a8.jpg"
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            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.1.5 Moore-Penrose 广义逆"
                    }
                ],
                "level": 2
            },
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        },
        {
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                        "content": "定义 2.5 (共轭转置)"
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                ],
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        },
        {
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                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n\\times m"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵，则定义  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{H} = (\\overline{A})^{T}"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 2.6 (Moore-Penrose 广义逆)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m \\times n"
                    },
                    {
                        "type": "text",
                        "content": " 阶矩阵，若存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n \\times m"
                    },
                    {
                        "type": "text",
                        "content": " 阶矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G"
                    },
                    {
                        "type": "text",
                        "content": " 满足："
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "1. AGA = A"
                    }
                ]
            },
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        },
        {
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                        "type": "text",
                        "content": "2.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "GAG = G"
                    }
                ]
            },
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        },
        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "3.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(AG)^{H} = AG"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "4.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(GA)^{H} = GA"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的M-P广义逆"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.11 (M-P 广义逆的存在唯一性)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A_{m\\infty n}"
                    },
                    {
                        "type": "text",
                        "content": "  的 M-P 广义逆总是存在且唯一的，且若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{1},\\cdots,\\beta_{r}"
                    },
                    {
                        "type": "text",
                        "content": "  是 A 列向量的极大线性无关组，记   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B=\\left[\\beta_{1},\\cdots,\\beta_{r}\\right]"
                    },
                    {
                        "type": "text",
                        "content": " ，分解 A=BC，则 M-P 广义逆的具体表达式为："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A ^ {+} = C ^ {H} (C C ^ {H}) ^ {- 1} (B ^ {H} B) ^ {- 1} B ^ {H}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/38a7226d6dd86c6efa5f43eac50229bf9d9dd64d1e53fe6cbdc96358261aa442.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 存在性：设 A 的秩为 r，取   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{1},\\cdots,\\beta_{r}"
                    },
                    {
                        "type": "text",
                        "content": "  为 A 列向量的极大线性无关组，此时"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = (\\alpha_ {1}, \\dots , \\alpha_ {n}) = (\\beta_ {1}, \\dots , \\beta_ {r}) \\left( \\begin{array}{c c c} c _ {1 1} & \\dots & c _ {1 n} \\\\ \\vdots & & \\vdots \\\\ c _ {r 1} & \\dots & c _ {r n} \\end{array} \\right) = B C",
                "math_type": "latex",
                "image_source": {
                    "path": "images/4141aa8fa8e9d8fd8ce2a301c7ec9697a2f6f3cd8f183a8584133766eff962f2.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "此时   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(BC)=r\\leq r(C)\\leq\\min\\{r,n\\}\\leq r"
                    },
                    {
                        "type": "text",
                        "content": " ，得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(C)=r"
                    },
                    {
                        "type": "text",
                        "content": "  行满秩，B 显然列满秩，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B^{H}B,CC^{H}"
                    },
                    {
                        "type": "text",
                        "content": "  是 r 阶可逆矩阵 (根据   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A^{H}A),r(AA^{H})"
                    },
                    {
                        "type": "text",
                        "content": "  的性质)。令"
                    }
                ]
            },
            "bbox": [
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                902,
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        },
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            "type": "equation_interline",
            "content": {
                "math_content": "G = C ^ {H} (C C ^ {H}) ^ {- 1} (B ^ {H} B) ^ {- 1} B ^ {H}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/799651516b69a7112eb34920086e3c13e9cac064a7b757eac5f4196fc1e59b92.jpg"
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            "bbox": [
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "可验证  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G"
                    },
                    {
                        "type": "text",
                        "content": " 满足上述所有条件"
                    }
                ]
            },
            "bbox": [
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        },
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            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.1 矩阵基础"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "17"
                    }
                ]
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        }
    ],
    [
        {
            "type": "paragraph",
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "唯一性：设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X"
                    },
                    {
                        "type": "text",
                        "content": " 是另一个广义逆，则进行下面步骤，本质是用  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AXA = A"
                    },
                    {
                        "type": "text",
                        "content": " 消去  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X"
                    }
                ]
            },
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            "content": {
                "math_content": "\\begin{array}{l} \\boldsymbol {X} = \\boldsymbol {X A X} = \\boldsymbol {X} (\\boldsymbol {A X}) ^ {\\mathrm{H}} = \\boldsymbol {X X} ^ {\\mathrm{H}} \\boldsymbol {A} ^ {\\mathrm{H}} = \\boldsymbol {X X} ^ {\\mathrm{H}} (\\boldsymbol {A G A}) ^ {\\mathrm{H}} \\\\ \\quad = \\boldsymbol {X X} ^ {\\mathrm{H}} \\boldsymbol {A} ^ {\\mathrm{H}} (\\boldsymbol {A G}) ^ {\\mathrm{H}} = \\boldsymbol {X} (\\boldsymbol {A X}) ^ {\\mathrm{H}} (\\boldsymbol {A G}) ^ {\\mathrm{H}} = \\boldsymbol {X A X A G} \\\\ \\quad = \\boldsymbol {X A G} = \\boldsymbol {X} (\\boldsymbol {A G A}) \\boldsymbol {G} = (\\boldsymbol {X A}) ^ {\\mathrm{H}} (\\boldsymbol {G A}) ^ {\\mathrm{H}} \\boldsymbol {G} \\\\ \\quad = (\\boldsymbol {G A X A}) ^ {\\mathrm{H}} \\boldsymbol {G} = (\\boldsymbol {G A}) ^ {\\mathrm{H}} \\boldsymbol {G} = (\\boldsymbol {G A}) \\boldsymbol {G} \\\\ \\quad = \\boldsymbol {G}. \\end{array}",
                "math_type": "latex",
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习2.3计算M-P逆  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(1)A = \\left( \\begin{array}{ccc}1 & 0 & -1\\\\ 1 & 2 & 0\\\\ 0 & 2 & 1 \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " 的M-P逆"
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                ]
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                        "type": "text",
                        "content": "解(1) 先进行分解，得到"
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            "content": {
                "math_content": "B = \\left( \\begin{array}{c c} 1 & 0 \\\\ 1 & 1 \\\\ 0 & 1 \\end{array} \\right), C = \\left( \\begin{array}{c c c} 1 & 0 & - 1 \\\\ 0 & 2 & 1 \\end{array} \\right)",
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                    "path": "images/17014278ed0e87e3acb286a66667a79f7c8e2eaef189f8222a68f970bdce2ab7.jpg"
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "最终结果得到"
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        {
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            "content": {
                "math_content": "A ^ {+} = \\frac {1}{9} \\left( \\begin{array}{c c c} 3 & 2 & - 1 \\\\ 0 & 2 & 2 \\\\ - 3 & - 1 & 2 \\end{array} \\right)",
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                "image_source": {
                    "path": "images/9d412c59fd6479bbbb0cccd0391bc327c94ab79eccfd9e4af90c9541ccee2322.jpg"
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            "bbox": [
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                        "type": "text",
                        "content": "定理 2.12 (最小二乘解的表达式)"
                    }
                ]
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                        "type": "equation_inline",
                        "content": "AX = b"
                    },
                    {
                        "type": "text",
                        "content": " 最小二乘解为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X = A^{+}b + (E_{n} - A^{+}A)Y"
                    },
                    {
                        "type": "text",
                        "content": " ，其中  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{+}b"
                    },
                    {
                        "type": "text",
                        "content": " 是唯一的极小最小二乘解。"
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                ]
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                        "content": "2.2 分块矩阵"
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                ],
                "level": 2
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                "title_content": [
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                        "type": "text",
                        "content": "2.2.1 常见矩阵拆分技巧"
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                ],
                "level": 2
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                        "type": "text",
                        "content": "定理 2.13 (点态分解)"
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                "level": 2
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                "paragraph_content": [
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                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n \\times n"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵，则"
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                ]
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            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c c c} a _ {1} b _ {1} & a _ {1} b _ {2} & \\dots & a _ {1} b _ {n} \\\\ a _ {2} b _ {1} & a _ {2} b _ {2} & \\dots & a _ {2} b _ {n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a _ {n} b _ {1} & a _ {n} b _ {2} & \\dots & a _ {n} b _ {n} \\end{array} \\right] \\Rightarrow A = \\left( \\begin{array}{c} a _ {1} \\\\ a _ {2} \\\\ \\vdots \\\\ a _ {n} \\end{array} \\right) \\left( \\begin{array}{c c c c} b _ {1} & b _ {2} & \\dots & b _ {n} \\end{array} \\right)\\tag{❤️}",
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                        "content": "定理 2.14 (  "
                    },
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                        "type": "equation_inline",
                        "content": "A^{T}A"
                    },
                    {
                        "type": "text",
                        "content": " 的和式表达)"
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                ],
                "level": 2
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        {
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                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n\\times m"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵，将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 做下面的行划分，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T A = \\sum_{i = 1}^{n}\\alpha_i^T\\alpha_i"
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                ]
            },
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            "content": {
                "math_content": "A = \\left( \\begin{array}{c} \\alpha_ {1} \\\\ \\alpha_ {2} \\\\ \\vdots \\\\ \\alpha_ {s} \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/7ce6246e9253ce4b7f1b5fafaa828f85e5cbb78bd0bb4b6492c794cdddec8d47.jpg"
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                "page_header_content": [
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                        "content": "2.2 分块矩阵"
                    }
                ]
            },
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        },
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            "type": "page_number",
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                "page_number_content": [
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                        "type": "text",
                        "content": "18"
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    ],
    [
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                    {
                        "type": "text",
                        "content": "#"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
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                        "type": "text",
                        "content": "2.2.2 分块初等变换"
                    }
                ],
                "level": 2
            },
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        },
        {
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                        "type": "text",
                        "content": "定义 2.7 (分块矩阵初等变换)"
                    }
                ],
                "level": 2
            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "分块矩阵初等变换分为行变换和列变换："
                    }
                ]
            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 行变换: (1) 左乘矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P"
                    },
                    {
                        "type": "text",
                        "content": ", 加到另一行 (2) 互换两行位置 (3) 将可逆矩阵左乘至一行"
                    }
                ]
            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 列变换：(1) 右乘矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P"
                    },
                    {
                        "type": "text",
                        "content": "，加到另一列 (2) 互换两列位置 (3) 将可逆矩阵右乘至一列"
                    }
                ]
            },
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        },
        {
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                        "content": "定义 2.8 (分块初等矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对应三种分块初等变换有三种分块初等矩阵，常用的即加至另一行/列的初等分块矩阵："
                    }
                ]
            },
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        },
        {
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            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c} E & 0 \\\\ P & E \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/e0572bf311bf75ba9ecefe4ffbfa0fbae5d83aaacc8d03603f1597602f405118.jpg"
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            "bbox": [
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                        "type": "text",
                        "content": "定理 2.15 (初等变换的分块初等矩阵表达)"
                    }
                ],
                "level": 2
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "分块初等矩阵左乘一个矩阵即做行变换，右乘即做列变换："
                    }
                ]
            },
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        },
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            "content": {
                "math_content": "\\left[ \\begin{array}{c c} E & 0 \\\\ P & E \\end{array} \\right] \\left[ \\begin{array}{c c} A _ {1} & A _ {2} \\\\ A _ {3} & A _ {4} \\end{array} \\right] = \\left[ \\begin{array}{c c} A _ {1} & A _ {2} \\\\ P A _ {1} + A _ {3} & P A _ {2} + A _ {4} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/89968b81662e68df7857617f72c2c45c5fc7c0cbaa87e08bcc52f08281d17606.jpg"
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            },
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        },
        {
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            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.2.3 分块打洞技巧总结"
                    }
                ],
                "level": 2
            },
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        },
        {
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            "content": {
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                        "content": "定理 2.16 (分块打洞)"
                    }
                ],
                "level": 2
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                    {
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                        "content": "对于分块矩阵   "
                    },
                    {
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                        "content": "\\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix}"
                    },
                    {
                        "type": "text",
                        "content": "  可通过分块初等变换转换为："
                    }
                ]
            },
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        {
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            "content": {
                "math_content": "\\left[ \\begin{array}{c c} A & B \\\\ O & D - C A ^ {- 1} B \\end{array} \\right] \\quad \\left[ \\begin{array}{c c} A - B D ^ {- 1} C & O \\\\ O & D \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/e08b8cfe502a9a9ea0fa30d9f0e18e8e325bf8b2e492cca60b2830602e10f0ab.jpg"
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            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "记忆方式：除去被减的位置，减去的内容顺序都是顺时针转的。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "推论 2.1 (两种特殊形式)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对称形式和归纳形式："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 对称形式:"
                    }
                ]
            },
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        {
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            "content": {
                "math_content": "\\left[ \\begin{array}{c c} A & B \\\\ B ^ {T} & D \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A & O \\\\ O & D - B ^ {T} A ^ {- 1} B \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/60b2877854bb00cf3be83112d26c54b1194de4f79617ef8692eadc02781d3db2.jpg"
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            },
            "bbox": [
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                        "content": "归纳形式："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c} A & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A & \\alpha \\\\ 0 & a _ {n n} - \\beta^ {T} A ^ {- 1} \\alpha \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/aa6d574a3c92015b9bd5176178b9b0acfb1dbc34d3f649ae58d30db7cee1017b.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 2.4 分块矩阵与数学归纳法 (1)n 阶方阵 A 的顺序主子式非零，证明：存在 n 阶下三角矩阵 B 使得 BA 是上三角阵"
                    }
                ]
            },
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                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶实方阵,  "
                    },
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                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 所有顺序主子式大于 0 , 且非主对角元素为负, 证明:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 的元素均为正数"
                    }
                ]
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                        "type": "text",
                        "content": "证明  "
                    },
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                        "content": "(1)n = 1"
                    },
                    {
                        "type": "text",
                        "content": " 显然成立。设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n - 1"
                    },
                    {
                        "type": "text",
                        "content": " 时成立，考虑如下分块，由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 顺序主子式非零，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 顺序主子式非零，存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n - 1"
                    },
                    {
                        "type": "text",
                        "content": " 阶下三角矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B_{1}A_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 为上三角阵，因此做对应分块初等变换"
                    }
                ]
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            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] \\Rightarrow \\left[ \\begin{array}{c c} B _ {1} & \\\\ & 1 \\end{array} \\right] \\cdot \\left[ \\begin{array}{c c} E _ {n - 1} & O \\\\ - \\beta^ {T} A _ {1} ^ {- 1} & 1 \\end{array} \\right] \\cdot \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] = \\left[ \\begin{array}{c c} B _ {1} A _ {1} & B _ {1} \\alpha \\\\ O & a _ {n n} - \\beta^ {T} A _ {1} ^ {- 1} \\alpha \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/172d4a43320540d1767a676e6c68490890c885922ff4eaf4fe0f83d73631dd58.jpg"
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            },
            "bbox": [
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "右侧即一个上三角矩阵，把两个初等矩阵的乘积视作 B 即可。"
                    }
                ]
            },
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                        "content": "2.2 分块矩阵"
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                        "type": "text",
                        "content": "❤️"
                    }
                ]
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                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n = 1"
                    },
                    {
                        "type": "text",
                        "content": " 显然。设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n - 1"
                    },
                    {
                        "type": "text",
                        "content": " 时成立，对  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 做下述分块，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 元素都是正的"
                    }
                ]
            },
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            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] \\Rightarrow \\left[ \\begin{array}{c c} E _ {n - 1} & O \\\\ - \\beta^ {T} A _ {1} ^ {- 1} & 1 \\end{array} \\right] \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {n - 1} & - A _ {1} ^ {- 1} \\alpha \\\\ O & 1 \\end{array} \\right] = \\left[ \\begin{array}{c c} A _ {1} & O \\\\ O & a _ {n n} - \\beta^ {T} A _ {1} ^ {- 1} \\alpha \\end{array} \\right]",
                "math_type": "latex",
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                        "content": "根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A| > 0, |A_1| > 0"
                    },
                    {
                        "type": "text",
                        "content": " 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{nn} - \\beta^T A_1^{-1}\\alpha > 0"
                    },
                    {
                        "type": "text",
                        "content": " 。记上述等式为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "BAC = D"
                    },
                    {
                        "type": "text",
                        "content": " ，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1} = CD^{-1}B"
                    },
                    {
                        "type": "text",
                        "content": " ，根据矩阵乘法乘开即可获得结论。"
                    }
                ]
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        {
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            "content": {
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                        "content": "2.2.4 分块矩阵行列式"
                    }
                ],
                "level": 2
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        },
        {
            "type": "title",
            "content": {
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                        "type": "text",
                        "content": "定理 2.17 (分块对角矩阵的行列式)"
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                ],
                "level": 2
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "根据 Laplace 多行展开定理可以得到:"
                    }
                ]
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            "content": {
                "math_content": "\\left| \\begin{array}{c c} A & 0 \\\\ B & C \\end{array} \\right| = | A | | B |,",
                "math_type": "latex",
                "image_source": {
                    "path": "images/274e8dbb4d128c7aa9218bacfcaed7c4609632547f027060725f67919bf85b92.jpg"
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            "content": {
                "math_content": "\\left| \\begin{array}{c c c c} A _ {1 1} & 0 & \\dots & 0 \\\\ A _ {2 1} & A _ {2 2} & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ A _ {n 1} & A _ {n 2} & \\dots & A _ {n n} \\end{array} \\right| = | A _ {1 1} | | A _ {2 2} | \\dots | A _ {n n} |\\tag{❤️}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f9c38dc74f4e6283699b67d8ad114a4337e48403950e8c4ca992a6f30b84b53c.jpg"
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            "content": {
                "title_content": [
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                        "type": "text",
                        "content": "定理 2.18 (一般分块矩阵的行列式)"
                    }
                ],
                "level": 2
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                    {
                        "type": "text",
                        "content": "设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, D"
                    },
                    {
                        "type": "text",
                        "content": " 分别为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r, s"
                    },
                    {
                        "type": "text",
                        "content": " 阶矩阵， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, D"
                    },
                    {
                        "type": "text",
                        "content": " 可逆，则"
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                ]
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            "content": {
                "math_content": "\\left| \\begin{array}{c c} A & B \\\\ C & D \\end{array} \\right| = | D | | A - B D ^ {- 1} C | = | A | | D - C A ^ {- 1} B |",
                "math_type": "latex",
                "image_source": {
                    "path": "images/ad2f6b3e73b37a001f263883aa1f6b3df7030d6f4d98beaf91c2af158c675b22.jpg"
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                        "type": "text",
                        "content": "证明 (1) 如果  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D"
                    },
                    {
                        "type": "text",
                        "content": " 可逆，则把  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B"
                    },
                    {
                        "type": "text",
                        "content": " 消掉 (左乘  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "BD^{-1}"
                    },
                    {
                        "type": "text",
                        "content": ")，取行列式即可"
                    }
                ]
            },
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        {
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            "content": {
                "math_content": "\\left[ \\begin{array}{c c} A & B \\\\ C & D \\end{array} \\right] \\Rightarrow \\left[ \\begin{array}{c c} A - B D ^ {- 1} C & 0 \\\\ C & D \\end{array} \\right]",
                "math_type": "latex",
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                    "path": "images/b9e60f682b78bf3a222b28cb0eb1f0a1368bea35865da353df0a45963169d5ca.jpg"
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            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.19 (行列式计算技巧 1: 配凑分块矩阵)"
                    }
                ],
                "level": 2
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                        "type": "text",
                        "content": "矩阵 A 可分解为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = J - u^{T}u"
                    },
                    {
                        "type": "text",
                        "content": " ，要计算   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A|"
                    },
                    {
                        "type": "text",
                        "content": " ，则可以构造分块矩阵："
                    }
                ]
            },
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            "content": {
                "math_content": "\\left| \\begin{array}{c c} J & u \\\\ u ^ {T} & 1 \\end{array} \\right| = | 1 | \\cdot | J - u ^ {T} u | = | J | \\cdot | 1 - u J ^ {- 1} u ^ {T} | \\Rightarrow | A | = | J | \\cdot | 1 - u J ^ {- 1} u ^ {T} |",
                "math_type": "latex",
                "image_source": {
                    "path": "images/84a745e25997074861fcd994ebda2ce9c69fb8be88c2bd6c68c114880eb36553.jpg"
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                        "content": "注意这里  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u"
                    },
                    {
                        "type": "text",
                        "content": " 是列向量"
                    }
                ]
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                        "type": "text",
                        "content": "2.2.5 分块矩阵求逆"
                    }
                ],
                "level": 2
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "一般分块矩阵求逆都先转换为分块对角矩阵，具体如下。"
                    }
                ]
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            "type": "title",
            "content": {
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                        "type": "text",
                        "content": "定理 2.20 (上三角分块矩阵求逆)"
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "当下面的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}, A_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 都是方阵，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 可逆当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}, A_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 都可逆，且逆矩阵如下："
                    }
                ]
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        {
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            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c} A _ {1} & A _ {3} \\\\ O & A _ {2} \\end{array} \\right] \\Rightarrow A ^ {- 1} = \\left[ \\begin{array}{c c} A _ {1} ^ {- 1} & - A _ {1} ^ {- 1} A _ {3} A _ {2} ^ {- 1} \\\\ 0 & A _ {2} ^ {- 1} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/e09e0790ec5a58301224af5aab7cfa6ec41334f8a9c105c63c213dc2923c718d.jpg"
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            },
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        {
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            "content": {
                "paragraph_content": [
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                        "type": "text",
                        "content": "证明 根据分块矩阵初等变换可知"
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c} E & - A _ {2} ^ {- 1} A _ {3} \\\\ O & E \\end{array} \\right] \\left[ \\begin{array}{c c} A _ {1} & A _ {3} \\\\ O & A _ {2} \\end{array} \\right] = \\left[ \\begin{array}{c c} A _ {1} & O \\\\ O & A _ {2} \\end{array} \\right] \\Rightarrow \\left[ \\begin{array}{c c} E & - A _ {2} ^ {- 1} A _ {3} \\\\ O & E \\end{array} \\right] ^ {- 1} \\left[ \\begin{array}{c c} A _ {1} & A _ {3} \\\\ O & A _ {2} \\end{array} \\right] ^ {- 1} = \\left[ \\begin{array}{c c} A _ {1} ^ {- 1} & O \\\\ O & A _ {2} ^ {- 1} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/68a8bb2c47f79499d65377001e7dffc9187f176ce2a04e95cb5da5eb181386f9.jpg"
                }
            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此可以得到结论。"
                    }
                ]
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                        "content": "2.2 分块矩阵"
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    ],
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            "content": {
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                        "type": "text",
                        "content": "定理 2.21 (一般分块矩阵可逆条件)"
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                ],
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                        "content": "分块矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "H = \\left( \\begin{array}{ll}A & B\\\\ C & D \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " 可逆当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A - BD^{-1}C"
                    },
                    {
                        "type": "text",
                        "content": " 和  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D - CA^{-1}B"
                    },
                    {
                        "type": "text",
                        "content": " 都可逆"
                    }
                ]
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            "content": {
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                        "type": "text",
                        "content": "练习2.5分块矩阵求逆  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 时  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n, m"
                    },
                    {
                        "type": "text",
                        "content": " 阶可逆方阵，求  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\left[ \\begin{array}{cc}O & A\\\\ B & O \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": " 的逆矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解 先做交换行的变换:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\left[ \\begin{array}{ll} O & A \\\\ B & O \\end{array} \\right]\\left[ \\begin{array}{ll} O & E \\\\ E & O \\end{array} \\right] = \\left[ \\begin{array}{l l} A & \\\\ & B \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": ", 两侧求逆即可得到:"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c} O & A \\\\ B & O \\end{array} \\right] ^ {- 1} = \\left[ \\begin{array}{c c} A ^ {- 1} & O \\\\ O & B ^ {- 1} \\end{array} \\right] \\left[ \\begin{array}{c c} O & E \\\\ E & O \\end{array} \\right] = \\left[ \\begin{array}{c c} O & B ^ {- 1} \\\\ A ^ {- 1} & O \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/cad423abf52dbd17a425ea4e3413778c994d102f24698f7f089c93b6ec11e2ab.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.3 矩阵的秩"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.3.1 秩的概念"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 2.9 (矩阵的秩)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "矩阵的秩等于:"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 矩阵行列向量的秩"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 最大非零子式的阶数"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.3.2 初等变换标准型与矩阵分解"
                    }
                ],
                "level": 2
            },
            "bbox": [
                94,
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                394,
                521
            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.22 (初等变换标准型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是秩为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r"
                    },
                    {
                        "type": "text",
                        "content": " 的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "s \\times n"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵，则存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "s"
                    },
                    {
                        "type": "text",
                        "content": " 阶可逆矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P"
                    },
                    {
                        "type": "text",
                        "content": " 和  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶可逆矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q"
                    },
                    {
                        "type": "text",
                        "content": " 使得"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = P \\left[ \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right] Q",
                "math_type": "latex",
                "image_source": {
                    "path": "images/dc25ae12ce18560cf138f2aaba6567bac90693caad314d7a1eef1e0a0957e8f6.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 如果一个题目中没给出条件，或者只给出了矩阵的秩，则往往需要用到初等变换标准型"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.23 (行列满秩的标准型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m\\times n"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 列满秩当且仅当存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m"
                    },
                    {
                        "type": "text",
                        "content": " 阶可逆矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = P\\left[ \\begin{array}{c}E_{n}\\\\ O \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": " 。  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 行满秩当且仅当存在 "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶可逆矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q"
                    },
                    {
                        "type": "text",
                        "content": " ，使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\left[ \\begin{array}{ll}E_m & O \\end{array} \\right]Q"
                    }
                ]
            },
            "bbox": [
                112,
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                885,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 由于列满秩，可以用行初等变换消成阶梯型，再从下到上消回去即可 (若列不满秩，则阶梯型可能消不干净，因此需要列变换)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.3 矩阵的秩"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "21"
                    }
                ]
            },
            "bbox": [
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                965
            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.3.3 利用初等变换标准型进行矩阵分解"
                    }
                ],
                "level": 2
            },
            "bbox": [
                92,
                85,
                453,
                102
            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.24 (矩阵分解)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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                278,
                133
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "s\\times n"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\left[ \\begin{array}{cc}E_r & O\\\\ O & O \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": " 可分解为"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right] = \\left[ \\begin{array}{c} E _ {r} \\\\ O \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {r} & O \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/16a83b77dac8afd4c55f9666913b955ccc672abcf408de08414bd92d71a35c74.jpg"
                }
            },
            "bbox": [
                371,
                180,
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                221
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若为方阵时，可以分解为："
                    }
                ]
            },
            "bbox": [
                115,
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                237
            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "{\\left[ \\begin{array}{l l} E _ {r} & O \\\\ O & O \\end{array} \\right]} = {\\left[ \\begin{array}{l l} E _ {r} & O \\\\ O & O \\end{array} \\right]} {\\left[ \\begin{array}{l l} E _ {r} & O \\\\ O & O \\end{array} \\right]}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/5884b522f8c0bb0f12ae67b81d7c1ea46e69eaf0b9e4ec04171baabe95d179d1.jpg"
                }
            },
            "bbox": [
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                235,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 2.6 行列满秩分解 (1) 证明：任意非零矩阵都可以分解为列满秩和行满秩矩阵的乘积"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B_{1}, B_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 为数域  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P"
                    },
                    {
                        "type": "text",
                        "content": " 上  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "s \\times n"
                    },
                    {
                        "type": "text",
                        "content": " 列满秩矩阵，证明：存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P"
                    },
                    {
                        "type": "text",
                        "content": " 上  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "s"
                    },
                    {
                        "type": "text",
                        "content": " 阶可逆矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C"
                    },
                    {
                        "type": "text",
                        "content": "，使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B_{2} = CB_{1}"
                    },
                    {
                        "type": "text",
                        "content": " (即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B_{1}, B_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 只需要行变换就可以相互转换)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C_{1}, C_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 为P上  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "s \\times n"
                    },
                    {
                        "type": "text",
                        "content": " 行满秩矩阵，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C_{1}, C_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 可以仅通过列变换相互转换。"
                    }
                ]
            },
            "bbox": [
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                367
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 使用等价标准型:"
                    }
                ]
            },
            "bbox": [
                90,
                370,
                287,
                385
            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = P \\left[ \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right] Q = P \\left[ \\begin{array}{c} E _ {r} \\\\ O \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {r} & O \\end{array} \\right] Q",
                "math_type": "latex",
                "image_source": {
                    "path": "images/43f6c8f66bd82a48a51ec705cf3ebf38644aba00e4853d8287b97564128aff38.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 根据列满秩初等标准型， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists P"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "PB_{1} = \\left[ \\begin{array}{c} E_{n} \\\\ O \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": "，同理存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "QB_{2} = \\left[ \\begin{array}{c} E_{n} \\\\ O \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": "，结论成立。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 和 (2) 一样"
                    }
                ]
            },
            "bbox": [
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                231,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 2.7 幂等矩阵分解 (1) 证明：任意方阵可以分解为可逆矩阵与幂等矩阵乘积"
                    }
                ]
            },
            "bbox": [
                58,
                505,
                677,
                520
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 利用等价标准型得到:"
                    }
                ]
            },
            "bbox": [
                90,
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                319,
                538
            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = P \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) Q = P Q Q ^ {- 1} \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) Q",
                "math_type": "latex",
                "image_source": {
                    "path": "images/6739ab1c8cc174ef77a7fd3baf6da82fcddc5e9213d95cd4f181edbbfc09d4ff.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "PQ 为可逆矩阵，后面为幂等"
                    }
                ]
            },
            "bbox": [
                90,
                583,
                310,
                599
            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.3.4 秩不等式"
                    }
                ],
                "level": 2
            },
            "bbox": [
                94,
                621,
                233,
                638
            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理 2.1 (乘可逆矩阵不改变秩)"
                    }
                ],
                "level": 2
            },
            "bbox": [
                122,
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                354,
                668
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "P, Q"
                    },
                    {
                        "type": "text",
                        "content": " 为可逆矩阵，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = r(PAQ)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 相当于做了一系列初等变换，因此不改变秩"
                    }
                ]
            },
            "bbox": [
                90,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 2.8 非方阵情况 A, B 分别为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m \\times n, n \\times s"
                    },
                    {
                        "type": "text",
                        "content": "  矩阵，证明："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(1) 若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = n"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(AB) = r(B)"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(B) = n"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(AB) = r(A)"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 存在可逆矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "PA = \\left[ \\begin{array}{c} A_{1} \\\\ O \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": "，这里  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 为可逆矩阵，因此"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "r (A B) = r (\\left[ \\begin{array}{c} A _ {1} \\\\ O \\end{array} \\right] B) = r \\binom{A _ {1} B}{O} = r (A _ {1} B) = r (B)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/9eac1271f0c15bf15f28e9948d57aae66a414c65b99532a06e59e59c96ecb214.jpg"
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 同理 (转置即可)"
                    }
                ]
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.3 矩阵的秩"
                    }
                ]
            },
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        },
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            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "22"
                    }
                ]
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        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理 2.2 (分块矩阵的秩)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\begin{array}{l} \\bullet r \\left( \\begin{array}{c c} A & O \\\\ O & B \\end{array} \\right) = r (A) + r (B) \\\\ \\bullet r \\left( \\begin{array}{c c} A & O \\\\ C & B \\end{array} \\right) \\geq r (A) + r (B) \\end{array}",
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                "image_source": {
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        {
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            "content": {
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                    {
                        "type": "text",
                        "content": "证明 (1) 根据初等变换等价标准型有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P_{1}AQ_{1} = \\left( \\begin{array}{cc}E_{r_1} & O\\\\ O & O \\end{array} \\right),P_{2}BQ_{2} = \\left( \\begin{array}{cc}E_{r_2} & O\\\\ O & O \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": "，因此"
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left( \\begin{array}{c c} P _ {1} & O \\\\ O & P _ {2} \\end{array} \\right) \\left( \\begin{array}{c c} A & O \\\\ O & B \\end{array} \\right) \\left( \\begin{array}{c c} Q _ {1} & O \\\\ O & Q _ {2} \\end{array} \\right) = \\left( \\begin{array}{c c c c} E _ {r _ {1}} & O & O & O \\\\ O & O & O & O \\\\ O & O & E _ {r _ {2}} & O \\\\ O & O & O & O \\end{array} \\right)",
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 根据初等变换等价标准型有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P_{1}AQ_{1} = \\left( \\begin{array}{cc}E_{r_1} & O\\\\ O & O \\end{array} \\right), P_{2}BQ_{2} = \\left( \\begin{array}{cc}E_{r_2} & O\\\\ O & O \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": "，因此"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left( \\begin{array}{c c} P _ {1} & O \\\\ O & P _ {2} \\end{array} \\right) \\left( \\begin{array}{c c} A & O \\\\ C & B \\end{array} \\right) \\left( \\begin{array}{c c} Q _ {1} & O \\\\ O & Q _ {2} \\end{array} \\right) = \\left( \\begin{array}{c c c c} E _ {r _ {1}} & O & O & O \\\\ O & O & O & O \\\\ C _ {1 1} & C _ {1 2} & E _ {r _ {2}} & O \\\\ C _ {2 1} & C _ {2 2} & O & O \\end{array} \\right) \\to \\left( \\begin{array}{c c c c} E _ {r _ {1}} & O & O & O \\\\ O & O & O & O \\\\ O & O & E _ {r _ {2}} & O \\\\ O & C _ {2 2} & O & O \\end{array} \\right)",
                "math_type": "latex",
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.25 (矩阵秩基本不等式)"
                    }
                ],
                "level": 2
            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "•   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(AB) \\leq \\min\\{r(A), r(B)\\}"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "•   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A+B) \\leq r(A) + r(B)"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- Sylvester 不等式:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵, 则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) + r(B) \\leq r(AB) + n"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- Frobenius 不等式:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B, C"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵, 则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(ABC) \\geq r(AB) + r(BC) - r(B)"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "s \\times n, n \\times m"
                    },
                    {
                        "type": "text",
                        "content": " 阶矩阵， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB = O"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) + r(B) \\neq n"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m \\times n"
                    },
                    {
                        "type": "text",
                        "content": " 的实矩阵，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A^T A) = r(A)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B"
                    },
                    {
                        "type": "text",
                        "content": " 写为行向量形式:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B = \\left( \\begin{array}{c} \\beta_{1} \\\\ \\beta_{2} \\\\ \\vdots \\\\ \\beta_{n} \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": ", 则"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A B = \\left( \\begin{array}{c c c} a _ {1 1} & \\dots & a _ {1 n} \\\\ \\vdots & & \\vdots \\\\ a _ {n 1} & \\dots & a _ {n n} \\end{array} \\right) \\left( \\begin{array}{c} \\beta_ {1} \\\\ \\vdots \\\\ \\beta_ {n} \\end{array} \\right) = \\left( \\begin{array}{c} a _ {1 1} \\beta_ {1} + \\dots + a _ {1 n} \\beta_ {n} \\\\ \\vdots \\\\ a _ {n 1} \\beta_ {1} + \\dots + a _ {n n} \\beta_ {n} \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
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        },
        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此 AB 每个行向量都可被 B 行向量线性表出，从而   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(AB) \\leq r(B)"
                    },
                    {
                        "type": "text",
                        "content": " ，同理可知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(AB) \\leq r(A)"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 用列向量角度，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A + B"
                    },
                    {
                        "type": "text",
                        "content": " 的列向量显然可以被  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 的列向量线性表出，从而结论成立。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 是 (4) 的推论"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(4)Frobenius: 即构造   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(ABC) + r(B) \\geq r(AB) + r(BC)"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c} A B C & \\\\ & B \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A B C & O \\\\ B C & B \\end{array} \\right] \\to \\left[ \\begin{array}{c c} O & - A B \\\\ B C & B \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
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            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "根据分块矩阵秩不等式可知"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 2.9 秩不等式的推广 (1) A, B 为 n 阶方阵，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A - ABA) = r(A) + r(E - BA) - n"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.3 矩阵的秩"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
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                        "type": "text",
                        "content": "23"
                    }
                ]
            },
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        }
    ],
    [
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            "type": "paragraph",
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB = BA = O, r(A^2) = r(A)"
                    },
                    {
                        "type": "text",
                        "content": "，证明： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A + B) = r(A) + r(B)"
                    },
                    {
                        "type": "text",
                        "content": " 证明 (1) 等号需要构造单位矩阵，即等式中的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 用  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "E_n"
                    },
                    {
                        "type": "text",
                        "content": " 代替："
                    }
                ]
            },
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        },
        {
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            "content": {
                "math_content": "\\left[ \\begin{array}{c c} A & \\\\ & E _ {n} - B A \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A & O \\\\ B A & E _ {n} - B A \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A & A \\\\ B A & E _ {n} \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A - A B A & O \\\\ O & E _ {n} \\end{array} \\right]",
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "从右往左推非常难，因此一般从无  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "E_{n}"
                    },
                    {
                        "type": "text",
                        "content": " 的一边推另一边(后面的练习也会体现这一点)"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 注意从复杂的一侧  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A + B"
                    },
                    {
                        "type": "text",
                        "content": " 开始，否则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 不知道怎么搞，"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left( \\begin{array}{c c} A + B & O \\\\ O & O \\end{array} \\right) \\to \\left( \\begin{array}{c c} A + B & O \\\\ A ^ {2} + A B & O \\end{array} \\right) = \\left( \\begin{array}{c c} A + B & O \\\\ A ^ {2} & O \\end{array} \\right) \\to \\left( \\begin{array}{c c} A + B & A ^ {2} \\\\ A ^ {2} & A ^ {3} \\end{array} \\right)",
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = r(A^2)"
                    },
                    {
                        "type": "text",
                        "content": "，因此矩阵方程  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^2 X = A"
                    },
                    {
                        "type": "text",
                        "content": " 有解，故  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists P"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^2 P = A"
                    },
                    {
                        "type": "text",
                        "content": "，故第二列可右乘  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P"
                    },
                    {
                        "type": "text",
                        "content": " 消去第一列："
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left( \\begin{array}{c c} A + B & A ^ {2} \\\\ A ^ {2} & A ^ {3} \\end{array} \\right) \\to \\left( \\begin{array}{c c} B & A ^ {2} \\\\ O & A ^ {3} \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/a6dd5f2639e6dbeb01291b300d27cb65394d5c413612cca973921aaad593cbf2.jpg"
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此根据分块矩阵的秩得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A + B) \\geq r(A^3) + r(B) = r(A) + r(B)"
                    },
                    {
                        "type": "text",
                        "content": "，而不等式另一侧显然."
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 2.10 用秩不等式研究矩阵性质 (1) 证明零矩阵:   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{m \\times n}, B_{n \\times s}, r(B) = n"
                    },
                    {
                        "type": "text",
                        "content": " ，证明：若 AB = O，则 A = O"
                    }
                ]
            },
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 判定可逆:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 为奇数,  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶矩阵,  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^2 = O"
                    },
                    {
                        "type": "text",
                        "content": ", 证明:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB - BA"
                    },
                    {
                        "type": "text",
                        "content": " 不可逆证明 (1) 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) + r(B) \\leq r(AB) + n"
                    },
                    {
                        "type": "text",
                        "content": ", 而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(B) = n"
                    },
                    {
                        "type": "text",
                        "content": ", 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = 0"
                    },
                    {
                        "type": "text",
                        "content": ", 即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = O"
                    }
                ]
            },
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n=2k+1"
                    },
                    {
                        "type": "text",
                        "content": " ，根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{2}=O"
                    },
                    {
                        "type": "text",
                        "content": " 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "2r(A)\\leq2k+1+0"
                    },
                    {
                        "type": "text",
                        "content": " ，从而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A)\\leq k"
                    },
                    {
                        "type": "text",
                        "content": " ，此时"
                    }
                ]
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "r (A B - B A) \\leq r (A B) + r (B A) \\leq r (A) + r (A) \\leq 2 k",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1d63973af282db4877a1a1cd51bb4937c622975f9c03fade1d8048bbda929693.jpg"
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此不可逆"
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.3.5 空间分解法研究矩阵秩"
                    }
                ],
                "level": 2
            },
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            "content": {
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                        "type": "text",
                        "content": "定理 2.26 (多项式矩阵的秩关系)"
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            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x) = f_{1}(x)f_{2}(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是多项式，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f_{1}(x),f_{2}(x)) = 1"
                    },
                    {
                        "type": "text",
                        "content": " ，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(A) = O"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(f_{1}(A)) + r(f_{2}(A)) = n"
                    },
                    {
                        "type": "text",
                        "content": " 。常见的推论如下："
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^2 = A"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A - E) + r(A) = n"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^2 = E"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A - E) + r(A + E) = n"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f_{1}(x), f_{2}(x)) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x)f_{1}(x) + v(x)f_{2}(x) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(A)f_{1}(A) + v(A)f_{2}(A) = E_{n}"
                    },
                    {
                        "type": "text",
                        "content": "，根据分块矩阵初等变换："
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c} f _ {1} (A) & \\\\ & f _ {2} (A) \\end{array} \\right] \\to \\left[ \\begin{array}{c c} f _ {1} (A) & f _ {1} (A) u (A) + v (A) f _ {2} (A) \\\\ O & f _ {2} (A) \\end{array} \\right] = \\left[ \\begin{array}{c c} f _ {1} (A) & E _ {n} \\\\ O & f _ {2} (A) \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1d699b251deaa6c11d09a40177bc177e23dec763c26736cd8a206d64e0050d5c.jpg"
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "再做初等变换："
                    }
                ]
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c} f _ {1} (A) & E _ {n} \\\\ O & f _ {2} (A) \\end{array} \\right] \\to \\left[ \\begin{array}{c c} O & E _ {n} \\\\ - f _ {2} (A) f _ {1} (A) & f _ {2} (A) \\end{array} \\right] \\to \\left[ \\begin{array}{c c} O & E _ {n} \\\\ - f (A) & O \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/9d9d039fc16d93d2c2b0ae7809e1d051940960448ccbcdd225e60e4ea0c6381d.jpg"
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            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(f_{1}(A))+r(f_{2}(A))=n+r(f(A))"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 分块矩阵、多项式互素、空间直和本质上是一体的。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 2.11 从线性空间角度解决难题 A, B 是 P 上 n 阶矩阵，AB = BA，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) + r(B) \\geq r(AB) + r(A + B)"
                    },
                    {
                        "type": "text",
                        "content": "  证明 设 AX = 0, BX = 0 的解空间分别为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}, V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " ，设 ABX = BAX = 0 的解空间为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W_{1}"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A + B)X = 0"
                    },
                    {
                        "type": "text",
                        "content": "  的解空间为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 。显然   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\subseteq W_{1}, V_{2} \\subseteq W_{1}"
                    },
                    {
                        "type": "text",
                        "content": " ，从而   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} + V_{2} \\subseteq W_{1}"
                    },
                    {
                        "type": "text",
                        "content": " ，显然   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\cap V_{2} \\subseteq W_{2}"
                    },
                    {
                        "type": "text",
                        "content": " ，根据维数公式："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "d i m V _ {1} + d i m V _ {2} = d i m (V _ {1} + V _ {2}) + d i m (V _ {1} \\cap V _ {2}) \\leq d i m W _ {1} + d i m W _ {2}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/bf3ce2b16774a4cf189116a2fe8f9fb977364eca39a92e59e048a3a268eabe36.jpg"
                }
            },
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        },
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            "type": "page_number",
            "content": {
                "page_number_content": [
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                        "type": "text",
                        "content": "24"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.3 矩阵的秩"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此推出："
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(n - r (A)) + (n - r (B)) \\leq (n - r (A B)) + (n - r (A + B))",
                "math_type": "latex",
                "image_source": {
                    "path": "images/b3672f5e69bd8ec259182128fd88749c240a1c4db02b51bbc2b0e999e60a6075.jpg"
                }
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "化简得到结论。"
                    }
                ]
            },
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        },
        {
            "type": "image",
            "content": {
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                "image_caption": [],
                "image_footnote": []
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 这种  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A), r(B), r(A + B), r(AB)"
                    },
                    {
                        "type": "text",
                        "content": " 相关问题绝大多数从线性空间角度都能迎刃而解"
                    }
                ]
            },
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.4 矩阵的迹"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.27 (矩阵迹的性质)"
                    }
                ],
                "level": 2
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵，则"
                    }
                ]
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\begin{array}{l} \\bullet \\operatorname{tr} (A) = \\operatorname{tr} (k A) \\\\ \\bullet \\operatorname{tr} (A + B) = \\operatorname{tr} (A) + \\operatorname{tr} (B) \\\\ \\bullet \\operatorname{tr} (A B) = \\operatorname{tr} (B A) \\end{array}",
                "math_type": "latex",
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                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为实方阵，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = O"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\operatorname{tr}(A^T A) = 0"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 显然"
                    }
                ]
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.5 几种技巧矩阵"
                    }
                ],
                "level": 2
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        {
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                        "type": "text",
                        "content": "2.5.1 初等矩阵"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 2.10 (初等矩阵)"
                    }
                ],
                "level": 2
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            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "单位矩阵通过一次初等变换得到的矩阵称为初等矩阵。记作  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P(i,j), P(i(c)), P(i,j(k))"
                    },
                    {
                        "type": "text",
                        "content": " 表示  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i,j"
                    },
                    {
                        "type": "text",
                        "content": " 行互换， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i"
                    },
                    {
                        "type": "text",
                        "content": " 行乘  "
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                    {
                        "type": "equation_inline",
                        "content": "c"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P_{ij} = k"
                    },
                    {
                        "type": "text",
                        "content": " 的矩阵"
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                ]
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.28 (初等矩阵乘法)"
                    }
                ],
                "level": 2
            },
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "AP"
                    },
                    {
                        "type": "text",
                        "content": " 相当于做一次列初等变换，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "PA"
                    },
                    {
                        "type": "text",
                        "content": " 相当于做一次行初等变换，"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.5.2 基本矩阵：仅一个元素为1的矩阵"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 2.11 (基本矩阵)"
                    }
                ],
                "level": 2
            },
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        {
            "type": "paragraph",
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "基本矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "E_{ij}"
                    },
                    {
                        "type": "text",
                        "content": " 表示  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n \\times n"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵，但是仅有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(i,j)"
                    },
                    {
                        "type": "text",
                        "content": " 元素为1，其余元素为0的矩阵。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.29 (基本矩阵乘法)"
                    }
                ],
                "level": 2
            },
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "基本矩阵乘法有以下性质："
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AE_{ij}"
                    },
                    {
                        "type": "text",
                        "content": " 将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的第  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i"
                    },
                    {
                        "type": "text",
                        "content": " 列移动到第  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "j"
                    },
                    {
                        "type": "text",
                        "content": " 列，"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "E_{ij}A"
                    },
                    {
                        "type": "text",
                        "content": " 将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的第  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "j"
                    },
                    {
                        "type": "text",
                        "content": " 行移动到第  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i"
                    },
                    {
                        "type": "text",
                        "content": " 行。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 基础矩阵分解：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "E_{ij} = e_i e_j^T"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 基本矩阵相乘：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "E_{ik}E_{kj}=E_{ij}"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k\\neq l"
                    },
                    {
                        "type": "text",
                        "content": " 时  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "E_{ik}E_{lj}=0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.4 矩阵的迹"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "25"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习2.12用基础矩阵研究幂次(1)设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\left( \\begin{array}{cccc}0 & 1 & 0 & \\dots & 0\\\\ 0 & 0 & 1 & \\dots & 0\\\\ \\vdots & \\vdots & \\vdots & & \\vdots \\\\ 0 & 0 & 0 & \\dots & 1\\\\ 1 & 0 & 0 & \\dots & 0 \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " ，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^k = \\left( \\begin{array}{cc}O & E_{n - k}\\\\ E_k & O \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶上三角阵且主对角上元素均为0，证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^n = O"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 写为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = (e_{n}, e_{1}, \\cdots, e_{n-1})"
                    },
                    {
                        "type": "text",
                        "content": " ，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{2} = (A e_{n}, A e_{1}, \\cdots, A e_{n-1})"
                    },
                    {
                        "type": "text",
                        "content": " 。显然   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Ae_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  为 A 的第 i 列，因此"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A ^ {2} = (e _ {n - 1}, e _ {n}, e _ {1} \\dots , e _ {n - 2})",
                "math_type": "latex",
                "image_source": {
                    "path": "images/31cbce50d38115ee00ed323ae5aa1c8e8da87c7fee2cd2b809d6aa8492cbff59.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "以此类推容易得到结论"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A=\\sum_{i<j}a_{ij}E_{ij}"
                    },
                    {
                        "type": "text",
                        "content": " ，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "j\\neq k"
                    },
                    {
                        "type": "text",
                        "content": " 时  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "E_{ij}E_{kl}=O"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{n}"
                    },
                    {
                        "type": "text",
                        "content": " 展开式中非零项只能是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "E_{ij_{1}}E_{j_{1}j_{2}}\\cdots E_{j_{n-1}j_{n}}"
                    },
                    {
                        "type": "text",
                        "content": " 这样的项，但满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1\\leq i<j_{1}<\\cdots<j_{n}\\leq n"
                    },
                    {
                        "type": "text",
                        "content": " 的项不存在，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{n}=O"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习2.13用二次型研究矩阵性质(1)证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶对称矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为零矩阵当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in \\mathbb{R}^n"
                    },
                    {
                        "type": "text",
                        "content": " 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha^T A\\alpha = 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是反对称矩阵当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in \\mathbb{R}^n"
                    },
                    {
                        "type": "text",
                        "content": " 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha^T A\\alpha = 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 左推右显然。先设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = e_i"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "e_i^T Ae_i = a_{ii} = 0"
                    },
                    {
                        "type": "text",
                        "content": "，再取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = e_i + e_j"
                    },
                    {
                        "type": "text",
                        "content": "，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "0 = (e _ {i} + e _ {j}) ^ {T} A (e _ {i} + e _ {j}) = a _ {i i} + a _ {j j} + a _ {i j} + a _ {j i}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/7441b6264b062dcfea3e5b321d507d7d073fc877f3ba8c290ace665e25f0858c.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{ij} = a_{ji}"
                    },
                    {
                        "type": "text",
                        "content": "，因此得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{ij} = 0, \\forall i, j"
                    },
                    {
                        "type": "text",
                        "content": " (2) 左推右用  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha^T A\\alpha = (\\alpha^T A\\alpha)^T = -\\alpha^T A\\alpha"
                    },
                    {
                        "type": "text",
                        "content": " 即可。右推左：根据条件得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha^T A^T\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha^T (A + A^T)\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": "，由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A + A^T"
                    },
                    {
                        "type": "text",
                        "content": " 为对称矩阵，根据上题结论得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A + A^T = O"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T = -A"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.5.3 置换矩阵"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 2.12 (置换矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "每行只有一个元素为 1，每列也只有一个元素为 1，则称该矩阵为置换矩阵。一般写为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P = [\\epsilon_{i_{1}}, \\cdots, \\epsilon_{i_{n}}] = [\\epsilon_{j_{1}}^{T}, \\cdots, \\epsilon_{j_{n}}^{T}]^{T}"
                    },
                    {
                        "type": "text",
                        "content": " ，其中   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  表示仅第 i 元素为 1 的列向量，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i_{k}, j_{k}"
                    },
                    {
                        "type": "text",
                        "content": "  分别表示列、行的重排顺序。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.30 (置换矩阵是正交矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "置换矩阵是正交矩阵。"
                    }
                ]
            },
            "bbox": [
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                278,
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "证明 证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^T P = I"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.31 (置换矩阵的作用)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "左乘置换矩阵相当于把行原本的   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "j_{1}, j_{2}, \\cdots, j_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  行换到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1, 2, \\cdots, n"
                    },
                    {
                        "type": "text",
                        "content": "  行，右乘相当于把列原本的   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i_{1}, \\cdots, i_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  列换到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1, 2, \\cdots, n"
                    },
                    {
                        "type": "text",
                        "content": "  列。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.32 (置换矩阵正交相似)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "i_{1}\\cdots i_{n}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1\\sim n"
                    },
                    {
                        "type": "text",
                        "content": " 的排列，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A=(a_{ij})"
                    },
                    {
                        "type": "text",
                        "content": " ，取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P=[\\epsilon_{i_{1}},\\cdots,\\epsilon_{i_{n}}]"
                    },
                    {
                        "type": "text",
                        "content": " ，则："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "P ^ {T} A P = P ^ {- 1} A P = \\left[ \\begin{array}{c c c c} a _ {i _ {1} i _ {1}} & a _ {i _ {1} i _ {2}} & \\dots & a _ {i _ {1} i _ {n}} \\\\ a _ {i _ {2} i _ {1}} & a _ {i _ {2} i _ {2}} & \\dots & a _ {i _ {2} i _ {n}} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a _ {i _ {n} i _ {1}} & a _ {i _ {n} i _ {2}} & \\dots & a _ {i _ {n} i _ {n}} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/2039e242ee35df159602ce6d8607ed71c713524e36f2f5d4cb5485122dac104e.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "相当于将行列均按新顺序排列了一遍(先把行放到对应位置，再把列放到对应位置)，且是一种正交变换。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.5 几种技巧矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "26"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 2.14 置换矩阵正交相似例子   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i_{1}\\cdots i_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1,2,\\cdots,n"
                    },
                    {
                        "type": "text",
                        "content": "  的排列，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A=\\mathrm{diag}\\{\\lambda_{1},\\cdots,\\lambda_{n}\\},B=\\mathrm{diag}\\{\\lambda_{i_{1}},\\cdots,\\lambda_{i_{n}}\\}"
                    },
                    {
                        "type": "text",
                        "content": " ，证明 A,B 是正交相似的"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 取   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P = [\\epsilon_{i_{1}}, \\cdots, \\epsilon_{i_{n}}]"
                    },
                    {
                        "type": "text",
                        "content": "  即可。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 2.13 (特殊的置换矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "特殊置换矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "K = \\left[ \\begin{array}{cccc}  &  &  & 1 \\\\  &  & 1 &  \\\\  & \\ddots &  &  \\\\ 1 &  &  &  \\\\ \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "K^T AK"
                    },
                    {
                        "type": "text",
                        "content": " 的功能是绕中心转180度(行列全部倒排)"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.6 特殊矩阵性质总结"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.6.1 对角矩阵"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.33 (对角矩阵乘法)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 右乘一个对角阵，则每列乘上对角相应元素。 "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 左乘对角阵，则每行乘上对角相应元素。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定理 2.34 (对角矩阵求逆)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\operatorname{diag} \\left\\{d _ {1}, \\dots , d _ {n} \\right\\} ^ {- 1} = \\operatorname{diag} \\left\\{d _ {1} ^ {- 1}, \\dots , d _ {n} ^ {- 1} \\right\\}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/015ecd01ccc2ca94070189ae3c471a4821c14c969860c3c78bb3becab1e2960c.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.6.2 上、下三角矩阵"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.35 (上下三角乘法、逆、伴随)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 是上三角矩阵，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB"
                    },
                    {
                        "type": "text",
                        "content": " 仍然是上三角矩阵，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB"
                    },
                    {
                        "type": "text",
                        "content": " 对角元为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{11}b_{11},\\dots ,a_{nn}b_{nn}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 也是上三角阵，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 对角元为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{11}^{-1}, \\cdots, a_{nn}^{-1}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{*}"
                    },
                    {
                        "type": "text",
                        "content": " 也是上三角阵，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{*}"
                    },
                    {
                        "type": "text",
                        "content": " 的对角元为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{22}a_{33}\\cdots a_{nn}, a_{11}a_{33}\\cdots a_{nn}, \\cdots, a_{11}a_{22}\\cdots a_{n-1,n-1}"
                    },
                    {
                        "type": "text",
                        "content": " (也是特征值)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 令  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C = AB"
                    },
                    {
                        "type": "text",
                        "content": "，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "c_{ij} = \\sum_{k=1}^{n} a_{ik} b_{kj} = \\sum_{k=1}^{i-1} a_{ik} b_{kj} + \\sum_{k=i}^{n} a_{ik} b_{kj}"
                    },
                    {
                        "type": "text",
                        "content": "，当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i > j"
                    },
                    {
                        "type": "text",
                        "content": " 时，显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "c_{ij} = 0"
                    },
                    {
                        "type": "text",
                        "content": "。而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "c_{ii} = \\sum_{k=1}^{i-1} a_{ik} b_{ki} + a_{ii} b_{ii} + \\sum_{k=i+1}^{n} a_{ik} b_{ki} = a_{ii} b_{ii}"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 根据行变换可以将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A, I) \\to (I, A^{-1})"
                    },
                    {
                        "type": "text",
                        "content": "，当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为上三角时，先用最后一行消去  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 最后一列其余行的元素，以此类推可以发现  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 为上三角，根据矩阵乘法显然有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 对角为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 对角的逆。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{*} = |A|A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 即可"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.36 (n 个上三角矩阵的乘积)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A_{1},\\cdots,A_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  时 n 个对角为 0 的 n 阶上三角矩阵，证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}A_{2}\\cdots A_{n}=O"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 用数学归纳法，设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n - 1"
                    },
                    {
                        "type": "text",
                        "content": " 时成立，取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{n} = \\left[ \\begin{array}{cc}B_{n - 1} & *\\\\ 0 & 0 \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": " ，此时"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A _ {1} \\dots A _ {n} = \\left[ \\begin{array}{c c} B _ {1} \\dots B _ {n - 1} & * \\\\ 0 & 0 \\end{array} \\right] \\cdot \\left[ \\begin{array}{c c} B _ {n} & * \\\\ 0 & 0 \\end{array} \\right] = \\left[ \\begin{array}{c c} 0 & * \\\\ 0 & 0 \\end{array} \\right] \\left[ \\begin{array}{c c} B _ {n} & * \\\\ 0 & 0 \\end{array} \\right] = O",
                "math_type": "latex",
                "image_source": {
                    "path": "images/4b9e84f5be933b23cfd104d6de6295324f70c20ac352128f2aa6f889b6a31ed4.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.6 特殊矩阵性质总结"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "27"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.6.3 实对称矩阵"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 2.14 (实对称矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "每个元素都是实数，且满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T = A"
                    },
                    {
                        "type": "text",
                        "content": " 的矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 称为实对称矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.37 (实对称矩阵的矩阵性质)"
                    }
                ],
                "level": 2
            },
            "bbox": [
                122,
                179,
                379,
                193
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "实对称矩阵 A 具有以下性质："
                    }
                ]
            },
            "bbox": [
                115,
                199,
                332,
                212
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 也是实对称矩阵"
                    }
                ]
            },
            "bbox": [
                141,
                217,
                315,
                231
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(AA^{-1})^T = E_n"
                    },
                    {
                        "type": "text",
                        "content": "，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A^{-1})^T A^T = E_n"
                    },
                    {
                        "type": "text",
                        "content": "，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 实对称，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A^{-1})^T A = E_n"
                    },
                    {
                        "type": "text",
                        "content": "，这说明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A^{-1})^T = A^{-1}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.38 (实对称矩阵的特征值性质)"
                    }
                ],
                "level": 2
            },
            "bbox": [
                122,
                277,
                396,
                292
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "实对称矩阵有如下性质："
                    }
                ]
            },
            "bbox": [
                115,
                297,
                295,
                311
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 每个特征值都是实数"
                    }
                ]
            },
            "bbox": [
                139,
                316,
                312,
                329
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 不同特征值的特征向量正交"
                    }
                ]
            },
            "bbox": [
                139,
                334,
                363,
                348
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 一定正交相似 (相似且合同) 于对角矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 共轭转置法：设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda \\in \\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 是特征值， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\xi \\in \\mathbb{C}^n"
                    },
                    {
                        "type": "text",
                        "content": " 是特征向量，一方面根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A\\xi = \\lambda\\xi"
                    },
                    {
                        "type": "text",
                        "content": " 两侧同时左乘  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\overline{\\xi}^T"
                    },
                    {
                        "type": "text",
                        "content": " 得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\overline {{\\xi}} ^ {T} A \\xi = \\lambda \\overline {{\\xi}} ^ {T} \\xi",
                "math_type": "latex",
                "image_source": {
                    "path": "images/b6c60f4dc6c798418f975b06b43ccdef6a009431723e1b2b6b0c731c5f05c412.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "另一方面直接对  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A\\xi = \\lambda \\xi"
                    },
                    {
                        "type": "text",
                        "content": " 取共轭转置，再同时右乘  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\xi"
                    },
                    {
                        "type": "text",
                        "content": " 得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\overline {{\\xi}} ^ {T} A = \\overline {{\\xi}} ^ {T} \\overline {{\\lambda}} \\Rightarrow \\overline {{\\xi}} ^ {T} A \\xi = \\overline {{\\lambda}} \\overline {{\\xi}} ^ {T} \\xi",
                "math_type": "latex",
                "image_source": {
                    "path": "images/4492c3a81de7f11f74ecd146ac78f0cea6a70872f65c440f20459bbc071f9e62.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此比较得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda = \\overline{\\lambda}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 不同特征值的特征向量正交：设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1}, \\lambda_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 是两个互异特征值， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\xi_{1}, \\xi_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 分别属于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1}, \\lambda_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 的实特征向量，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\lambda_ {1} (\\xi_ {1}, \\xi_ {2}) = (\\lambda_ {1} \\xi_ {1}) ^ {T} \\xi_ {2} = (A \\xi_ {1}) ^ {T} \\xi_ {2} = \\xi_ {1} ^ {T} (A \\xi_ {2}) = \\lambda_ {2} (\\xi_ {1}, \\xi_ {2})",
                "math_type": "latex",
                "image_source": {
                    "path": "images/63098a6f74d4931fa9f306ebc79efcaf439525bf249c053f97a83b8028a959cf.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "从而得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\lambda_{2} - \\lambda_{1})(\\xi_{1},\\xi_{2}) = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_1\\neq \\lambda_2"
                    },
                    {
                        "type": "text",
                        "content": " ，从而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\xi_1,\\xi_2) = 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 做数学归纳法，n=1 显然，假设 n-1 成立，任取特征值   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " ，对应单位特征向量   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}"
                    },
                    {
                        "type": "text",
                        "content": " ，将   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}"
                    },
                    {
                        "type": "text",
                        "content": "  扩充为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "R^{n}"
                    },
                    {
                        "type": "text",
                        "content": "  的一组标准正交基   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，令   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T=(\\alpha_{1},\\cdots,\\alpha_{n})"
                    },
                    {
                        "type": "text",
                        "content": " ，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A T = T \\left( \\begin{array}{c c} \\lambda & \\alpha^ {T} \\\\ 0 & A _ {n - 1} \\end{array} \\right) \\Rightarrow T ^ {T} A T = \\left( \\begin{array}{c c} \\lambda & \\alpha^ {T} \\\\ 0 & A _ {n - 1} \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/aba43288da80d53e4823b02a3a964ef6e4c7a063085dd094cdbe1aa3c72b309b.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T^T A T"
                    },
                    {
                        "type": "text",
                        "content": " 对称，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha^T = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，再根据归纳假设，存在正交矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P_{1}^{T}A_{n - 1}P_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 为对角阵，因此令  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P = \\mathrm{diag}(1,P_1)"
                    },
                    {
                        "type": "text",
                        "content": " ，此时"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "P ^ {T} T ^ {T} A T P = \\left( \\begin{array}{c c} \\lambda & 0 \\\\ 0 & P ^ {T} A _ {n - 1} P \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/7940771d2854e893b91183463c955a4461dd3904c5b5dec1da79b4a262a3794a.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "TP = Q"
                    },
                    {
                        "type": "text",
                        "content": " ，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q"
                    },
                    {
                        "type": "text",
                        "content": " 也是正交阵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.6.4 反对称矩阵"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.39 (反对称矩阵的行列式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "奇数阶反对称矩阵的行列式必为 0。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T = -A"
                    },
                    {
                        "type": "text",
                        "content": " 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A| = (-1)^n |A|"
                    },
                    {
                        "type": "text",
                        "content": "，当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 为奇数时  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A| = -|A|"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A| = 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "28"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.6 特殊矩阵性质总结"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.40 (反对称矩阵的矩阵性质)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为反对称矩阵，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 为反对称矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 类似实对称矩阵的"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.41 (反对称矩阵的二次型)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为反对称矩阵当且仅当任意  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X \\in \\mathbb{R}^n"
                    },
                    {
                        "type": "text",
                        "content": "，有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^T A X = 0"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "证明 在基础矩阵技巧一节中给出过证明"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.42 (反对称矩阵的特征值、特征向量)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "反对称矩阵 A 的特征值是 0 或者纯虚数。且纯虚数对应特征向量的实部和虚部的实向量等长、正交。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda = a + bi"
                    },
                    {
                        "type": "text",
                        "content": "，对应特征向量  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = u + vi"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A(u + vi) = (a + bi)(u + vi)"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Au + iAv = (au - bv) + (bu + av)i"
                    },
                    {
                        "type": "text",
                        "content": "，实虚部分离得到："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A u = a u - b v, A v = b u + a v \\Rightarrow u ^ {T} A u = a u ^ {T} u - b u ^ {T} v, v ^ {T} A v = b v ^ {T} u + a v ^ {T} v",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1dd0b95f580867a494405a09efd12ff86e3c1f024b0007647393545c2961561e.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u^T Au + v^T Av = a(|u|^2 + |v|^2)"
                    },
                    {
                        "type": "text",
                        "content": "，由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u^T Au \\in \\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 是一个数，因此"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "u ^ {T} A u = (u ^ {T} A u) ^ {T} = u ^ {T} A ^ {T} u = - u ^ {T} A u \\Rightarrow u ^ {T} A u = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/690928faf123318e59b24ae8978cf87f900d09d4a5b047468f992050f120daca.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "同理有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "v^{T}Av = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a(|u|^2 +|v|^2) = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u + vi\\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " ，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda = a + bi = bi"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha + \\beta i"
                    },
                    {
                        "type": "text",
                        "content": " 是特征值  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "ki"
                    },
                    {
                        "type": "text",
                        "content": " 的特征向量，即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A(\\alpha + i\\beta) = ki(\\alpha + i\\beta)"
                    },
                    {
                        "type": "text",
                        "content": "，对比实部虚部，得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A \\alpha = - k \\beta , A \\beta = k \\alpha",
                "math_type": "latex",
                "image_source": {
                    "path": "images/eeb17b6ba2f69a17afe312208661ec4d46297b9114b22cef57da1210c9d01bf5.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha^T A\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha^T A\\alpha = \\alpha^T (-k\\beta) = -k\\alpha^T\\beta = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，这说明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha, \\beta"
                    },
                    {
                        "type": "text",
                        "content": " 正交，再由于"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\alpha^ {T} A \\beta + \\beta^ {T} A \\alpha = \\left\\{ \\begin{array}{l} \\alpha^ {T} A \\beta - (\\alpha^ {T} A \\beta) ^ {T} = 0 \\\\ k (\\alpha^ {T} \\alpha - \\beta^ {T} \\beta) \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/35193e50b9ad697b074c84721fda525209c3367205ee27644eeb2294354abb1a.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha^T\\alpha -\\beta^T\\beta = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，即模长相等"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.43 (反对称矩阵的标准型)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶反对称矩阵，则"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 正交相似于如下准对角阵"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\operatorname{diag} \\left\\{\\left( \\begin{array}{c c} 0 & b _ {1} \\\\ - b _ {1} & 0 \\end{array} \\right), \\dots , \\left( \\begin{array}{c c} 0 & b _ {s} \\\\ - b _ {s} & 0 \\end{array} \\right), 0, \\dots , 0 \\right\\}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/4b5043f67927c114e14e0228544402877f891560f44d41f232d3278e65f3b094.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 合同于标准型"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\operatorname{diag} \\left\\{\\left( \\begin{array}{c c} 0 & 1 \\\\ - 1 & 0 \\end{array} \\right), \\dots , \\left( \\begin{array}{c c} 0 & 1 \\\\ - 1 & 0 \\end{array} \\right), 0, \\dots , 0 \\right\\}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/57a2ef9518e71225e453a9647984a1a3b65ffd4d766d94ba3226d7501410b08c.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.44 (将矩阵拆为对称 + 反对称)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\frac {1}{2} (A + A ^ {T}) + \\frac {1}{2} (A - A ^ {T})",
                "math_type": "latex",
                "image_source": {
                    "path": "images/0544469c84ba536c307b98df0d340a1c424a810903330e6a387e0c76febde432.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.6 特殊矩阵性质总结"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "29"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.6.5 Hermite 矩阵"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定义 2.15 (Hermite 矩阵)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\overline{A}^T = A"
                    },
                    {
                        "type": "text",
                        "content": " 的矩阵称为Hermite矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定理 2.45 (Hermite 矩阵的性质)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "Hermite 矩阵的特征值都是实数"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.6.6 正交矩阵"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定义 2.16 (正交矩阵)"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "TT^{T} = I"
                    },
                    {
                        "type": "text",
                        "content": " ，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T"
                    },
                    {
                        "type": "text",
                        "content": " 为正交矩阵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.46 (正交矩阵的性质)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶正交矩阵，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 求逆:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1} = A^{T}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}, A^T, AB"
                    },
                    {
                        "type": "text",
                        "content": " 都是正交矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "S = \\left[ \\begin{array}{ll}A & 0\\\\ 0 & B \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": " 也是正交矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 正交当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的列向量是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{R}^n"
                    },
                    {
                        "type": "text",
                        "content": " 的一组标准正交基。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 显然"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(AB)^T AB = I"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 直接用  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "S^T S"
                    },
                    {
                        "type": "text",
                        "content": " 验证即可"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(4)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A=\\left[\\alpha_{1},\\cdots,\\alpha_{n}\\right]"
                    },
                    {
                        "type": "text",
                        "content": " ，计算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{T}A=I"
                    },
                    {
                        "type": "text",
                        "content": " 可得出结论。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 2.15 正交矩阵练习 b 是单位列向量，证明存在 n 阶对称正交矩阵 A，使得 b 为 A 的第一列证明"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.47 (分块上三角正交矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\left[ \\begin{array}{ll}A_{1} & A_{2}\\\\ 0 & A_{4} \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": " 是正交矩阵，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{2} = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1},A_{4}"
                    },
                    {
                        "type": "text",
                        "content": " 是正交矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 根据分块矩阵转置以及乘法可以得到结论"
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A ^ {T} A = \\left[ \\begin{array}{c c} A _ {1} ^ {T} & 0 \\\\ A _ {2} ^ {T} & A _ {4} ^ {T} \\end{array} \\right] \\left[ \\begin{array}{c c} A _ {1} & A _ {2} \\\\ 0 & A _ {4} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/c91c41cba495f4f8293789d5a195919a06bd58e2a0ba8e8241d19a87e38bf387.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.48 (正交矩阵的行列式与特征值)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶正交矩阵，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值模长为1，复特征值成对出现， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A| = \\pm 1"
                    },
                    {
                        "type": "text",
                        "content": " ，进一步地"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A| = -1: -1"
                    },
                    {
                        "type": "text",
                        "content": " 一定是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是奇数阶的，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A| = 1"
                    },
                    {
                        "type": "text",
                        "content": " ，则1一定是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 复特征值的特征向量实部和虚部模长相等且正交"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.6 特殊矩阵性质总结"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "30"
                    }
                ]
            },
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 主命题:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A\\alpha = \\lambda \\alpha"
                    },
                    {
                        "type": "text",
                        "content": "，两侧取共轭得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A\\overline{\\alpha} = \\overline{\\lambda}\\overline{\\alpha}"
                    },
                    {
                        "type": "text",
                        "content": "，两侧取转置得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\overline{\\alpha}^T A^T = \\overline{\\lambda}\\overline{\\alpha}^T"
                    },
                    {
                        "type": "text",
                        "content": "，两侧右乘  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha"
                    },
                    {
                        "type": "text",
                        "content": " 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\overline{\\alpha}^T A^{-1}\\alpha = \\overline{\\lambda}\\overline{\\alpha}^T\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\frac{1}{\\lambda}\\overline{\\alpha}^T\\alpha = \\overline{\\lambda}\\overline{\\alpha}^T\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda \\overline{\\lambda} = 1"
                    },
                    {
                        "type": "text",
                        "content": "，即模长为 1。根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A^T A| = 1"
                    },
                    {
                        "type": "text",
                        "content": " 可知行列式为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\pm 1"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A|=-1"
                    },
                    {
                        "type": "text",
                        "content": "  情况：根据   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A|"
                    },
                    {
                        "type": "text",
                        "content": "  是特征值乘积可知至少有 1 个 -1"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A|=1"
                    },
                    {
                        "type": "text",
                        "content": " 情况：同理"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记在计算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值时，如果  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特殊性需要  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T"
                    },
                    {
                        "type": "text",
                        "content": " 来体现，则经常两侧取共轭转置(例如实对称、反对称、正交矩阵)。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.49 (n 阶正交矩阵的标准型)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶正交矩阵，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 总是正交相似于"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\operatorname{diag} \\left\\{\\lambda_ {1}, \\dots , \\lambda_ {r}, \\left( \\begin{array}{c c} \\cos \\theta_ {1} & - \\sin \\theta_ {1} \\\\ \\sin \\theta_ {1} & \\cos \\theta_ {1} \\end{array} , \\dots , \\left( \\begin{array}{c c} \\cos \\theta_ {m} & - \\sin \\theta_ {m} \\\\ \\sin \\theta_ {m} & \\cos \\theta_ {m} \\end{array} \\right)\\right) \\right\\}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/5f37b7a2e3cc3d4f9c6d638f4e15dcec0cbc0680c28bfb365da2026d5fb17181.jpg"
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            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.6.7 Unitary 矩阵"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
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                        "type": "text",
                        "content": "定理 2.50 (Unitary 矩阵的性质)"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "Unitary 矩阵的特征值模长均为 1"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.6.8 幂零矩阵"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 2.17 (幂零矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若存在整数 l，使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{l}=O"
                    },
                    {
                        "type": "text",
                        "content": " ，则称 A 为幂零矩阵。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.51 (幂零矩阵充要条件)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是幂零矩阵当且仅当"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "1. 特征值:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值只有 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "2. 特征多项式:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征多项式为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{n}"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "3. 最小多项式：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的最小多项式为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^{k}, k \\leq n"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "4. 迹 (重点): 对   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall k > 0"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\operatorname{tr}(A^{k}) = 0"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "2.6.9 幂等矩阵"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 2.18 (幂等矩阵)"
                    }
                ],
                "level": 2
            },
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        },
        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^2 = A"
                    },
                    {
                        "type": "text",
                        "content": " ，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为幂等矩阵"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.52 (幂等矩阵充要条件)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是幂等矩阵当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) + r(A - E) = n"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^2 = A"
                    },
                    {
                        "type": "text",
                        "content": "，即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A(A - E) = O"
                    },
                    {
                        "type": "text",
                        "content": "，根据空间分解研究矩阵秩定理（具体见线性空间-直和与空间分解）"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "n = n - r (A) + n - r (A - E)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/8eb718b4e14c9c57761a10d5f749a160993076b143eeaf9c2017965addfadd9d.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) + r(A - E) = n"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.6 特殊矩阵性质总结"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "31"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.53 (幂等矩阵的性质)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶幂等矩阵，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值只有1和0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 一定可以对角化，且标准型为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathrm{diag}\\{E_r, O\\}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "r (A) = \\operatorname{tr} (A)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/96b8da93301667a4c288b5239303195409f61048e8cf106f57428629465d6f8d.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值，故  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A\\alpha = \\lambda \\alpha"
                    },
                    {
                        "type": "text",
                        "content": " ， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^2\\alpha = \\lambda^2\\alpha"
                    },
                    {
                        "type": "text",
                        "content": " ，由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = A^2"
                    },
                    {
                        "type": "text",
                        "content": " ，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda^2 = \\lambda"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda = 0,1"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 根据最小多项式理论，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(A)=A(A-E)=O"
                    },
                    {
                        "type": "text",
                        "content": " ，因此最小多项式只有一次项乘积，故可对角化。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 根据秩、迹在相似下的不变性可知"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.54 (加减乘法的幂等性)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 为两个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶幂等矩阵，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "1.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A + B"
                    },
                    {
                        "type": "text",
                        "content": " 为幂等矩阵当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB = BA = O"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "2.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A - B"
                    },
                    {
                        "type": "text",
                        "content": " 为幂等矩阵当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB = BA = B"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "3. 若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB = BA"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB"
                    },
                    {
                        "type": "text",
                        "content": " 为幂等矩阵。反之不一定"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "4. 如 E-A-B 可逆，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A)=r(B)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 右推左：由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB = BA = O"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A + B)^2 = A^2 + AB + BA + B^2 = A + B"
                    },
                    {
                        "type": "text",
                        "content": "。左推右： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A + B)^2 = A + AB + BA + B = A + B"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB + BA = O"
                    },
                    {
                        "type": "text",
                        "content": "，得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A B = A B ^ {2} = (- B A) B = - B A B = B (B A) = B A",
                "math_type": "latex",
                "image_source": {
                    "path": "images/e7e249912058cd5a21f4e1f15a54d05602ec4a0766047d1f30b47e89d008baf7.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 2.16 幂等矩阵分解 (1) 证明：任意方阵都可以分解为可逆矩阵与幂等矩阵的乘积"
                    }
                ]
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 利用等价标准型得到:"
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            "type": "equation_interline",
            "content": {
                "math_content": "A = P \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) Q = P Q Q ^ {- 1} \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) Q",
                "math_type": "latex",
                "image_source": {
                    "path": "images/571745df6e27fbb51c254eda31b7807526073fd9b9340cee3a2dcc42983b79ea.jpg"
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "PQ 为可逆矩阵，后面为幂等"
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 2.55 (幂等矩阵列的性质)"
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                "level": 2
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                        "type": "equation_inline",
                        "content": "A_{1},\\cdots,A_{m}"
                    },
                    {
                        "type": "text",
                        "content": "  为 n 阶方阵，且   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}+A_{2}+\\cdots+A_{m}=E_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，则下面三个命题等价："
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                        "type": "text",
                        "content": "-  "
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                    {
                        "type": "equation_inline",
                        "content": "A_{1}, \\cdots, A_{m}"
                    },
                    {
                        "type": "text",
                        "content": " 为幂等矩阵"
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            "content": {
                "math_content": "\\bullet \\mathrm{r} (A _ {1}) + \\dots + \\mathrm{r} (A _ {m}) = n",
                "math_type": "latex",
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                    "path": "images/292c46fc6be909ad85bd0552a117bab77b15ebe36b01e0ce9c70c6707c799b62.jpg"
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                        "type": "text",
                        "content": "-  "
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                    {
                        "type": "equation_inline",
                        "content": "\\forall i \\neq j"
                    },
                    {
                        "type": "text",
                        "content": " 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{i}A_{j} = 0"
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                        "type": "text",
                        "content": "证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(1) \\Rightarrow (2)"
                    },
                    {
                        "type": "text",
                        "content": ": 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A_{i}) = \\operatorname{tr}(A_{i})"
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                    {
                        "type": "text",
                        "content": ", 因此"
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            "content": {
                "math_content": "r \\left(A _ {1}\\right) + \\dots + r \\left(A _ {m}\\right) = \\operatorname{tr} \\left(A _ {1}\\right) + \\dots + \\operatorname{tr} \\left(A _ {m}\\right) = \\operatorname{tr} \\left(A _ {1} + \\dots + A _ {m}\\right) = \\operatorname{tr} \\left(E _ {n}\\right) = n",
                "math_type": "latex",
                "image_source": {
                    "path": "images/25edb87510ef17afbc4c253c1d456f2f719ffaba0c028b25ea7e225fe40cd36d.jpg"
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                        "type": "text",
                        "content": "(2)  "
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                    {
                        "type": "equation_inline",
                        "content": "\\Leftarrow"
                    },
                    {
                        "type": "text",
                        "content": " (3): 比较经典，先通过分块矩阵的初等变换凑出 E:"
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            "type": "equation_interline",
            "content": {
                "math_content": "\\left( \\begin{array}{c c c c c} A _ {1} & & & & \\\\ & A _ {2} & & & \\\\ & & A _ {3} & & \\\\ & & & \\ddots & \\\\ & & & & A _ {m} \\end{array} \\right) \\left( \\begin{array}{c c c c c} A _ {1} & A _ {2} & A _ {3} & \\dots & A _ {m} \\\\ A _ {2} & & & & \\\\ & & A _ {3} & & \\\\ & & & \\ddots & \\\\ & & & & A _ {m} \\end{array} \\right) \\to \\left( \\begin{array}{c c c c c} E & A _ {2} & A _ {3} & \\dots & A _ {m} \\\\ A _ {2} & A _ {2} & & \\\\ A _ {3} & & A _ {3} & \\\\ \\vdots & & & \\ddots \\\\ A _ {m} & & & A _ {m} \\end{array} \\right)",
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                "image_source": {
                    "path": "images/e019a1420ecde0db8b1a4c8ead179b0121fda08cac3b3721aae4f7d08c2e3bd8.jpg"
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            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "2.6 特殊矩阵性质总结"
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                        "content": "32"
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            "type": "paragraph",
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "再通过 E 消去其他部分:"
                    }
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            "content": {
                "math_content": "\\rightarrow \\left(\\begin{array}{c c c c c}E&O&O&\\dots&O\\\\A _ {2}&A _ {2} - A _ {2} ^ {2}&- A _ {2} A _ {3}&\\dots&- A _ {2} A _ {m}\\\\A _ {3}&- A _ {3} A _ {2}&A _ {3} - A _ {3} ^ {2}&\\dots&- A _ {3} A _ {m}\\\\\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\A _ {m}&- A _ {m} A _ {2}&- A _ {m} A _ {3}&\\dots&A _ {m} - A _ {m} ^ {2}\\end{array}\\right)\\rightarrow \\left(\\begin{array}{c c c c c}E&O&O&\\dots&O\\\\O&A _ {2} - A _ {2} ^ {2}&- A _ {2} A _ {3}&\\dots&- A _ {2} A _ {m}\\\\O&- A _ {3} A _ {2}&A _ {3} - A _ {3} ^ {2}&\\dots&- A _ {3} A _ {m}\\\\\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\O&- A _ {m} A _ {2}&- A _ {m} A _ {3}&\\dots&A _ {m} - A _ {m} ^ {2}\\end{array}\\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/9018ef00cfc5f131768eaa2c0172228c013e076c13b26e1e53e8863111a0f5f1.jpg"
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "根据   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A_{1}) + r(A_{2}) + \\cdots + r(A_{m}) = n"
                    },
                    {
                        "type": "text",
                        "content": " ，得到"
                    }
                ]
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left( \\begin{array}{c c c c} A _ {2} - A _ {2} ^ {2} & - A _ {2} A _ {3} & \\dots & - A _ {2} A _ {m} \\\\ - A _ {3} A _ {2} & A _ {3} - A _ {3} ^ {2} & \\dots & - A _ {3} A _ {m} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ - A _ {m} A _ {2} & - A _ {m} A _ {3} & \\dots & A _ {m} - A _ {m} ^ {2} \\end{array} \\right) = O",
                "math_type": "latex",
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                    "path": "images/816315ac81638ea5ebeca4aef9eee21379466aa1e087086c5d4b60a73c0cd84e.jpg"
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                    {
                        "type": "text",
                        "content": "这说明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall i \\neq j"
                    },
                    {
                        "type": "text",
                        "content": "  有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{i}^{2} = A_{i}"
                    },
                    {
                        "type": "text",
                        "content": " ，且   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{i}A_{j} = O(i, j = 2, 3, \\cdots, m)"
                    },
                    {
                        "type": "text",
                        "content": " ，再对   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1} + \\cdots + A_{m} = E_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  两边同乘   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  得到\n\n "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}A_{i} + A_{i} = A_{i}"
                    }
                ]
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}A_{i} = O"
                    },
                    {
                        "type": "text",
                        "content": " ，结论成立"
                    }
                ]
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            "type": "paragraph",
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                        "type": "text",
                        "content": "(3)  "
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                    {
                        "type": "equation_inline",
                        "content": "\\Rightarrow(1)"
                    },
                    {
                        "type": "text",
                        "content": " ：直接对   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}+\\cdots+A_{m}=E_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  两边右乘   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  得到"
                    }
                ]
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A _ {1} A _ {i} + A _ {2} A _ {i} + \\dots + A _ {m} A _ {i} = A _ {i} ^ {2} = A _ {i}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/cbf27f6bc2ded7abaf68d7cf88d5bb5370f3f466b9f0e90fd09910866bf49785.jpg"
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            "bbox": [
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此都是幂等矩阵"
                    }
                ]
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            "type": "image",
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                "image_footnote": []
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 此定理在 ZJU2020 中考过"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
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                        "type": "text",
                        "content": "2.6.10 对角占优矩阵"
                    }
                ],
                "level": 2
            },
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                "page_header_content": [
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                        "type": "text",
                        "content": "2.6 特殊矩阵性质总结"
                    }
                ]
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                        "content": "33"
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    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "第3章 行列式"
                    }
                ],
                "level": 2
            },
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        {
            "type": "title",
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                        "type": "text",
                        "content": "3.1 行列式基本概念与性质"
                    }
                ],
                "level": 2
            },
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        {
            "type": "title",
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                        "type": "text",
                        "content": "定理 3.1 (行列式初等变换)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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                "paragraph_content": [
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                        "type": "text",
                        "content": "矩阵初等变换相对行列式的变换为:"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 互换两行/列：取反号"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 同行/列可同时提出公因式"
                    }
                ]
            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 倍加：不改变行列式大小"
                    }
                ]
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        },
        {
            "type": "title",
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                        "type": "text",
                        "content": "定理 3.2 (行列式分拆)"
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                ],
                "level": 2
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            "content": {
                "paragraph_content": [
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                        "type": "text",
                        "content": "行列式可以关于一行或者一列进行分拆，即"
                    }
                ]
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            "content": {
                "math_content": "\\left| \\begin{array}{c c c c} a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ b _ {s 1} + c _ {s 1} & b _ {s 2} + c _ {s 2} & \\dots & b _ {s n} + c _ {s n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right| = \\left| \\begin{array}{c c c c} a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ b _ {s 1} & b _ {s 2} & \\dots & b _ {s n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right| + \\left| \\begin{array}{c c c c} a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ c _ {s 1} & c _ {s 2} & \\dots & c _ {s n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right|.",
                "math_type": "latex",
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                        "content": "3.2 行列式展开"
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        {
            "type": "title",
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                        "content": "定义 3.1 (余子式与代数余子式)"
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                "level": 2
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "将矩阵 A 去除第 i 行，第 j 列后矩阵的行列式称为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{ij}"
                    },
                    {
                        "type": "text",
                        "content": "  的余子式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{ij}"
                    },
                    {
                        "type": "text",
                        "content": " ，代数余子式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{ij} := (-1)^{i+j} M_{ij}"
                    }
                ]
            },
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        {
            "type": "title",
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                        "content": "定理 3.3 (行列式展开)"
                    }
                ],
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        {
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "固定列  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "j"
                    },
                    {
                        "type": "text",
                        "content": " ，则按列展开为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A| = \\sum_{i=1}^{n} a_{ij} A_{ij}"
                    },
                    {
                        "type": "text",
                        "content": " 。固定行  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i"
                    },
                    {
                        "type": "text",
                        "content": " ，则按行展开为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A| = \\sum_{j=1}^{n} a_{ij} A_{ij}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 3.4 (不同行相乘为 0)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的第  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i"
                    },
                    {
                        "type": "text",
                        "content": " 行元素与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k"
                    },
                    {
                        "type": "text",
                        "content": " 行的代数余子式乘积和为0:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\sum_{j=1}^{n} a_{ij} A_{kj} = 0"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "3.3 特殊的行列式"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "3.3.1 爪型行列式"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 3.5 (正爪型行列式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "正爪型行列式通过对角消去一条直边，再按三角行列式展开"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 3.6 (异爪型行列式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "异爪型行列式(一横两斜)分以下几种情况:"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 行和为0型：全部加到第一行，按第一列展开"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 两对角各自相等型：按斜直夹边展开，获得递推公式"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习3.1异爪型行列式(1)计算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A| = \\left| \\begin{array}{cccccc}0 & -1 & -1 & \\dots & -1 & -1\\\\ 1 & 1 & 0 & \\dots & 0 & 0\\\\ 0 & 1 & 1 & \\dots & 0 & 0\\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 0 & 0 & 0 & \\dots & 1 & 1 \\end{array} \\right|"
                    }
                ]
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "s(x) = \\begin{cases} \\frac{x}{|x|}, & x \\neq 0 \\\\ 0, & x = 0 \\end{cases}"
                    },
                    {
                        "type": "text",
                        "content": "，已知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = (a_{ij})"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{ij} = s(i - j)"
                    },
                    {
                        "type": "text",
                        "content": "，求  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A|"
                    }
                ]
            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解(1)按最后一列展开，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D_{n} = (-1)^{n + 2} + D_{n - 1}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 为奇数时  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D_{n} = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 为偶数时  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D_{n} = 1"
                    }
                ]
            },
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        },
        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 上下相减后变为 (1)"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "3.3.2 大对角型 (三斜线)"
                    }
                ],
                "level": 2
            },
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        },
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            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "3.3.3 行和相同型"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
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                    {
                        "type": "text",
                        "content": "定理 3.7 (行和相同型)"
                    }
                ],
                "level": 2
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        },
        {
            "type": "paragraph",
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "行和相同型行列式分以下两种情况："
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 内部大量相同：加到第一列，提出公因式，消去内部大量元素"
                    }
                ]
            },
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 循环型：上下相减 (例题见循环型一节)"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 3.2 内部大量相同型 计算行列式"
                    }
                ]
            },
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            "content": {
                "math_content": "\\left| \\begin{array}{c c c c c} a & b & b & \\dots & b \\\\ b & a & b & \\dots & b \\\\ b & b & a & \\dots & b \\\\ \\vdots & \\vdots & & & \\vdots \\\\ b & b & b & \\dots & a \\end{array} \\right|",
                "math_type": "latex",
                "image_source": {
                    "path": "images/fe14f2132824e32541083e2ebc7447a757e014decd350bfefad1340f8294ebb4.jpg"
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        {
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解加到第一列，提出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "[a + (n - 1)b]"
                    },
                    {
                        "type": "text",
                        "content": "，用第一列消去其他元素，最终得到答案  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(a - b)^{n - 1}[a + (n - 1)b]"
                    }
                ]
            },
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        },
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                        "type": "text",
                        "content": "3.3.4 循环型"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "循环型一般无论形态都是上下相减，一般只有行和相同型和反对称型两种。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习3.3循环型 计算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A|, |B|"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "| A | = \\left| \\begin{array}{c c c c c c} 0 & 1 & 2 & \\dots & n - 2 & n - 1 \\\\ 1 & 0 & 1 & \\dots & n - 3 & n - 2 \\\\ 2 & 1 & 0 & \\dots & n - 4 & n - 3 \\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ n - 2 & n - 3 & n - 4 & \\dots & 0 & 1 \\\\ n - 1 & n - 2 & n - 3 & \\dots & 1 & 0 \\end{array} \\right| \\quad | B | = \\left| \\begin{array}{c c c c c c} 1 & 2 & 3 & \\dots & n - 1 & n \\\\ n & 1 & 2 & \\dots & n - 2 & n - 1 \\\\ n - 1 & n & 1 & \\dots & n - 3 & n - 2 \\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 3 & 4 & 5 & \\dots & 1 & 2 \\\\ 2 & 3 & 4 & \\dots & n & 1 \\end{array} \\right|",
                "math_type": "latex",
                "image_source": {
                    "path": "images/4497bda787cbe3a2fcb411a2edd771435e305c195b9b26dfe3e0070df829f32e.jpg"
                }
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "3.3 特殊的行列式"
                    }
                ]
            },
            "bbox": [
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                        "content": "35"
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    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解(1)依次上面一行减下面一行, 变成如下行列式, 再每一列加上第一列即可"
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left| \\begin{array}{c c c c c c} - 1 & 1 & 1 & \\dots & 1 & 1 \\\\ - 1 & - 1 & 1 & \\dots & 1 & 1 \\\\ - 1 & - 1 & - 1 & \\dots & 1 & 1 \\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ - 1 & - 1 & - 1 & \\dots & - 1 & 1 \\\\ n - 1 & n - 2 & n - 3 & \\dots & 1 & 0 \\end{array} \\right|",
                "math_type": "latex",
                "image_source": {
                    "path": "images/472d84f145d78c66ab717f28c664dc9fd36ffb540ec16ef608257cd68c93acf2.jpg"
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 依次上面一行减下面一行，然后将最后一列的 -1 倍加到每一列。"
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                        "content": "3.3.5 行列式加边法"
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                ],
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                        "type": "text",
                        "content": "定理 3.8 (行列式加边法)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
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                    {
                        "type": "text",
                        "content": "计算矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的行列式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A|"
                    },
                    {
                        "type": "text",
                        "content": " 时，可加上一边："
                    }
                ]
            },
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        {
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            "content": {
                "math_content": "| A | = \\left| \\begin{array}{c c c c} a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ a _ {2 1} & a _ {2 2} & \\dots & a _ {2 n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right| = \\left| \\begin{array}{c c c c c} 1 & 1 & 1 & 1 & 1 \\\\ 0 & a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ 0 & a _ {2 1} & a _ {2 2} & \\dots & a _ {2 n} \\\\ \\vdots & \\vdots & \\vdots & & \\vdots \\\\ 0 & a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right|",
                "math_type": "latex",
                "image_source": {
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        {
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                "content": "",
                "image_caption": [],
                "image_footnote": []
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 行列式加边法往往在内部元素大量相同，但行和不同的情况。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "3.4 Cramer 法则"
                    }
                ],
                "level": 2
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                    {
                        "type": "text",
                        "content": "定理 3.9 (Cramer 法则)"
                    }
                ],
                "level": 2
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 个方程的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 元线性方程组  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX = B"
                    },
                    {
                        "type": "text",
                        "content": " ，系数矩阵行列式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A|\\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " ，则其唯一解为"
                    }
                ]
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left(\\frac {| B _ {1} |}{| A |}, \\frac {| B _ {2} |}{| A |}, \\dots , \\frac {| B _ {n} |}{| A |}\\right)",
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                "image_source": {
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                "paragraph_content": [
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                        "type": "text",
                        "content": "其中  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 将第  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i"
                    },
                    {
                        "type": "text",
                        "content": " 列换为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B"
                    },
                    {
                        "type": "text",
                        "content": " 组成的矩阵。"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "3.4 Cramer 法则"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
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                        "content": "36"
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                ]
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        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "第 4 章 线性方程组"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "4.1 线性方程组解的判定定理"
                    }
                ],
                "level": 2
            },
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            "content": {
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                        "content": "4.1.1 解存在判定定理"
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                        "content": "定理 4.1 (解存在充要条件)"
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        {
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            "content": {
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                    {
                        "type": "text",
                        "content": "K 上线性方程组 Ax=b 有解的充要条件为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A)=r(\\overline{A})"
                    },
                    {
                        "type": "text",
                        "content": " ，这里   "
                    },
                    {
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                        "content": "\\overline{A} := [A, b]"
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                ]
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        },
        {
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            "content": {
                "title_content": [
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                        "type": "text",
                        "content": "4.1.2 线性方程组同解"
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                ],
                "level": 2
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        },
        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定理 4.2 (同解定理)"
                    }
                ]
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A_{s\\times n},B_{m\\times n}"
                    },
                    {
                        "type": "text",
                        "content": " 为两个矩阵，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX = 0,BX = 0"
                    },
                    {
                        "type": "text",
                        "content": " 同解当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A,B"
                    },
                    {
                        "type": "text",
                        "content": " 行向量等价"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 4.1 具体应用 (1) 已知下面两个线性方程组同解，求 a, b, c 的值，且求通解"
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                ]
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            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} x _ {1} + 2 x _ {2} + 3 x _ {3} = 0 \\\\ 2 x _ {1} + 3 x _ {2} + 5 x _ {3} = 0 \\\\ x _ {1} + x _ {2} + a x _ {3} = 0 \\end{array} \\right. \\quad \\left\\{ \\begin{array}{l} x _ {1} + b x _ {2} + c x _ {3} = 0 \\\\ 2 x _ {1} + b ^ {2} x _ {2} + (c + 1) x _ {3} = 0 \\end{array} \\right.",
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                "image_source": {
                    "path": "images/e8795e596a650f2a33aaed876fe0307f1d76bd9dd14fca29c0fcd88f737f2b82.jpg"
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                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{s \\times n}, B_{t \\times n}"
                    },
                    {
                        "type": "text",
                        "content": " 为两个矩阵， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX = 0, BX = 0"
                    },
                    {
                        "type": "text",
                        "content": " 的解空间分别为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}, V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，证明： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\subseteq V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当存在矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C_{t \\times s}"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B = CA"
                    }
                ]
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        {
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            "content": {
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                    {
                        "type": "text",
                        "content": "证明 (1) 设方程组为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX = 0, BX = 0"
                    },
                    {
                        "type": "text",
                        "content": "，即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX = 0"
                    },
                    {
                        "type": "text",
                        "content": " 与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\left( \\begin{array}{l} A \\\\ B \\end{array} \\right) X = 0"
                    },
                    {
                        "type": "text",
                        "content": " 同解，做初等行变换即可。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}\\subseteq V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当AX=0与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\left(\\begin{array}{l}A\\\\ B\\end{array}\\right)X=0"
                    },
                    {
                        "type": "text",
                        "content": " 同解。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
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                        "type": "text",
                        "content": "4.1.3 解个数判定定理"
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                ],
                "level": 2
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        },
        {
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                        "type": "text",
                        "content": "定理 4.3 (解个数判定定理)"
                    }
                ],
                "level": 2
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若 K 上 Ax = b 满足   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = r(\\overline{A})"
                    },
                    {
                        "type": "text",
                        "content": " :"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = n"
                    },
                    {
                        "type": "text",
                        "content": " ：则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Ax = b"
                    },
                    {
                        "type": "text",
                        "content": " 有唯一解"
                    }
                ]
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        },
        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) < n"
                    },
                    {
                        "type": "text",
                        "content": "：则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Ax = b"
                    },
                    {
                        "type": "text",
                        "content": " 有无穷解"
                    }
                ]
            },
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        {
            "type": "title",
            "content": {
                "title_content": [
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                        "type": "text",
                        "content": "4.2 线性方程组的求解"
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                ],
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                        "type": "text",
                        "content": "4.2.1  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Ax = 0"
                    },
                    {
                        "type": "text",
                        "content": " 求解"
                    }
                ]
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
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                        "type": "text",
                        "content": "定义 4.1 (自由未知量)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "消元后多余的 0 = 0 位置对应的变量   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{r+1}, \\cdots, x_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  称为方程组自由未知量"
                    }
                ]
            },
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                        "type": "text",
                        "content": "定理 4.4 (自由变量的个数)"
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            },
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        },
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "自由变量的个数为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n - r(A)"
                    }
                ]
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        },
        {
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            "content": {
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                        "type": "text",
                        "content": "定义 4.2 (基础解系)"
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                ],
                "level": 2
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        },
        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "让自由未知量  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{r - 1 + i}"
                    },
                    {
                        "type": "text",
                        "content": " 取1，其余取0，构造解向量  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\eta_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 称为基础解系："
                    }
                ]
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            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} \\eta_ {1} = (*, \\dots , *, 1, 0, \\dots , 0) \\\\ \\eta_ {2} = (*, \\dots , *, 0, 1, \\dots , 0) \\\\ \\vdots \\\\ \\eta_ {n - r} = (*, \\dots , *, 0, 0, \\dots , 1) \\end{array} \\right.",
                "math_type": "latex",
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            "content": {
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                    {
                        "type": "text",
                        "content": "笔记 如果只有一个自由未知数，则一般取 1 而非 0，否则无法表示出基础解系，只是要给特解。"
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                        "type": "text",
                        "content": "定理 4.5 (齐次线性方程组的解)"
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                ]
            },
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                    {
                        "type": "text",
                        "content": "齐次线性方程组  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Ax = 0"
                    },
                    {
                        "type": "text",
                        "content": " 的任意解是其基础解系的线性组合"
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                ]
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                        "type": "text",
                        "content": "4.2.2  "
                    },
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                        "content": "Ax = b"
                    },
                    {
                        "type": "text",
                        "content": " 求解"
                    }
                ],
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            "type": "title",
            "content": {
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                        "content": "定理 4.6 (非齐次线性方程组的解)"
                    }
                ],
                "level": 2
            },
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        },
        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "非齐次线性方程组 Ax = b 的任意解是其特解   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\gamma_{0}"
                    },
                    {
                        "type": "text",
                        "content": "  加上齐次方程组 Ax = 0 基础解系线性组合，即   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = \\gamma_{0} + L\\{\\eta_{1}, \\cdots, \\eta_{n-r}\\}"
                    }
                ]
            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 4.2 求解非齐次线性方程组 (1) 讨论以下线性方程组解的情况"
                    }
                ]
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            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} x _ {1} + x _ {2} - x _ {3} = 1 \\\\ 2 x _ {1} + 3 x _ {2} + k x _ {3} = 3 \\\\ x _ {1} + k x _ {2} + 3 x _ {3} = 2 \\end{array} \\right.",
                "math_type": "latex",
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                        "type": "text",
                        "content": "(2) 已知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\left( \\begin{array}{ccc}1 & -1 & -1\\\\ 4 & t & 1\\\\ -2 & 2 & t \\end{array} \\right),b = \\left( \\begin{array}{c}1\\\\ 2\\\\ -t \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " ，当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "t"
                    },
                    {
                        "type": "text",
                        "content": " 为何值时， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX = b"
                    },
                    {
                        "type": "text",
                        "content": " 无解，有唯一解，有无穷多解，并给出有解时对应的解。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解(1)对增广矩阵做初等变换得到"
                    }
                ]
            },
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            "content": {
                "math_content": "\\left[ \\begin{array}{c c c c} 1 & 1 & - 1 & 1 \\\\ 0 & 1 & k + 2 & 1 \\\\ 0 & 0 & (k + 3) (k - 2) & k - 2 \\end{array} \\right]",
                "math_type": "latex",
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                    "path": "images/20234c9e2271a28174a29dda9b3f3a133826b014be1c4ebeb06f5ed61e8367e0.jpg"
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "先讨论解的情况：有解当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = r(\\overline{A})"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k = -3"
                    },
                    {
                        "type": "text",
                        "content": " 无解，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k = 2"
                    },
                    {
                        "type": "text",
                        "content": " 有无穷多解。其余情况有唯一解。  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k = 2"
                    },
                    {
                        "type": "text",
                        "content": " 时特解以及齐次基础解系："
                    }
                ]
            },
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            "content": {
                "math_content": "\\gamma = \\left( \\begin{array}{c} 0 \\\\ 1 \\\\ 0 \\end{array} \\right), y = \\left( \\begin{array}{c} 5 \\\\ - 4 \\\\ 1 \\end{array} \\right) \\Rightarrow x = \\gamma + k _ {1} y",
                "math_type": "latex",
                "image_source": {
                    "path": "images/2d7b97173307741a4d5e8f2435cae54c14905de27eba62f7ca98f3753dff5ba9.jpg"
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "唯一解情况直接解出结果即可。"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
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                        "type": "text",
                        "content": "4.2 线性方程组的求解"
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                        "content": "38"
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 做增广矩阵初等变换得到"
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            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c c c} 1 & - 1 & - 1 & 1 \\\\ 0 & t + 4 & 5 & - 2 \\\\ 0 & 0 & t - 2 & 2 - t \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1c9e18dc69640471e15d9a1c3db53767a774e96a3309d327147b6aa99a02757a.jpg"
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            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "t = -4 时无解。t = 2 无穷多解，其他情况有唯一解。具体求解略去。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "4.2.3 线性方程组公共解的计算"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 4.3 公共解的情况 (1) 线性方程组 I 的通解为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(1,1,0,0)^{T} + k_{1}(1,0,-1,0)^{T} + k_{2}(2,3,0,1)^{T}"
                    },
                    {
                        "type": "text",
                        "content": " ，线性方程组 II 如下，若两个方程组有无穷组公共解，求 a, b 的值并表示出所有公共解"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} 7 x _ {1} - 6 x _ {2} + 3 x _ {3} = b \\\\ 8 x _ {1} - 9 x _ {2} + a x _ {4} = 7 \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/cafe4fb86dd8fa9de5e78af821482df8b2178f5539dd9f4f8728a0637567e809.jpg"
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解 由于有无穷多组解，因此有无穷组  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{1}, k_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(1 + k_{1} + 2k_{2}, 1 + 3k_{2}, -k_{1}, k_{2})"
                    },
                    {
                        "type": "text",
                        "content": " 为方程组 II 的解。代入方程化简得到"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} 4 k _ {1} - 4 k _ {2} = b - 1 \\\\ 8 k _ {1} + (a - 1 1) k _ {2} = 8 \\end{array} \\right. \\Rightarrow \\left\\{ \\begin{array}{l} 4 k _ {1} - 4 k _ {2} = b - 1 \\\\ 0 k _ {1} + (a - 3) k _ {2} = 1 0 - 2 b \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f3760b1f45397f04d54138ef96009e1cd35e29173843065bb3fe8e658b28c849.jpg"
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "需要有无穷多解，得有自由位置量，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a = 3, b = 5"
                    },
                    {
                        "type": "text",
                        "content": " ， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{1} = k_{2} + 1"
                    },
                    {
                        "type": "text",
                        "content": " ，解系为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(2,1, -1,0)^{T} + k_{2}(3,3, -1,1)^{T}"
                    },
                    {
                        "type": "text",
                        "content": " 笔记若给两个完整的线性方程组，则先求出一个的基础解系，再往另一个方程中代。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "4.3 最小二乘解"
                    }
                ],
                "level": 2
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "4.3.1  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T A"
                    },
                    {
                        "type": "text",
                        "content": " 的研究"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 4.7 (  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{T}A"
                    },
                    {
                        "type": "text",
                        "content": "  与 A 的秩)"
                    }
                ],
                "level": 2
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "设 A 是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m \\times n"
                    },
                    {
                        "type": "text",
                        "content": "  的实矩阵，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A^{T}A) = r(A)"
                    },
                    {
                        "type": "text",
                        "content": " 。复数域有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A\\overline{A}^{T}) = r(\\overline{A}^{T}A) = r(A)"
                    },
                    {
                        "type": "text",
                        "content": " ，但不一定有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A^{T}A) = r(A)"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 只需要考虑 AX = 0 和   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{T}AX = 0"
                    },
                    {
                        "type": "text",
                        "content": "  解的情况。首先根据 AX = 0 可以推出   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{T}AX = 0"
                    },
                    {
                        "type": "text",
                        "content": " 。其次   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{T}AX = 0"
                    },
                    {
                        "type": "text",
                        "content": "  可推出   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^{T}A^{T}AX = (AX)^{T}(AX) = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，而 AX 是一个实向量，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha^{T}\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": "  当且仅当   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，因此 AX = 0。故它们同解，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n - r(A) = n - r(A^{T}A)"
                    },
                    {
                        "type": "text",
                        "content": "  得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = r(A^{T}A)"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX = 0"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\overline{A}^T A X = 0"
                    },
                    {
                        "type": "text",
                        "content": "。若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\overline{A}^T A X = 0"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\overline{X}^T \\overline{A}^T A X = (\\overline{A X})^T A X = 0"
                    },
                    {
                        "type": "text",
                        "content": "，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A X = 0"
                    },
                    {
                        "type": "text",
                        "content": " 笔记该定理最重要的结论是最小二乘解一定存在。"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "4.4A^{T}A"
                    },
                    {
                        "type": "text",
                        "content": "  与 A 性质 (1) A 是 R 上任意   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "s \\times n"
                    },
                    {
                        "type": "text",
                        "content": "  矩阵，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(AA^{T}A) = r(A)"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(AA^{T}A)\\leq r(A)"
                    },
                    {
                        "type": "text",
                        "content": " ，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(AA^{T}A)\\geq r(A^{T}AA^{T}A)=r(A^{T}A)=r(A)"
                    }
                ]
            },
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "4.3.2 正规方程组"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 4.3 (最小二乘解)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对一个无解的线性方程组 (矛盾方程组) AX = b，若向量   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x_{1}, \\cdots, x_{s}) \\in \\mathbb{R}^{s}"
                    },
                    {
                        "type": "text",
                        "content": "  使得距离"
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "d = \\sum_ {i = 1} ^ {n} (a _ {i 1} x _ {1} + \\dots + a _ {i s} x _ {s} - b _ {i}) ^ {2}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/31526a73a268eb9c93f7017642ae5087f2418b2dc5bb3180db1e59a0b2af6402.jpg"
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            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "最小，则称该向量为最小二乘解。"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "4.3 最小二乘解"
                    }
                ]
            },
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "39"
                    }
                ]
            },
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        }
    ],
    [
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            "type": "image",
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 4.8 (正规方程组)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于任一方程组  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX = b"
                    },
                    {
                        "type": "text",
                        "content": " ，最小二乘解一定存在，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_0"
                    },
                    {
                        "type": "text",
                        "content": " 是最小二乘解当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T A X = A^T b"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{s}"
                    },
                    {
                        "type": "text",
                        "content": "  是 A 的列向量，定义   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W=L(\\alpha_{1},\\cdots,\\alpha_{s})"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{0}"
                    },
                    {
                        "type": "text",
                        "content": "  是最小二乘解当且仅当   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX_{0}"
                    },
                    {
                        "type": "text",
                        "content": "  是 b 在上的投影，即   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(b-AX_{0})\\perp\\alpha_{i}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\alpha_ {i} ^ {T} (b - A X _ {0}) \\Rightarrow A ^ {T} (b - A X _ {0}) = 0 \\Rightarrow A ^ {T} A X _ {0} = A ^ {T} b",
                "math_type": "latex",
                "image_source": {
                    "path": "images/de97ee9df0c7d86de2c80237dbe25e963fbb03593e99b9b0ce8b2d14644d3798.jpg"
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            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 4.9 (正规方程组解的存在性)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "方程组  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T A X = A^T b"
                    },
                    {
                        "type": "text",
                        "content": " 一定有解。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 因为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A^T A) = r(A)"
                    },
                    {
                        "type": "text",
                        "content": "，先将增广矩阵拆开得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r([A^T A, A^T b]) = r(A^T [A, b])"
                    },
                    {
                        "type": "text",
                        "content": "，用秩不等式得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A^T [A, b]) \\leq r(A^T) = r(A) = r(A^T A)"
                    },
                    {
                        "type": "text",
                        "content": "，而显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r([A^T A, A^T b]) \\geq r(A^T A)"
                    },
                    {
                        "type": "text",
                        "content": "，因此增广矩阵的秩等于系数矩阵的秩，综上得到最小二乘法解存在。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 4.10 (凸闭集最佳逼近元的唯一性)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{H}"
                    },
                    {
                        "type": "text",
                        "content": " 是Hilbert空间，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M\\subset \\mathcal{H}"
                    },
                    {
                        "type": "text",
                        "content": " 是凸闭集，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall x\\in \\mathcal{H}"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M"
                    },
                    {
                        "type": "text",
                        "content": " 中存在唯一最佳逼近。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 一般而言最小二乘解不唯一，除非是凸闭集。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "4.3 最小二乘解"
                    }
                ]
            },
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "40"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "第 5 章 线性空间与线性变换"
                    }
                ],
                "level": 1
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            "type": "title",
            "content": {
                "title_content": [
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                        "type": "text",
                        "content": "5.1 线性空间"
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                "level": 2
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            "type": "title",
            "content": {
                "title_content": [
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                        "type": "text",
                        "content": "5.1.1 线性空间的基本概念"
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                ],
                "level": 2
            },
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        },
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                "title_content": [
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                        "content": "定义 5.1 (线性空间)"
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                "level": 2
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 是一个数域，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 是非空集合，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 上定义了二元运算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "+"
                    },
                    {
                        "type": "text",
                        "content": " 和数乘，若这两种运算满足："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "二元运算：(1) 交换 (2) 结合 (3) 零元 (4) 负元"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 数乘：(1)一元(2)结合  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(c_{1}c_{2})\\alpha = c_{1}(c_{2}\\alpha)(3)"
                    },
                    {
                        "type": "text",
                        "content": " 向量分配  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(c_{1} + c_{2})\\alpha = c_{1}\\alpha + c_{2}\\alpha (4)"
                    },
                    {
                        "type": "text",
                        "content": " 数分配  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "c(\\alpha +\\beta) = c\\alpha +c\\beta"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 上的线性空间"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.1.2 线性相关、线性无关与向量组的秩"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
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                        "type": "text",
                        "content": "定义 5.2 (线性相关和线性无关)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  为向量组，若存在不全为0的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{1},\\cdots,k_{n}"
                    },
                    {
                        "type": "text",
                        "content": " 使得"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "k _ {1} \\alpha_ {1} + k _ {2} \\alpha_ {2} + \\dots + k _ {n} \\alpha_ {n} = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1420b66271dc124cd311e88a62fd71eb86c608c2871478ba25e4bb1481dc74bd.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则称它们线性相关。若该等式成立当且仅当   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{1}=k_{2}=\\cdots=k_{n}=0"
                    },
                    {
                        "type": "text",
                        "content": " ，则称线性无关。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 5.1 证明线性无关 (1) 幂零基: A 为 V 上的线性变换,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\xi \\in V"
                    },
                    {
                        "type": "text",
                        "content": " , 若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{k-1}\\xi \\neq 0, A^{k}\\xi = 0"
                    },
                    {
                        "type": "text",
                        "content": " , 证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\xi, A\\xi, \\cdots, A^{k-1}\\xi"
                    },
                    {
                        "type": "text",
                        "content": "  线性无关\n\n证明 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{0}\\xi + k_{1}A\\xi + \\cdots + k_{n-1}A^{k-1}\\xi = 0"
                    },
                    {
                        "type": "text",
                        "content": " , 两侧同时作用   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{k-1}"
                    },
                    {
                        "type": "text",
                        "content": "  得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{0}A^{k-1}\\xi = 0"
                    },
                    {
                        "type": "text",
                        "content": " , 因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{0} = 0"
                    },
                    {
                        "type": "text",
                        "content": " , 以此类推得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{0}, k_{1}, \\cdots, k_{n-1} = 0"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.1 (极大线性无关组的计算)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "给定一列向量   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，将它们作为矩阵的列向量拼成一个矩阵，对矩阵做初等行变换化为阶梯型矩阵，阶梯头所在列对应的向量为极大线性无关组。(其实按行也可以，按列更不容易出错，因为换行不影响结果)"
                    }
                ]
            },
            "bbox": [
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 5.2 极大线性无关组的计算 计算下列向量组的极大线性无关组"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} \\alpha_ {1} = (1, - 1, 2, 4) \\\\ \\alpha_ {2} = (0, 3, 1, 2) \\\\ \\alpha_ {3} = (3, 0, 7, 1 4) \\\\ \\alpha_ {4} = (1, - 1, 2, 0) \\\\ \\alpha_ {5} = (2, 1, 5, 6) \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/c823ad2669ec2a3191860e43e4a8a4e702017da84744be1221af0fa2539c4a85.jpg"
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    ],
    [
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解 拼成一个矩阵，并做初等行变换："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[\\begin{array}{c c c c c}1&0&3&1&2\\\\- 1&3&0&- 1&1\\\\2&1&7&2&5\\\\4&2&1 4&0&6\\end{array}\\right]\\rightarrow \\left[\\begin{array}{c c c c c}1&0&3&1&- 2\\\\0&1&1&0&1\\\\0&0&0&1&1\\\\0&0&0&0&0\\end{array}\\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/8af117f6e16fc8d2878a7426559b1a3b838e045fde50dc79d543f0d9af1ca831.jpg"
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            },
            "bbox": [
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此极大线性无关组为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\alpha_{2},\\alpha_{4}"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
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                        "type": "text",
                        "content": "定义 5.3 (向量组的秩)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  为向量组，称其中极大线性无关组的向量个数为它们的秩。"
                    }
                ]
            },
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.2 (源泉定理及其推论)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{r}"
                    },
                    {
                        "type": "text",
                        "content": "  可以被   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{1},\\cdots,\\beta_{s}"
                    },
                    {
                        "type": "text",
                        "content": "  线性表出，r>s，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{r}"
                    },
                    {
                        "type": "text",
                        "content": "  线性相关。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.1.3 线性空间同构"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定义 5.4 (同构)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若两个代数系统之间存在双射，且该双射保持代数系统的运算，则称两个代数系统同构。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.3 (有限维线性空间的同构映射)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 上线性空间， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}, \\cdots, \\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 的一组基，构造线性映射  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f"
                    },
                    {
                        "type": "text",
                        "content": "，将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in V"
                    },
                    {
                        "type": "text",
                        "content": " 映射到其坐标，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V \\to \\mathbb{P}^{n}"
                    },
                    {
                        "type": "text",
                        "content": " 的同构映射"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.4 (用坐标研究向量组)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": " 是 V 的一组基，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{n\\times s}"
                    },
                    {
                        "type": "text",
                        "content": " ，向量组  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{1},\\cdots,\\beta_{s}"
                    },
                    {
                        "type": "text",
                        "content": " 满足"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(\\beta_ {1}, \\dots , \\beta_ {s}) = (\\alpha_ {1}, \\dots , \\alpha_ {n}) A",
                "math_type": "latex",
                "image_source": {
                    "path": "images/901cd3f21bb3454749576188be17e1f3b39afde8767b8ec8d7e63d5ba9932b6f.jpg"
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            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\dim(\\beta_{1},\\cdots,\\beta_{s})=r(A)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = [X_{1}, \\cdots, X_{s}]"
                    },
                    {
                        "type": "text",
                        "content": "  即   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{1}, \\cdots, \\beta_{s}"
                    },
                    {
                        "type": "text",
                        "content": "  的坐标，因此 A 的列向量与   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{1}, \\cdots, \\beta_{s}"
                    },
                    {
                        "type": "text",
                        "content": "  等价，从而   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = \\dim(\\beta_{1}, \\cdots, \\beta_{s})"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "推论 5.1 (线性无关情况)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  线性无关，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\beta_{1},\\cdots,\\beta_{n})=(\\alpha_{1},\\cdots,\\alpha_{n})A"
                    },
                    {
                        "type": "text",
                        "content": " ，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  线性无关当且仅当 A 可逆。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习5.3同构的应用  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(1)\\beta_{1} = \\alpha_{1} + \\alpha_{2},\\beta_{2} = \\alpha_{1} + \\alpha_{3},\\beta_{3} = \\alpha_{1} + \\alpha_{2}"
                    },
                    {
                        "type": "text",
                        "content": " ，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\alpha_{2},\\alpha_{3}"
                    },
                    {
                        "type": "text",
                        "content": " 和  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{1},\\beta_{2},\\beta_{3}"
                    },
                    {
                        "type": "text",
                        "content": " 等价"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 证明 n 为奇数时，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": " 线性无关当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}+\\alpha_{2},\\cdots,\\alpha_{n-1}+\\alpha_{n},a_{n}+\\alpha_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 线性无关"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{1} = \\alpha_{2} + \\cdots + \\alpha_{n}, \\beta_{2} = \\alpha_{1} + \\alpha_{3} + \\cdots + \\alpha_{n}, \\beta_{n} = \\alpha_{1} + \\cdots + \\alpha_{n-1}"
                    },
                    {
                        "type": "text",
                        "content": " ，证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}, \\cdots, \\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  和   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{1}, \\cdots, \\beta_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  有相同的秩"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 如下同构映射至坐标, 显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 可逆, 因此等价"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(\\beta_ {1}, \\beta_ {2}, \\beta_ {3}) = (\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}) \\left[ \\begin{array}{c c c} 1 & 1 & 1 \\\\ 1 & & 1 \\\\ 0 & 1 \\end{array} \\right] := (\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}) A",
                "math_type": "latex",
                "image_source": {
                    "path": "images/0f71ce02ab30b3d173ed7b87cf71611abe4dbc70dc940761a8b78a5e036961f4.jpg"
                }
            },
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        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.1 线性空间"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "42"
                    }
                ]
            },
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        }
    ],
    [
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 将后者设为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{1}, \\cdots, \\beta_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，如下线性表出"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(\\beta_ {1}, \\dots , \\beta_ {n}) = (\\alpha_ {1}, \\dots , \\alpha_ {n}) \\left[ \\begin{array}{c c c c c} 1 & & & & 1 \\\\ 1 & 1 & & & \\\\ & 1 & \\ddots & & \\\\ & & \\ddots & 1 & \\\\ & & & 1 & 1 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f7afcdc9a91123ab8b435ded0edaed5377e3aa21e5062eab6d81f9b61acee21f.jpg"
                }
            },
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        },
        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "用 Gauss 消元法可看出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 为奇数时矩阵满秩，因此等价。"
                    }
                ]
            },
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        },
        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 同理列矩阵，用加边法可算出行列式非零，因此 A 可逆，即等价。"
                    }
                ]
            },
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.1.4 基、坐标"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 5.5 (基与坐标)"
                    }
                ],
                "level": 2
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "V 是线性空间，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1},\\cdots,\\epsilon_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  是 V 中线性无关的向量组，若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall\\alpha\\in V"
                    },
                    {
                        "type": "text",
                        "content": "  可被该向量组线性表出，则称该向量组为 V 的一组基。且若"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\alpha = (\\epsilon_ {1}, \\dots , \\epsilon_ {n}) X",
                "math_type": "latex",
                "image_source": {
                    "path": "images/b832663a7574a99a8f51edb13e93c10d8e9f80161bddcd45b946ea1ac2b9b8fd.jpg"
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则称 X 为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "  在   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1},\\cdots,\\epsilon_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  基下的坐标"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 坐标的计算：直接根据   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = (\\epsilon_{1}, \\cdots, \\epsilon_{n})X"
                    },
                    {
                        "type": "text",
                        "content": "  解线性方程组 (或者用初等变换求逆)"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "X = (\\epsilon_ {1}, \\dots , \\epsilon_ {n}) ^ {- 1} \\alpha",
                "math_type": "latex",
                "image_source": {
                    "path": "images/9c60eefe3deb09898636d266b269ba3045cfa03493ad737eaef8cdf33a78860f.jpg"
                }
            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 5.4 证明基 给定   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}=(1,1,\\cdots,1,1)"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{2}=(1,1,\\cdots,1,0)"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\cdots"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{n}=(1,0,\\cdots,0)"
                    },
                    {
                        "type": "text",
                        "content": " ，证明其是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^{n}"
                    },
                    {
                        "type": "text",
                        "content": "  的一组基证明 当成列向量排成一个矩阵，满秩即可"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 5.5 坐标的计算 (1) 已知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1}, \\epsilon_{2}, \\epsilon_{3}"
                    },
                    {
                        "type": "text",
                        "content": "  是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^{3}"
                    },
                    {
                        "type": "text",
                        "content": "  的一组基，计算   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = (1, 2, 1)^{T}"
                    },
                    {
                        "type": "text",
                        "content": "  在这组基下的坐标"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\epsilon_ {1} = (1, 1, 1) ^ {T}, \\epsilon_ {2} = (1, 1, - 1) ^ {T}, \\epsilon_ {3} = (1, - 1, - 1) ^ {T}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/b9ccc22f7991d08a1b905851ee20075f209e2ca6a2003c03e85a7e9bb6cb13cf.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解(1)只需要解方程组  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = k_{1}\\epsilon_{1} + k_{2}\\epsilon_{2} + k_{3}\\epsilon_{3}"
                    },
                    {
                        "type": "text",
                        "content": "，即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "[\\epsilon_1,\\epsilon_2,\\epsilon_3]X = \\alpha"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c c} 1 & 1 & 1 \\\\ 1 & 1 & - 1 \\\\ 1 & - 1 & - 1 \\end{array} \\right] X = \\left[ \\begin{array}{c} 1 \\\\ 2 \\\\ 1 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/8b1e105458bcd2db14b82b02b20a5d0b9e76fe96c89532130edd2283bc14a44d.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{1} = 1, k_{2} = \\frac{1}{2}, k_{3} = -\\frac{1}{2}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习5.6难度进阶(1)求非零向量  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\in \\mathbb{P}^4"
                    },
                    {
                        "type": "text",
                        "content": " ，使得其在下面两组基中有相同的坐标"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} \\epsilon_ {1} = (1, 0, 0, 0) ^ {T} \\\\ \\epsilon_ {2} = (0, 1, 0, 0) ^ {T} \\\\ \\epsilon_ {3} = (0, 0, 1, 0) ^ {T} \\\\ \\epsilon_ {4} = (0, 0, 0, 1) ^ {T} \\end{array} \\right., \\quad \\left\\{ \\begin{array}{l} \\eta_ {1} = (2, 1, - 1, 1) ^ {T} \\\\ \\eta_ {2} = (0, 3, 1, 0) ^ {T} \\\\ \\eta_ {3} = (5, 3, 2, 1) ^ {T} \\\\ \\eta_ {4} = (6, 6, 1, 3) ^ {T} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/20e997fc4e47468ba3bf30256867404efbfcc53c567f9dd2bd3c07b3cc3a735d.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解 (1) 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha"
                    },
                    {
                        "type": "text",
                        "content": " 在第一组基下的坐标为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x_{1}, x_{2}, x_{3}, x_{4})^{T}"
                    },
                    {
                        "type": "text",
                        "content": ", 则由于同坐标"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\alpha = x _ {1} \\eta_ {1} + x _ {2} \\eta_ {2} + x _ {3} \\eta_ {3} + x _ {4} \\eta_ {4}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/6c589d0d3f7a2644a682bc2e21b0b8378bf4a82866114b7fff9d180c2eb0beca.jpg"
                }
            },
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        },
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            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "43"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.1 线性空间"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解线性方程组   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\left[\\eta_{1},\\eta_{2},\\eta_{3},\\eta_{4}\\right]X=\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "  即可 (注意将   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "  减到左边):"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{r r r r} 1 & 0 & 5 & 6 \\\\ 1 & 2 & 3 & 6 \\\\ - 1 & 1 & 1 & 1 \\\\ 1 & 0 & 1 & 2 \\end{array} \\right] X = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/2c1e5bc672ee11a21630414900cc99f31837765f85ecf13f773b8e8069451982.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = k(1,1,1,-1)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.1.5 坐标变换"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 5.6 (过渡矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "给定  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\epsilon_{1},\\cdots,\\epsilon_{n}),(\\eta_{1},\\cdots,\\eta_{n})"
                    },
                    {
                        "type": "text",
                        "content": " 两组基，若存在矩阵T满足："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(\\epsilon_ {1}, \\dots , \\epsilon_ {n}) = (\\eta_ {1}, \\dots , \\eta_ {n}) T",
                "math_type": "latex",
                "image_source": {
                    "path": "images/3712ca51da41312b335f5eff9e80166884c57c0ae1d42670a4101c1abbafce00.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则称 T 为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\eta_{1},\\cdots,\\eta_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1},\\cdots,\\epsilon_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  的过渡矩阵 (不要搞反)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 过渡矩阵的计算：直接根据   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\epsilon_{1},\\cdots,\\epsilon_{n})=(\\eta_{1},\\cdots,\\eta_{n})T"
                    },
                    {
                        "type": "text",
                        "content": "  得到"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "T = (\\eta_ {1}, \\dots , \\eta_ {n}) ^ {- 1} (\\epsilon_ {1}, \\dots , \\epsilon_ {n})",
                "math_type": "latex",
                "image_source": {
                    "path": "images/8bf6516ff18907dc1641b6d6775b6f44228405a51e3251259ff072b92a656bd4.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "用初等变换求   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}B"
                    },
                    {
                        "type": "text",
                        "content": "  的方式直接求 T 即可。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.5 (坐标变换)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 维线性空间，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\eta_1,\\dots ,\\eta_n"
                    },
                    {
                        "type": "text",
                        "content": " 和  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_1,\\dots ,\\epsilon_n"
                    },
                    {
                        "type": "text",
                        "content": " 是空间两组基，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\eta_{1},\\dots ,\\eta_{n}) = (\\epsilon_{1},\\dots ,\\epsilon_{n})T"
                    },
                    {
                        "type": "text",
                        "content": " ，向量  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x\\in V"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_i,\\eta_i"
                    },
                    {
                        "type": "text",
                        "content": " 下的坐标分别为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X,Y"
                    },
                    {
                        "type": "text",
                        "content": " ，则"
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "X = T Y",
                "math_type": "latex",
                "image_source": {
                    "path": "images/798f3c7c07a99e9f52fce1324420a04ed3c1c39ebc693eb87fc11ad42e133d15.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x=(\\epsilon_{1},\\cdots,\\epsilon_{n})X=(\\eta_{1},\\cdots,\\eta_{n})Y=(\\epsilon_{1},\\cdots,\\epsilon_{n})TY"
                    },
                    {
                        "type": "text",
                        "content": "  即可。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 坐标变换用荡秋千记忆，即   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\eta_{1},\\cdots,\\eta_{n})=(\\epsilon_{1},\\cdots,\\epsilon_{n})T"
                    },
                    {
                        "type": "text",
                        "content": " ，分别对应 Y,X，则把 T 荡到另一边，即 TY=X。这与线性变换在不同基下的矩阵一样都需要记住。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 5.7 过渡矩阵的计算 给定两组基，求它们之间的过渡矩阵，并求   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\xi = (1, 0, 0, -1)"
                    },
                    {
                        "type": "text",
                        "content": "  在   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}, \\alpha_{2}, \\alpha_{3}, \\alpha_{4}"
                    },
                    {
                        "type": "text",
                        "content": "  下的坐标"
                    }
                ]
            },
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            "content": {
                "math_content": "\\begin{array}{l l l l} \\alpha_ {1} = (1, 1, 1, 1) & \\alpha_ {2} = (1, 1, - 1, - 1) & \\alpha_ {3} = (1, - 1, 1, - 1) & \\alpha_ {4} = (1, - 1, - 1, 1) \\\\ \\beta_ {1} = (1, 1, 0, 1) & \\beta_ {2} = (2, 1, 3, 1) & \\beta_ {3} = (1, 1, 0, 0) & \\beta_ {4} = (0, 1, - 1, - 1) \\end{array}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/00f045609bd6253625d1bcbcc3f730346fb9fde97696f8ad801e6aec19048274.jpg"
                }
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解(1)先取标准基   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1}, \\epsilon_{2}, \\epsilon_{3}, \\epsilon_{4}"
                    },
                    {
                        "type": "text",
                        "content": " ，设"
                    }
                ]
            },
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            "content": {
                "math_content": "(\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}, \\alpha_ {4}) = (\\epsilon_ {1}, \\epsilon_ {2}, \\epsilon_ {3}, \\epsilon_ {4}) A, (\\beta_ {1}, \\beta_ {2}, \\beta_ {3}, \\beta_ {4}) = (\\epsilon_ {1}, \\epsilon_ {2}, \\epsilon_ {3}, \\epsilon_ {4}) B",
                "math_type": "latex",
                "image_source": {
                    "path": "images/47cb9cef0a82540971f80da5eb3395af8bab5d6852cb2a2214c36bc001b3a429.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{r r r r} 1 & 1 & 1 & 1 \\\\ 1 & 1 & - 1 & - 1 \\\\ 1 & - 1 & 1 & - 1 \\\\ 1 & - 1 & - 1 & 1 \\end{array} \\right], B = \\left[ \\begin{array}{r r r r} 1 & 2 & 1 & 0 \\\\ 1 & 1 & 1 & 1 \\\\ 0 & 3 & 0 & - 1 \\\\ 1 & 1 & 0 & - 1 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/b72e3604c4989cf2854cabdd7a72363b0343bfa6943ed1ad4c3562c624896bb9.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由此得到"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(\\beta_ {1}, \\beta_ {2}, \\beta_ {3}, \\beta_ {4}) = (\\epsilon_ {1}, \\epsilon_ {2}, \\epsilon_ {3}, \\epsilon_ {4}) B = (\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}, \\alpha_ {4}) A ^ {- 1} B",
                "math_type": "latex",
                "image_source": {
                    "path": "images/7401a0c821b96e398e58556f1a66b70a2e16d76dbba3e4a42fa4236b6b6e796b.jpg"
                }
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.1 线性空间"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "44"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "综上得到过渡矩阵为 (具体计算时用行初等变换:   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A|B) \\rightarrow (E|A^{-1}B)"
                    },
                    {
                        "type": "text",
                        "content": " )"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A ^ {- 1} B = \\frac {1}{4} \\left[ \\begin{array}{r r r r} 3 & 7 & 2 & - 1 \\\\ 1 & - 1 & 2 & 3 \\\\ - 1 & 3 & 0 & - 1 \\\\ 1 & - 1 & 0 & - 1 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/eb4ef16469b3ed4834524257af667f4d1d3c601ecc81ad0d4227ed32707bae51.jpg"
                }
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\xi"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\epsilon_1, \\epsilon_2, \\epsilon_3, \\epsilon_4)"
                    },
                    {
                        "type": "text",
                        "content": " 下的坐标即本身，因此在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4)"
                    },
                    {
                        "type": "text",
                        "content": " 下坐标为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}\\xi"
                    },
                    {
                        "type": "text",
                        "content": " (具体计算时也和  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A|B)"
                    },
                    {
                        "type": "text",
                        "content": " 摆在一起)，结果为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(0, \\frac{1}{2}, \\frac{1}{2}, 0)^T"
                    }
                ]
            },
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.1.6 和与交、维数定理"
                    }
                ],
                "level": 2
            },
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 5.7 (线性空间和与交)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V_{1}, V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 为线性空间，则"
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "V _ {1} \\cap V _ {2} := \\{\\alpha : \\alpha \\in V _ {1}, \\alpha \\in V _ {2} \\}, V _ {1} + V _ {2} := \\{\\alpha = \\alpha_ {1} + \\alpha_ {2}: \\alpha_ {1} \\in V _ {1}, \\alpha_ {2} \\in V _ {2} \\}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/e937da3c34f815def7e23c112820ba0fd3718f66b05d07db094dcd1afdde0547.jpg"
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "且显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\cap V_{2}, V_{1} + V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 都是子空间。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "image",
            "content": {
                "image_source": {
                    "path": "images/6704a6fa582ccd0f7b0270cfe13b761ae03c428e9e7a8137480049d96a04f59a.jpg"
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                "content": "",
                "image_caption": [],
                "image_footnote": []
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\cup V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 由于运算不封闭，一般而言不是子空间。注意分别  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\cup V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 和  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} + V_{2}"
                    }
                ]
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 5.8 和与交基本计算 (1) 已知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} = L(\\alpha_{1}, \\alpha_{2}, \\alpha_{3}), V_{2} = L(\\beta_{1}, \\beta_{2})"
                    },
                    {
                        "type": "text",
                        "content": " ，其中"
                    }
                ]
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            "content": {
                "math_content": "\\alpha_ {1} = \\left( \\begin{array}{c} 1 \\\\ 2 \\\\ 1 \\\\ 0 \\end{array} \\right), \\alpha_ {2} = \\left( \\begin{array}{c} - 1 \\\\ 1 \\\\ 1 \\\\ 1 \\end{array} \\right), \\alpha_ {3} = \\left( \\begin{array}{c} 0 \\\\ 3 \\\\ 2 \\\\ 1 \\end{array} \\right), \\beta_ {1} = \\left( \\begin{array}{c} 2 \\\\ - 1 \\\\ 0 \\\\ 1 \\end{array} \\right), \\beta_ {2} = \\left( \\begin{array}{c} 1 \\\\ - 1 \\\\ 3 \\\\ 7 \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/b32a88ae884b2d4983cf6b04456176a8fe7ea96d3e1e996bcb64ba248915913b.jpg"
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "求  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} + V_{2}, V_{1} \\cap V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 的基与维数"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 已知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W_{1} = L(\\alpha_{1}, \\alpha_{2}, \\alpha_{3}), W_{2} = L(\\beta_{1}, \\beta_{2})"
                    },
                    {
                        "type": "text",
                        "content": "，求  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W_{1} \\cap W_{2}, W_{1} + W_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 的基与维数。"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\alpha_ {1} = \\left( \\begin{array}{r} 1 \\\\ 2 \\\\ - 1 \\\\ - 3 \\end{array} \\right), \\alpha_ {2} = \\left( \\begin{array}{r} - 1 \\\\ - 1 \\\\ 2 \\\\ 1 \\end{array} \\right), \\alpha_ {3} = \\left( \\begin{array}{r} - 1 \\\\ - 3 \\\\ 0 \\\\ 5 \\end{array} \\right), \\beta_ {1} = \\left( \\begin{array}{r} - 1 \\\\ 0 \\\\ 4 \\\\ 2 \\end{array} \\right), \\beta_ {2} = \\left( \\begin{array}{r} 0 \\\\ 5 \\\\ 9 \\\\ - 1 4 \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/faabb4c5452764fb8cc5e8d3068c9f2b575f10f5f562a20df8266548794fa7b2.jpg"
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}+V_{2}=L(\\alpha_{1},\\alpha_{2},\\alpha_{3},\\beta_{1},\\beta_{2})"
                    },
                    {
                        "type": "text",
                        "content": " ，因此只需要排成列，做初等变换："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left( \\begin{array}{c c c c c} 1 & - 1 & 0 & 2 & 1 \\\\ 2 & 1 & 3 & - 1 & - 1 \\\\ 1 & 1 & 2 & 0 & 3 \\\\ 0 & 1 & 1 & 1 & 7 \\end{array} \\right) \\to \\left( \\begin{array}{c c c c c} 1 & 0 & 1 & 0 & - 1 \\\\ 0 & 1 & 1 & 0 & 4 \\\\ 0 & 0 & 0 & 1 & 3 \\\\ 0 & 0 & 0 & 0 & 0 \\end{array} \\right) := A",
                "math_type": "latex",
                "image_source": {
                    "path": "images/3f4386bd471dcc5b2e2c3d3f253c17019d3bae07e94244814acdbe4f2c7b36ed.jpg"
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此维数为3，基为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\alpha_{2},\\beta_{1}"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "任取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\in V_1 \\cap V_2"
                    },
                    {
                        "type": "text",
                        "content": "，设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = x_1\\alpha_1 + x_2\\alpha_2 + x_3\\alpha_3 = y_1\\beta_1 + y_2\\beta_2"
                    },
                    {
                        "type": "text",
                        "content": "，因此系数需要满足方程组："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "x _ {1} \\alpha_ {1} + x _ {2} \\alpha_ {2} + x _ {3} \\alpha_ {3} - y _ {1} \\beta_ {1} - y _ {2} \\beta_ {2} = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/d4a68e8b23e3efc03bdbfd12ed84458e93ed88dd2ac01ded5fc2d5bc8754de9e.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{4} = -y_{1}, x_{5} = y_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，即求解  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX = [\\alpha_{1}, \\alpha_{2}, \\alpha_{3}, \\beta_{1}, \\beta_{2}]X = 0"
                    },
                    {
                        "type": "text",
                        "content": "，解得基础解系为"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\eta_ {1} = \\left( \\begin{array}{c} - 1 \\\\ - 1 \\\\ 1 \\\\ 0 \\\\ 0 \\end{array} \\right), \\eta_ {2} = \\left( \\begin{array}{c} 1 \\\\ - 4 \\\\ 0 \\\\ - 3 \\\\ 1 \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/56969c6ee2784a483f4aaf1391f72415eb5517f0980e5c458cd2cb4ef25203f9.jpg"
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                    {
                        "type": "text",
                        "content": "5.1 线性空间"
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                ]
            },
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        }
    ],
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对应  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\cap V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 的基础解系为："
                    }
                ]
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "- \\alpha_ {1} - \\alpha_ {2} + \\alpha_ {3} = (0, 0, 0, 0) ^ {T}, \\alpha_ {1} - 4 \\alpha_ {2} = (5, - 2, - 3, - 4) ^ {T}",
                "math_type": "latex",
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                    "path": "images/419fab008837b540543b36c0d13541c7e1bae008fbeb044eed244c679d3f2e54.jpg"
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "只取非零的，故只能为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k(5, -2, -3, -4)"
                    },
                    {
                        "type": "text",
                        "content": "，维数为1，基即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1} - 4\\alpha_{2}"
                    },
                    {
                        "type": "text",
                        "content": "。"
                    }
                ]
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        {
            "type": "paragraph",
            "content": {
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                    {
                        "type": "text",
                        "content": "练习 5.9 和与并的区别 (1) 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}, V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "  是 P 上线性空间 V 的两个非平凡子空间，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists \\alpha \\in V"
                    },
                    {
                        "type": "text",
                        "content": "  使得   "
                    },
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                        "type": "equation_inline",
                        "content": "\\alpha \\notin V_{1}, \\alpha \\notin V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " \n\n(2)   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}, \\cdots, V_{s}"
                    },
                    {
                        "type": "text",
                        "content": "  是 V 的 s 个非平凡子空间，证明：V 中至少有一个向量不属于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}, \\cdots, V_{s}"
                    },
                    {
                        "type": "text",
                        "content": "  中的任意一个。"
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                ]
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                        "type": "text",
                        "content": "(3)  "
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                    {
                        "type": "equation_inline",
                        "content": "V_{1}, V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 是两个子空间，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} + V_{2} = V_{1} \\cup V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\subseteq V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 或者  "
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                        "type": "equation_inline",
                        "content": "V_{2} \\subseteq V_{1}"
                    }
                ]
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        {
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                        "type": "text",
                        "content": "证明 (1) 由于  "
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                    {
                        "type": "equation_inline",
                        "content": "V_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 非平凡， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists \\alpha \\notin V_{1}"
                    },
                    {
                        "type": "text",
                        "content": "，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\notin V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，则结论成立；否则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\in V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 非平凡，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists \\beta \\notin V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta \\notin V_{1}"
                    },
                    {
                        "type": "text",
                        "content": "，则结论成立；否则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta \\in V_{1}"
                    },
                    {
                        "type": "text",
                        "content": "，此时有"
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                "math_content": "\\alpha \\notin V _ {1}, \\beta \\in V _ {1}, \\alpha \\in V _ {2}, \\beta \\notin V _ {2}",
                "math_type": "latex",
                "image_source": {
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            "content": {
                "paragraph_content": [
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                        "type": "text",
                        "content": "因此  "
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                        "content": "\\gamma = \\alpha +\\beta \\notin V_1,\\gamma \\notin V_2"
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                        "content": "(2) 用归纳法"
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                        "type": "text",
                        "content": "(3) 右推左显然。左推右：设  "
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                    {
                        "type": "equation_inline",
                        "content": "V_{1} + V_{2} = V_{1} \\cup V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，但  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\notin V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists \\alpha_{1} \\in V_{1}, \\alpha_{1} \\notin V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，但"
                    }
                ]
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                "math_content": "\\forall \\alpha_ {2} \\in V _ {2}, \\alpha = \\alpha_ {1} + \\alpha_ {2} \\in V _ {1} + V _ {2}",
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                    "path": "images/fc8f5ccfb45346a6d12876f18cd3a010daf4e985d0976e31f13b827ae01c74d0.jpg"
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} + V_{2} = V_{1}\\cup V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha"
                    },
                    {
                        "type": "text",
                        "content": " 至少属于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1},V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 中的一个，显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\notin V_2"
                    },
                    {
                        "type": "text",
                        "content": " (根据加法封闭)，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\in V_{1}"
                    },
                    {
                        "type": "text",
                        "content": " ，故"
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            "content": {
                "math_content": "\\alpha_ {2} = \\alpha - \\alpha_ {1} \\in V _ {1}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/77ac8246199a51e306b618c664aa6e66f99aa3b6f4ea32a16302d1da85ba72d3.jpg"
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "综上得到  "
                    },
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                        "content": "V_{2} \\subseteq V_{1}"
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                        "content": "定理 5.6 (维数公式)"
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                        "type": "text",
                        "content": "设  "
                    },
                    {
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                        "content": "V_{1}, V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 是有限维线性空间  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 的两个子空间，则"
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                ]
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            "content": {
                "math_content": "\\dim V _ {1} + \\dim V _ {2} = \\dim (V _ {1} + V _ {2}) + \\dim (V _ {1} \\cap V _ {2})",
                "math_type": "latex",
                "image_source": {
                    "path": "images/cc8ae98a6235391e02e1cf83f633813d30774e44169143872c49b322a8e9f579.jpg"
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设小扩大。设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\cap V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "  的基为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}, \\cdots, \\alpha_{s}"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}, V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "  的基分别为："
                    }
                ]
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            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} \\alpha_ {1}, \\dots , \\alpha_ {s}, \\beta_ {s + 1}, \\dots , \\beta_ {m} \\\\ \\alpha_ {1}, \\dots , \\alpha_ {s}, \\gamma_ {s + 1}, \\dots , \\gamma_ {n} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f8ee15a002e996b220be87a8229f7994e6f857c8b8f5a07b78a91d908f977aa2.jpg"
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            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} + V_{2} = L(\\alpha_{1}, \\cdots, \\alpha_{s}, \\beta_{s+1}, \\cdots, \\beta_{m}, \\gamma_{s+1}, \\cdots, \\gamma_{n})"
                    },
                    {
                        "type": "text",
                        "content": " ，下面证明上面的向量全部线性无关。不妨设"
                    }
                ]
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        {
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            "content": {
                "math_content": "k _ {1} \\alpha_ {1} + \\dots + k _ {s} \\alpha_ {s} + k _ {s + 1} \\beta_ {s + 1} + \\dots + k _ {n} \\beta_ {n} + l _ {s + 1} \\gamma_ {s + 1} + \\dots + l _ {n} \\gamma_ {n} = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/2e68f1591b0148cd0ad1cfea16ea793dfea9db4dcf1eb35f925dc0493dd870b2.jpg"
                }
            },
            "bbox": [
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "l_{i}\\gamma_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 的部分移到右侧，此时左侧线性组合属于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}"
                    },
                    {
                        "type": "text",
                        "content": "，右侧属于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，因此都属于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\cap V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "。根据左侧得到："
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "k _ {1} \\alpha_ {1} + \\dots + k _ {s} \\alpha_ {s} + k _ {s + 1} \\beta_ {s + 1} + \\dots + k _ {n} \\beta_ {n} \\in V _ {1} \\cap V _ {2} \\Rightarrow k _ {s + 1} = k _ {s + 2} = \\dots = k _ {n} = 0 \\Rightarrow k _ {1} = \\dots = k _ {s} = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/d50e8d24f48c3924d58606f6d0c04e3e7f276327327445a27011c8bb09cad8c1.jpg"
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            },
            "bbox": [
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此左侧系数全为0，同理右侧系数也全为0"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习5.10维数公式相关(1)经典：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}, V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 是有限维线性空间  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 的子空间，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\dim (V_1 + V_2) = \\dim (V_1 \\cap V_2) + 1"
                    },
                    {
                        "type": "text",
                        "content": " ，证明：要么  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\subseteq V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，要么  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{2} \\subseteq V_{1}"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 根据维数公式得到"
                    }
                ]
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            "content": {
                "math_content": "\\dim (V _ {1}) + \\dim (V _ {2}) = \\dim (V _ {1} + V _ {2}) + \\dim (V _ {1} \\cap V _ {2}) = 2 \\dim (V _ {1} \\cap V _ {2}) + 1",
                "math_type": "latex",
                "image_source": {
                    "path": "images/5a614ad8230f507b9abebf1fbaf8ca74a2483a620e1d56c3f5f2a27f7c2dfa69.jpg"
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            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "等价于"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left(\\dim V _ {1} - \\dim (V _ {1} \\cap V _ {2})\\right) + \\left(\\dim V _ {2} - \\dim (V _ {1} \\cap V _ {2})\\right) = 1",
                "math_type": "latex",
                "image_source": {
                    "path": "images/2362fc99fc5ffa7c33be87776c53198b62ccd6d8c1199a0c5ccc6893491d9c2d.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于都是整数，只能一个0，一个1，因此结论成立。"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.1 线性空间"
                    }
                ]
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                        "content": "46"
                    }
                ]
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        }
    ],
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                        "content": "5.1.7 直和理论"
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                        "type": "text",
                        "content": "定义 5.8 (直和)"
                    }
                ],
                "level": 2
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            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V_{1}, V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 是线性空间  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 的子空间，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\cap V_{2} = \\{0\\}"
                    },
                    {
                        "type": "text",
                        "content": "，则记  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\oplus V_{2} := V_{1} + V_{2}"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 5.9 (补空间)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = V_{1} \\oplus V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 的补空间。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 补空间一般是不唯一的，因为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{m}"
                    },
                    {
                        "type": "text",
                        "content": "  扩充为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  的方式不唯一。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.7 (直和等价条件)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "W = V_{1} \\oplus V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W"
                    },
                    {
                        "type": "text",
                        "content": " 中任意向量  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = \\alpha_{1} + \\alpha_{2},\\alpha_{1}\\in V_{1},\\alpha_{2}\\in V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 唯一表出"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W"
                    },
                    {
                        "type": "text",
                        "content": " 中零向量被  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}, V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 中向量唯一表出:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "0 = 0 + 0"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\dim W = \\dim V_1 + \\dim V_2"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 5.11 判断直和 (1) 数域 P 上所有 n 阶矩阵组成线性空间   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = M_{n}(P)"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}"
                    },
                    {
                        "type": "text",
                        "content": "  表示所有对称矩阵组成的集合，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "  表示所有反对称矩阵组成的集合，证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}, V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "  是子空间且   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = V_{1} \\oplus V_{2}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 显然关于加法和数乘封闭，因此是子空间。首先证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = V_{1} + V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，由于矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 都可以写为："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\frac {A + A ^ {T}}{2} + \\frac {A ^ {T} - A}{2}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/2139e4088d01d4ce1996acbec9411bda8e79b5d2d6513378e4f6988e89b5bfc5.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "从而   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = V_{1} + V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = V_{1} \\oplus V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " ：可用  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\cap V_{2} = \\{\\}"
                    },
                    {
                        "type": "text",
                        "content": "，此时取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A \\in V_{1} \\cap V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = A^{T} = -A^{T}"
                    },
                    {
                        "type": "text",
                        "content": "，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{T} = A = O"
                    },
                    {
                        "type": "text",
                        "content": "，因此为直和。也可以用  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "O"
                    },
                    {
                        "type": "text",
                        "content": " 分解唯一，设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "O = B + C = B^{T} - C^{T}"
                    },
                    {
                        "type": "text",
                        "content": "，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "O^{T} = B^{T} + C^{T}"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B = C = O"
                    },
                    {
                        "type": "text",
                        "content": "。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.8 (多空间直和等价条件)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V_{1},\\cdots,V_{s}"
                    },
                    {
                        "type": "text",
                        "content": " 是 W 的子空间，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W=V_{1}+V_{2}+\\cdots+V_{s}"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W=V_{1}\\oplus V_{2}\\oplus\\cdots\\oplus V_{s}"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W"
                    },
                    {
                        "type": "text",
                        "content": " 中零向量分解唯一"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W"
                    },
                    {
                        "type": "text",
                        "content": " 中任意向量  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha"
                    },
                    {
                        "type": "text",
                        "content": " 分解唯一"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "V _ {i} \\cap \\sum_ {j \\neq i} V _ {j} = \\{0 \\}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/a9c704c2e3a934ddc716cf733a8f10d71ec6d59f4a88b50d59df774c3e0b8bb2.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\bullet \\dim W = \\dim V _ {1} + \\dots + \\dim V _ {s}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/8b89948b7c13d8704503ad658f1e23d08b063f81f32738d0d1c1f890ece3af02.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (3) 只证明第三条。第三条推直和：若"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "0 = \\alpha_ {1} + \\dots + \\alpha_ {n} \\Rightarrow \\alpha_ {i} = \\alpha_ {1} + \\dots + \\alpha_ {i - 1} + \\alpha_ {i + 1} + \\dots + \\alpha_ {n}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/024fcf5da9b7dd29622a241d52e7282c34a1facb97e44abe63506b0f969528a4.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{i} \\cap \\sum_{j \\neq i} V_{j} = \\{0\\}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{k} = 0, \\forall k = 1, 2, \\cdots, s"
                    },
                    {
                        "type": "text",
                        "content": " ，从而是直和"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "直和推第三条：若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\in V_{i} \\cap \\sum_{j \\neq i} V_{j}"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\in V_{i}"
                    },
                    {
                        "type": "text",
                        "content": "，且"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\alpha = \\alpha_ {1} + \\dots + \\alpha_ {i - 1} + \\alpha_ {i + 1} + \\dots + \\alpha_ {n} \\Rightarrow \\alpha_ {1} + \\dots + \\alpha_ {i - 1} + (- \\alpha) + \\alpha_ {i + 1} + \\dots + \\alpha_ {n} = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/9870a1bd3f6cea6b39d81cfbb9cf5fce41f6fed95a0b8dee89291cd9acf478ed.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\sum_{i=1}^{n} V_i"
                    },
                    {
                        "type": "text",
                        "content": " 为直和，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，即第三条结论成立。"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.1 线性空间"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "47"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.9 (直和合并定理)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = V_{1} \\oplus V_{2}, V_{1} = V_{11} \\oplus V_{12} \\oplus \\cdots \\oplus V_{1s}, V_{2} = V_{21} \\oplus \\cdots \\oplus V_{2t}"
                    },
                    {
                        "type": "text",
                        "content": " ，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "V = V _ {1 1} \\oplus \\dots V _ {1 s} \\oplus V _ {2 1} \\oplus \\dots \\oplus V _ {2 t}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/7994b1ce2f5b32ef9cf1cc6a6cab33081bd41773e04614c8e3a014eb2afbcbab.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 只证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = V_{1} \\oplus V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} = V_{11} \\oplus V_{12}"
                    },
                    {
                        "type": "text",
                        "content": "  时，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = V_{11} \\oplus V_{12} \\oplus V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "首先显然有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = V_{11} + V_{12} + V_2"
                    },
                    {
                        "type": "text",
                        "content": " 。设"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "0 = \\alpha_ {1 1} + \\alpha_ {1 2} + \\alpha_ {2} = (\\alpha_ {1 1} + \\alpha_ {1 2}) + \\alpha_ {2}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/73bae0b69db992a2f8229d7e889d62290593fb8d2571f0ac26ae1b9b479b62ab.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = V_{1} \\oplus V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，因此得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{11} + \\alpha_{12} = \\alpha_{2} = 0"
                    },
                    {
                        "type": "text",
                        "content": "，同理根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} = V_{11} \\oplus V_{12}"
                    },
                    {
                        "type": "text",
                        "content": " 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{11} = \\alpha_{12} = 0"
                    },
                    {
                        "type": "text",
                        "content": "，综上0元素分解唯一，为直和。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.1.8 商空间"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 5.10 (商空间的元素)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "W 是线性空间 V 的子空间，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\in V"
                    },
                    {
                        "type": "text",
                        "content": " ，定义"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\alpha + W := \\{\\alpha + \\omega : \\omega \\in W \\}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/3fb3adb9d6d373895816974f62b48edd9042c9872136d542a88a434953294216.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 5.11 (商空间)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定义  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V / W"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\{\\alpha + W : \\alpha \\in V\\}"
                    },
                    {
                        "type": "text",
                        "content": "，即线性空间子集组成的集合。可以证明其是一个线性空间"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 5.12 (自然映射)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定义  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\eta : V \\to V / W"
                    },
                    {
                        "type": "text",
                        "content": " 为自然映射"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.2 线性映射"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.2.1 线性映射的概念"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 5.13 (线性映射)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V, W"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 上的线性空间， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\varphi: V \\to W"
                    },
                    {
                        "type": "text",
                        "content": " 若满足："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\bullet \\varphi (\\alpha + \\beta) = \\varphi (\\alpha) + \\varphi (\\beta)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/37a354fc0b925e63f2f9701ec1ba1e87ce098a510d0e2f0ebe78ce7cb04cebf0.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\bullet \\varphi (c \\alpha) = c \\varphi (\\alpha)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f1e675f05c227d1ab42122fdcc7d3ada16ed54d4e442bee3812451e20e55a2a2.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\varphi"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W"
                    },
                    {
                        "type": "text",
                        "content": " 的线性映射，将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W"
                    },
                    {
                        "type": "text",
                        "content": " 的线性映射全体记为集合  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathrm{Hom}_{\\mathbb{P}}(V,W)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.2.2 核与像"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "核与像是最常见的不变子空间，因此在进一步研究线性映射之前，要先研究这两个特殊的集合。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 5.14 (核与像)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "U, V"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 上的线性空间， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f: U \\to V"
                    },
                    {
                        "type": "text",
                        "content": "，定义  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\operatorname{Ker}(f) = \\{\\alpha \\in U: f(\\alpha) = 0\\}"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\operatorname{Im}(f) = \\{f(\\alpha): \\alpha \\in U\\}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记像的维度对应着  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\alpha_{1},\\cdots,\\alpha_{n})A"
                    },
                    {
                        "type": "text",
                        "content": " 的秩，核的维度对应着线性方程组AX=0的解空间维数。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.2 线性映射"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "48"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.10 (核像维数公式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f \\in \\text{Hom}(U, V)"
                    },
                    {
                        "type": "text",
                        "content": " 其核与像满足："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\dim \\operatorname{Ker} (f) + \\dim \\operatorname{Im} (f) = \\dim (U)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/a61c5655fc3c4b8f4d3bfa2a4a6d6aefccb750d45518b48ab16b1cbc628aa2cb.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{r}"
                    },
                    {
                        "type": "text",
                        "content": "  是 Kerf 的基,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  是 U 的基, 而   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathrm{Im}f=L(f(\\alpha_{1}),\\cdots,f(\\alpha_{n}))=L(f(\\alpha_{r+1}),\\cdots,f(\\alpha_{n}))"
                    },
                    {
                        "type": "text",
                        "content": " , 下面证明线性无关。假设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{r+1}f(\\alpha_{r+1})+\\cdots+k_{n}f(\\alpha_{n})=0"
                    },
                    {
                        "type": "text",
                        "content": " , 根据线性性可知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(k_{r+1}\\alpha_{r+1}+\\cdots+k_{n}\\alpha_{n})=0"
                    },
                    {
                        "type": "text",
                        "content": " , 即   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{r+1}\\alpha_{r+1}+\\cdots+k_{n}\\alpha_{n}\\in\\mathrm{Ker}(f)"
                    },
                    {
                        "type": "text",
                        "content": " , 根据基的线性无关性, 显然   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{r+1},\\cdots,k_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  均为 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 该定理的证明会在题目中出现，主要方式是先设原空间的子空间的一组基，扩展成原空间的一组基，再作用 f 后与像空间进行联系。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "练习 5.12 核像维数公式推广证明 证明以下核像命题"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(1) 定义域为子空间:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 是有限维线性空间  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 的线性变换,  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 的子空间, 证明:"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\dim (\\mathcal {A} W) + \\dim (\\operatorname{Ker} \\mathcal {A} \\cap W) = \\dim (W)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/97fae0e0adcdb46fda91018b382f7a415bee04966cd09b088abeb27a18ddff79.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 复合映射：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\sigma,\\tau"
                    },
                    {
                        "type": "text",
                        "content": "  是有限维空间   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "U\\to V,V\\to W"
                    },
                    {
                        "type": "text",
                        "content": "  的线性映射，证明："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\dim (\\operatorname{Ker} (\\sigma)) + \\dim (\\operatorname{Im} (\\sigma) \\cap \\operatorname{Ker} (\\tau)) = \\dim (\\operatorname{Ker} (\\tau \\sigma))",
                "math_type": "latex",
                "image_source": {
                    "path": "images/ac0198a0c7fb574ce734f9bc6bf725a9ee15fc6a68542d30993280ebe501dc2e.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{r}"
                    },
                    {
                        "type": "text",
                        "content": "  是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\operatorname{Ker}\\mathcal{A}\\cap W"
                    },
                    {
                        "type": "text",
                        "content": "  的一组基，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  是 W 的一组基，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}W=L(\\mathcal{A}\\alpha_{r+1},\\cdots,\\mathcal{A}\\alpha_{n})"
                    },
                    {
                        "type": "text",
                        "content": " 。考虑 AW 的维数，设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{r+1}\\mathcal{A}\\alpha_{r+1}+\\cdots+k_{n}\\mathcal{A}\\alpha_{n}=0"
                    },
                    {
                        "type": "text",
                        "content": " ，则有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{r+1}\\alpha_{r+1}+\\cdots+k_{n}\\alpha_{n}\\in\\operatorname{Ker}\\mathcal{A}\\cap W"
                    },
                    {
                        "type": "text",
                        "content": " ，因此得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k_{r+1}=\\cdots=k_{n}=0"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 先得找原空间最小的空间，可知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathrm{Ker}(\\sigma) \\subseteq \\mathrm{Ker}(\\tau\\sigma)"
                    },
                    {
                        "type": "text",
                        "content": " ，因此设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}, \\cdots, \\alpha_{r}"
                    },
                    {
                        "type": "text",
                        "content": "  是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathrm{Ker}(\\sigma)"
                    },
                    {
                        "type": "text",
                        "content": "  的基，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}, \\cdots, \\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathrm{Ker}(\\tau\\sigma)"
                    },
                    {
                        "type": "text",
                        "content": "  的基。设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} = L(\\sigma\\alpha_{r+1}, \\cdots, \\sigma\\alpha_{n})"
                    },
                    {
                        "type": "text",
                        "content": " 。显然   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} \\subseteq \\mathrm{Im}(\\sigma) \\cap \\mathrm{Ker}(\\tau)"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "另一侧  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in \\mathrm{Im}(\\sigma) \\cap \\mathrm{Ker}(\\tau)"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists \\beta"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\sigma(\\beta) = \\alpha, \\tau(\\alpha) = 0"
                    },
                    {
                        "type": "text",
                        "content": "，这说明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\tau(\\sigma(\\beta)) = 0"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta \\in \\mathrm{Ker}(\\tau\\sigma)"
                    },
                    {
                        "type": "text",
                        "content": "，即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\in V_1"
                    },
                    {
                        "type": "text",
                        "content": "，最终得出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_1 = \\mathrm{Im}(\\sigma) \\cap \\mathrm{Ker}(\\tau)"
                    },
                    {
                        "type": "text",
                        "content": "。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.11 (核像与单射、满射)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\sigma"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "U\\to W"
                    },
                    {
                        "type": "text",
                        "content": " 的线性映射，则："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\sigma"
                    },
                    {
                        "type": "text",
                        "content": " 是满射当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\operatorname{Im}(\\sigma) = W"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\sigma"
                    },
                    {
                        "type": "text",
                        "content": " 是单射当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\operatorname{Ker}(\\sigma) = \\{\\mathbf{0}\\}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 推论：有限维线性空间上线性变换(非映射)是单射当且仅当其是满射(一般单射更好证)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.12 (核、像任意性)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P"
                    },
                    {
                        "type": "text",
                        "content": " 上  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 维线性空间， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}, V_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 是其子空间， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\dim V_{1} + \\dim V_{2} = n"
                    },
                    {
                        "type": "text",
                        "content": "，则存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 上的线性变换  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\operatorname{Ker} \\mathcal{A} = V_{1}, \\operatorname{Im} \\mathcal{A} = V_{2}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}"
                    },
                    {
                        "type": "text",
                        "content": "  的一组基为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{r}"
                    },
                    {
                        "type": "text",
                        "content": " ，扩充为 V 的一组基   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{r+1},\\cdots,\\beta_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "  的一组基，构造映射   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}(k_{1}\\alpha_{1}+\\cdots+k_{n}\\alpha_{n})=k_{r+1}\\beta_{r+1}+\\cdots+k_{n}\\beta_{n}"
                    },
                    {
                        "type": "text",
                        "content": " 。可证明 A 是线性变换，且核、像满足条件。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 5.13 计算核与像 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A: P^{4} \\rightarrow P^{3}"
                    },
                    {
                        "type": "text",
                        "content": " ，求 KerA, ImA"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\mathcal {A} \\left( \\begin{array}{c} x _ {1} \\\\ x _ {2} \\\\ x _ {3} \\\\ x _ {4} \\end{array} \\right) = \\left( \\begin{array}{c} x _ {1} - 3 x _ {2} + x _ {3} - 2 x _ {4} \\\\ - x _ {1} - 1 1 x _ {2} + 2 x _ {3} - 5 x _ {4} \\\\ 3 x _ {1} + 5 x _ {2} + x _ {4} \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/9ff1849d7feeec383b1a8e35e95c67764b74e6baeb0fb97d474e54a9918a285c.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\operatorname{Ker} \\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 相当于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX = 0"
                    },
                    {
                        "type": "text",
                        "content": " 的解空间， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\operatorname{Im} \\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 相当于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的列空间"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.2 线性映射"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "49"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.2.3 幂等变换的核与像"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "一般而言核与像虽然有维数关系，但不具备直和关系(可能相交也可能和不为全空间)，但是幂等变换是一个特例。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 5.15 (幂等变换)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若线性变换  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}^2 = \\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": "，则称其为幂等变换。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.13 (幂等变换核像空间分解)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 上的幂等变换，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "V = \\operatorname{Ker} \\mathcal {A} \\oplus \\operatorname{Im} \\mathcal {A}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/09cdf6f1ed6fe12cee14a14a0069305f440e24244649584b70e99e6d50de4102.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 先证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 是核与像的和:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in V"
                    },
                    {
                        "type": "text",
                        "content": ", 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = \\alpha - \\mathcal{A}\\alpha + \\mathcal{A}\\alpha"
                    },
                    {
                        "type": "text",
                        "content": ", 其中  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}\\alpha \\in \\operatorname{Im}\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": "。而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}(\\alpha - \\mathcal{A}\\alpha) = \\mathcal{A}\\alpha - \\mathcal{A}^2\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": ", 因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha - \\mathcal{A}\\alpha \\in \\operatorname{Ker}\\mathcal{A}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 证明直和:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in \\operatorname{Ker} \\mathcal{A} \\cap \\operatorname{Im} \\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": ", 则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A} \\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": " 且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists \\beta, \\mathcal{A} \\beta = \\alpha"
                    },
                    {
                        "type": "text",
                        "content": ", 两侧同时作用  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": ", 左侧得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}^2 \\beta = \\mathcal{A} \\beta = \\alpha"
                    },
                    {
                        "type": "text",
                        "content": ", 右侧  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A} \\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": ", 因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 5.14 经典幂等变换核像题 A, B 为 P 上线性空间中的幂等变换，证明："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A},\\mathcal{B}"
                    },
                    {
                        "type": "text",
                        "content": " 有相同的像当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{AB} = \\mathcal{B},\\mathcal{BA} = \\mathcal{A}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A},\\mathcal{B}"
                    },
                    {
                        "type": "text",
                        "content": " 有相同的核当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{AB} = \\mathcal{A},\\mathcal{BA} = \\mathcal{B}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 左推右:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in V"
                    },
                    {
                        "type": "text",
                        "content": " 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{B} \\alpha = \\in \\operatorname{Im}(\\mathcal{B})"
                    },
                    {
                        "type": "text",
                        "content": ", 根据条件即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{B} \\alpha \\in \\operatorname{Im} \\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": ", 因此根据幂等变换投影的性质 ( "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 不改变  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\operatorname{Im} \\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 中的向量) 得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\mathcal {A} (\\mathcal {B} \\alpha) = \\mathcal {B} \\alpha \\Rightarrow \\mathcal {A B} = \\mathcal {B}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/545d70d3114bbc8f8ea8c3ae159fff458f439917e23143bd2b0989f7df0c09ba.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "同理得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{BA} = \\mathcal{A}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "右推左： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\gamma \\in \\operatorname{Im}(\\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": "，有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}\\gamma = \\gamma"
                    },
                    {
                        "type": "text",
                        "content": "，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{B}\\mathcal{A}\\gamma = \\mathcal{A}\\gamma = \\gamma"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{B}\\gamma = \\gamma"
                    },
                    {
                        "type": "text",
                        "content": "，即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\gamma \\in \\operatorname{Im}(\\mathcal{B})"
                    },
                    {
                        "type": "text",
                        "content": "，另一侧同理"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 左推右：由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = \\operatorname{Im}(\\mathcal{A}) \\oplus \\operatorname{Ker}(\\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": " ，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in V"
                    },
                    {
                        "type": "text",
                        "content": "  有"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\alpha = \\alpha_ {1} + \\alpha_ {2}, \\alpha_ {1} \\in \\operatorname{Im} (\\mathcal {A}), \\alpha_ {2} \\in \\operatorname{Ker} \\mathcal {A}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/bf99ad0ec912fdb52e5145a6cf2e3b12a6b52fc47df6c1103048fb2f59371d62.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\alpha - \\alpha_{1} \\in \\operatorname{Ker} \\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha - \\alpha_{1} \\in \\operatorname{Ker} \\mathcal{B}"
                    },
                    {
                        "type": "text",
                        "content": "，故  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{B}(\\alpha - \\alpha_{1}) = 0"
                    },
                    {
                        "type": "text",
                        "content": "，即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{B} \\alpha = \\mathcal{B} \\alpha_{1}"
                    },
                    {
                        "type": "text",
                        "content": "。而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A} \\alpha = \\alpha_{1}"
                    },
                    {
                        "type": "text",
                        "content": "，得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(\\mathcal {B A}) \\alpha = \\mathcal {B} \\alpha_ {1} = \\mathcal {B} \\alpha",
                "math_type": "latex",
                "image_source": {
                    "path": "images/87a8277a95ffde1d7786e48dee28f8da70223d5b9244b20381ae371146b2d501.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此得出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{BA} = \\mathcal{B}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "右推左"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
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                        "content": "假设  "
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                        "content": "\\mathcal{A}"
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                        "content": " 为线性空间  "
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                        "content": "5.3 线性变换"
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                        "content": " 是数域  "
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                        "content": " 是  "
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                        "content": " 内全体线性变换组成集合  "
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                        "content": "。"
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                        "content": "练习 5.15 线性变换基本练习 (1) 若   "
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                        "content": "T_{1}, \\cdots, T_{m}"
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                        "content": "证明 (1) 令  "
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                        "content": "V_{ij} = \\{x \\in V : T_i x = T_j x\\}"
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                        "content": "，则显然  "
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                        "content": "\\forall T_i, T_j, \\exists \\beta"
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                        "content": " 使得  "
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                        "content": " 不属于任一非平凡子空间，故  "
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                        "type": "equation_inline",
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                        "content": " 不属于  "
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                        "content": "V_{ij}, \\forall i, j"
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                        "content": "，因此  "
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                        "content": "T_1 \\alpha, \\cdots, T_m \\alpha"
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                        "content": "线性变换 T 可逆当且仅当"
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                        "content": "-  "
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                    {
                        "type": "equation_inline",
                        "content": "T\\epsilon_{1}, \\cdots, T\\epsilon_{n}"
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                        "content": " 线性无关"
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                        "type": "equation_inline",
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                        "content": " 是单射"
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                        "content": "-  "
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                        "type": "equation_inline",
                        "content": "T"
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                        "type": "text",
                        "content": " 是满射"
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                        "content": "证明 (1) 从矩阵角度看即可"
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                        "type": "text",
                        "content": "(2) 左推右：若  "
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                        "content": "T\\alpha_{1} = T\\alpha_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，则两侧作用  "
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                        "content": "T^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 得到  "
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                        "content": "\\alpha_{1} = \\alpha_{2}"
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                        "content": "右推左：下证   "
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                        "content": "T\\epsilon_{1},\\cdots,T\\epsilon_{n}"
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                    {
                        "type": "text",
                        "content": "  线性无关，若   "
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                    {
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                        "content": "k_{1}T\\epsilon_{1}+\\cdots+k_{n}T\\epsilon_{n}=0"
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                "math_content": "T (k _ {1} \\epsilon_ {1} + \\dots + k _ {n} \\epsilon_ {n}) = 0",
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                        "type": "text",
                        "content": "由于 T 是单射，得到   "
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                        "content": "k_{1}\\epsilon_{1}+\\cdots+k_{n}\\epsilon_{n}=0"
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                    {
                        "type": "text",
                        "content": " ，因此   "
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                        "content": "k_{1}=\\cdots=k_{n}=0"
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                        "content": "(3) 左推右：设  "
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                        "content": " 可逆，则  "
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                        "content": "，存在  "
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                        "content": "T^{-1}\\alpha"
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                        "content": "，而  "
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                        "content": "\\alpha = T(T^{-1}\\alpha)"
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                        "content": "，因此是满射"
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                        "content": "右推左：由于满射，对任一一组基   "
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                        "content": "\\eta_{1},\\cdots,\\eta_{n}"
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                    {
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                        "content": " ，总存在   "
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                "math_content": "T \\epsilon_ {1} = \\eta_ {1}, \\dots , T \\epsilon_ {n} = \\eta_ {n}",
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                        "type": "text",
                        "content": "下面证明   "
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                        "content": "\\epsilon_{1},\\cdots,\\epsilon_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  线性无关，设   "
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                        "content": "k_{1}\\epsilon_{1}+\\cdots+k_{n}\\epsilon_{n}=0"
                    },
                    {
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                        "content": " ，则"
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                "math_content": "T (k _ {1} \\epsilon_ {1} + \\dots + k _ {n} \\epsilon_ {n}) = k _ {1} \\eta_ {1} + \\dots + k _ {n} \\eta_ {n} = 0",
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                        "content": "这说明   "
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                        "content": "k_{1}=\\cdots=k_{n}=0"
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                        "content": " ，因此可逆。"
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                        "content": "\\mathcal{A}"
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                    {
                        "type": "text",
                        "content": " 是  "
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                        "content": " 内的线性变换， "
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                        "content": "V = L(\\epsilon_1, \\epsilon_2, \\dots, \\epsilon_n)"
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                        "content": "，若  "
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                        "content": "(\\mathcal{A}(\\epsilon_1), \\dots, \\mathcal{A}(\\epsilon_n)) = (\\epsilon_1, \\dots, \\epsilon_n)A"
                    },
                    {
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                        "content": "，则  "
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                    },
                    {
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                        "content": " 称为  "
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                        "content": " 在  "
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                        "content": "\\epsilon_1, \\dots, \\epsilon_n"
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                        "content": " 基下的矩阵。"
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                        "content": "笔记 计算线性变换在基下的矩阵：根据   "
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                        "content": "(\\mathcal{A}\\epsilon_{1},\\cdots,\\mathcal{A}\\epsilon_{n})=(\\epsilon_{1},\\cdots,\\epsilon_{n})A"
                    },
                    {
                        "type": "text",
                        "content": "  推出"
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                "math_content": "A = (\\epsilon_ {1}, \\dots , \\epsilon_ {n}) ^ {- 1} (\\mathcal {A} \\epsilon_ {1}, \\dots , \\mathcal {A} \\epsilon_ {n})",
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "可以用矩阵求逆方式求解。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.3 线性变换"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "51"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习5.16线性变换在基下的矩阵已知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = (a_{ij}),i,j = 1,2,3"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_1,\\epsilon_2,\\epsilon_3"
                    },
                    {
                        "type": "text",
                        "content": " 下的矩阵，计算"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_3, \\epsilon_2, \\epsilon_1"
                    },
                    {
                        "type": "text",
                        "content": " 下的矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_1, k\\epsilon_2, \\epsilon_3"
                    },
                    {
                        "type": "text",
                        "content": " 下的矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_1 + \\epsilon_2, \\epsilon_2, \\epsilon_3"
                    },
                    {
                        "type": "text",
                        "content": " 下的矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解 根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T(\\epsilon_1, \\epsilon_2, \\epsilon_3) = (\\epsilon_1, \\epsilon_2, \\epsilon_3)A"
                    },
                    {
                        "type": "text",
                        "content": "，得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} T \\epsilon_ {1} = a _ {1 1} \\epsilon_ {1} + a _ {2 1} \\epsilon_ {2} + a _ {3 1} \\epsilon_ {3} \\\\ T \\epsilon_ {2} = a _ {1 2} \\epsilon_ {1} + a _ {2 2} \\epsilon_ {2} + a _ {3 2} \\epsilon_ {3} \\\\ T \\epsilon_ {3} = a _ {1 3} \\epsilon_ {1} + a _ {2 3} \\epsilon_ {2} + a _ {3 3} \\epsilon_ {3} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/dbfa17042bb12d546d7933a7ea5781816ac3ec1002494af47072000bf51aec0d.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(1) 重新排列得到:"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} T \\epsilon_ {3} = a _ {3 3} \\epsilon_ {3} + a _ {2 3} \\epsilon_ {2} + a _ {1 3} \\epsilon_ {1} \\\\ T \\epsilon_ {2} = a _ {3 2} \\epsilon_ {3} + a _ {2 2} \\epsilon_ {2} + a _ {1 2} \\epsilon_ {1} \\\\ T \\epsilon_ {1} = a _ {3 1} \\epsilon_ {3} + a _ {2 1} \\epsilon_ {2} + a _ {1 1} \\epsilon_ {1} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/c546c08d89048573d024824eded76f0c06b73132ee43c8d3bcceb206826e53a5.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "得到矩阵为"
                    }
                ]
            },
            "bbox": [
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                179,
                353
            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{l l l} a _ {3 3} & a _ {3 2} & a _ {3 1} \\\\ a _ {2 3} & a _ {2 2} & a _ {2 1} \\\\ a _ {1 3} & a _ {1 2} & a _ {1 1} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/61e602a6589197a43f3f0f43bfb192cad75f04e606a053c7cb0e9292e14df2cc.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)(3) 同理列开即可"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 5.17 幂零变换的矩阵 T 是 n 维线性空间 V 上的线性变换，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T^{n-1} \\neq O, T^n = O"
                    },
                    {
                        "type": "text",
                        "content": " ，证明：T 在某组基下的矩阵为"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c c c c} 0 & 0 & \\dots & 0 & 0 \\\\ 1 & 0 & \\dots & 0 & 0 \\\\ \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 0 & 0 & \\dots & 1 & 0 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/5e6c279fa4f3756d79c497c5bb7235e491bedcb1815bfc992464bdb8252b8021.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 首先   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha, T\\alpha, \\cdots, T^{n-1}\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "  均线性无关，因此它们是 V 的一组基，T 在这组基下的矩阵即目标。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.16 (线性变换后的坐标)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "  在   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\epsilon_{1},\\cdots,\\epsilon_{n})"
                    },
                    {
                        "type": "text",
                        "content": "  基下的坐标为 X，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "  在这组基下的坐标为 AX。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}\\alpha=\\mathcal{A}(x_{1}\\epsilon_{1}+\\cdots+x_{n}\\epsilon_{n})=\\mathcal{A}(\\epsilon_{1},\\cdots,\\epsilon_{n})X=(\\epsilon_{1},\\cdots,\\epsilon_{n})AX,"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.3.3 线性变换在不同基下的矩阵：相似关系"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.17 (线性变换在不同基下的矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{i},\\eta_{i}(i=1,\\cdots,n)"
                    },
                    {
                        "type": "text",
                        "content": " 是 V 两组基，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\eta_{1},\\cdots,\\eta_{n})=(\\epsilon_{1},\\cdots,\\epsilon_{n})T"
                    },
                    {
                        "type": "text",
                        "content": " 。若 A 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{i},\\eta_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 基下的矩阵分别为 A,B，则有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B=T^{-1}AT"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 由线性变换矩阵定义，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}(\\epsilon_{1},\\cdots,\\epsilon_{n})=(\\epsilon_{1},\\cdots,\\epsilon_{n})A,\\mathcal{A}(\\eta_{1},\\cdots,\\eta_{n})=(\\eta_{1},\\cdots,\\eta_{n})B"
                    },
                    {
                        "type": "text",
                        "content": " 。而   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}(\\eta_{1},\\cdots,\\eta_{n})=\\mathcal{A}(\\epsilon_{1},\\cdots,\\epsilon_{n})T=(\\epsilon_{1},\\cdots,\\epsilon_{n})AT"
                    },
                    {
                        "type": "text",
                        "content": " ，且   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}(\\eta_{1},\\cdots,\\eta_{n})=(\\epsilon_{1},\\cdots,\\epsilon_{n})TB"
                    },
                    {
                        "type": "text",
                        "content": "  上面两式取等号即可。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 记忆时根据   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\eta_{1},\\cdots,\\eta_{n})=(\\epsilon_{1},\\cdots,\\epsilon_{n})T"
                    },
                    {
                        "type": "text",
                        "content": " ，将   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\eta,\\epsilon"
                    },
                    {
                        "type": "text",
                        "content": "  基直接换成对应的矩阵 B,A，有 T 的一边添上   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T^{-1}"
                    },
                    {
                        "type": "text",
                        "content": "  即可，即   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B=T^{-1}AT"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.3 线性变换"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "52"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 5.18 线性变换在不同基下的矩阵 给定下述两组基，线性变换 T 满足   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T \\epsilon_{i} = \\eta_{i}"
                    },
                    {
                        "type": "text",
                        "content": " ,"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} \\epsilon_ {1} = (1, 0, 1) ^ {T}, \\epsilon_ {2} = (2, 1, 0) ^ {T}, \\epsilon_ {3} = (1, 1, 1) ^ {T} \\\\ \\eta_ {1} = (1, 2, - 1) ^ {T}, \\eta_ {2} = (2, 2, - 1) ^ {T}, \\eta_ {3} = (2, - 1, - 1) ^ {T} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/299c369b083f321c469ffcd8bbc92998f2447e1da6d180297863e2b3e0365d8c.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(1) 写出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1}, \\epsilon_{2}, \\epsilon_{3}"
                    },
                    {
                        "type": "text",
                        "content": " 到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\eta_{1}, \\eta_{2}, \\eta_{3}"
                    },
                    {
                        "type": "text",
                        "content": " 的过渡矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 写出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_1, \\epsilon_2, \\epsilon_3"
                    },
                    {
                        "type": "text",
                        "content": " 下的矩阵"
                    }
                ]
            },
            "bbox": [
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                186
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 写出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\eta_1, \\eta_2, \\eta_3"
                    },
                    {
                        "type": "text",
                        "content": " 下的矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "e_{1}, e_{2}, e_{3}"
                    },
                    {
                        "type": "text",
                        "content": " 下的矩阵为"
                    }
                ]
            },
            "bbox": [
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                223
            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(\\eta_ {1}, \\eta_ {2}, \\eta_ {3}) =",
                "math_type": "latex",
                "image_source": {
                    "path": "images/d307f4f7f5c5153b5b10d15c93ab7194d412a045c9c770ebec8e33ebb0f7d89a.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.18 (线性变换构造的等价类)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 相似的充要条件为它们是一个线性变换在两组基下的矩阵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.3.4 相似矩阵的性质"
                    }
                ],
                "level": 2
            },
            "bbox": [
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                368
            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.19 (相似矩阵的基本性质)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "K"
                    },
                    {
                        "type": "text",
                        "content": " 上矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 相似，且满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B = P^{-1}AP"
                    },
                    {
                        "type": "text",
                        "content": " ，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "•   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = r(B)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "•   "
                    },
                    {
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                        "content": "A^{m} \\sim B^{m}"
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                "paragraph_content": [
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                        "type": "text",
                        "content": "-  "
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                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 为多项式，则  "
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                    {
                        "type": "equation_inline",
                        "content": "f(A) \\sim f(B)"
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                        "type": "text",
                        "content": "若  "
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                    {
                        "type": "equation_inline",
                        "content": "A_{1} \\sim B_{1}, A_{2} \\sim B_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，不一定有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1} + A_{2} \\sim B_{1} + B_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，要求过渡矩阵相同才行"
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            "content": {
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                    {
                        "type": "text",
                        "content": "证明 (1)  "
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                    {
                        "type": "equation_inline",
                        "content": "r(A)=r(B)"
                    },
                    {
                        "type": "text",
                        "content": " ：因为   "
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                    {
                        "type": "equation_inline",
                        "content": "B=P^{-1}AP"
                    },
                    {
                        "type": "text",
                        "content": " ，而 P 满秩，等价于一系列初等变换，故不改变秩。"
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        {
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            "content": {
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                    {
                        "type": "text",
                        "content": "5.4 特征值、相似对角化"
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                        "type": "text",
                        "content": "5.4.1 特征值与特征向量"
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                        "type": "text",
                        "content": "定义 5.18 (特征值与特征向量)"
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                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 是数域  "
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                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 上线性空间  "
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                        "type": "equation_inline",
                        "content": "V"
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                    {
                        "type": "text",
                        "content": " 内的一个线性变换，若对于  "
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                        "type": "equation_inline",
                        "content": "\\lambda \\in \\mathbb{P}"
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                    {
                        "type": "text",
                        "content": "， "
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                    {
                        "type": "equation_inline",
                        "content": "\\exists \\xi \\neq 0"
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                    {
                        "type": "text",
                        "content": "，使得  "
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                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}\\xi = \\lambda \\xi"
                    },
                    {
                        "type": "text",
                        "content": "，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
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                    {
                        "type": "text",
                        "content": " 是  "
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                    {
                        "type": "text",
                        "content": " 的一个特征值， "
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                    {
                        "type": "equation_inline",
                        "content": "\\xi"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
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                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 属于  "
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                        "type": "equation_inline",
                        "content": "\\lambda"
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                        "content": " 的特征向量。"
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        {
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            "content": {
                "title_content": [
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                        "type": "text",
                        "content": "定义 5.19 (特征子空间)"
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                "level": 2
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        {
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                        "type": "text",
                        "content": "对于   "
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                    {
                        "type": "equation_inline",
                        "content": "\\lambda\\in P"
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                    {
                        "type": "text",
                        "content": " ，定义其特征子空间为所有满足   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A\\xi=\\lambda\\xi"
                    },
                    {
                        "type": "text",
                        "content": "  的特征向量   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\xi"
                    },
                    {
                        "type": "text",
                        "content": "  所组成的集合   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{\\lambda}"
                    },
                    {
                        "type": "text",
                        "content": " 。"
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                ]
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理 5.1 (特征子空间的性质)"
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                "level": 2
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                        "type": "text",
                        "content": "特征子空间  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{\\lambda}"
                    },
                    {
                        "type": "text",
                        "content": " 有如下性质："
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                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{\\lambda}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 的子空间。"
                    }
                ]
            },
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        {
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                        "type": "text",
                        "content": "- 对于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda \\in \\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": ",  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{\\lambda} \\neq \\{\\mathbf{0}\\}"
                    },
                    {
                        "type": "text",
                        "content": " 的充要条件为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|\\lambda E - A| = 0"
                    },
                    {
                        "type": "text",
                        "content": ", 其中  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 在任意基下的矩阵。"
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                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 只需要证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{\\lambda}"
                    },
                    {
                        "type": "text",
                        "content": " 对  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 内的数乘和加法封闭即可。"
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                ]
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 对于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda \\alpha = \\mathcal{A}\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "，可以取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 一组基，若  "
                    },
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                        "type": "equation_inline",
                        "content": "\\alpha"
                    },
                    {
                        "type": "text",
                        "content": " 在这组基下的坐标为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X"
                    },
                    {
                        "type": "text",
                        "content": "。则等价于求解  "
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                    {
                        "type": "equation_inline",
                        "content": "\\lambda X = AX"
                    },
                    {
                        "type": "text",
                        "content": "，其中  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 在这组基下的矩阵。等价于求解  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\lambda E - A)X = 0"
                    },
                    {
                        "type": "text",
                        "content": "，而线性方程组有非零解的充要条件为系数矩阵行列式非零。"
                    }
                ]
            },
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        },
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            "type": "page_number",
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                "page_number_content": [
                    {
                        "type": "text",
                        "content": "53"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.4 特征值、相似对角化"
                    }
                ]
            },
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    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 5.20 (特征多项式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于 P 上的矩阵 A，特征多项式定义为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(\\lambda) = |\\lambda E - A|"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.20 (相似矩阵的特征多项式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "相似矩阵有相同特征多项式。即对于同一个线性变换，特征多项式是不变的。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B = T^{-1}AT"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|\\lambda E - B| = |\\lambda E - T^{-1}AT| = |T^{-1}||\\lambda E - A||T| = |\\lambda E - A|"
                    },
                    {
                        "type": "text",
                        "content": "。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.21 (特征值基本性质)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1},\\cdots,\\lambda_{n}"
                    },
                    {
                        "type": "text",
                        "content": " 是矩阵 A 的特征值。则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\sum\\lambda_{i}=\\mathrm{tr}(A),\\quad\\prod\\lambda_{i}=|A|"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 用行列式展开与韦达定理证明。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 5.19 特征值与特征向量的基本练习 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1}, \\lambda_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 是 T 两个不同的特征值， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}, \\alpha_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 分别是对应的特征向量，证明： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1} + \\alpha_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 不是 T 的特征向量"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T"
                    },
                    {
                        "type": "text",
                        "content": " 是可逆线性变换，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值证明 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T\\alpha_{1} = \\lambda_{1}\\alpha_{1}, T\\alpha_{2} = \\lambda_{2}\\alpha_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，因此"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "T (\\alpha_ {1} + \\alpha_ {2}) = \\lambda_ {1} \\alpha_ {1} + \\lambda_ {2} \\alpha_ {2}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/3380aaa54ed450b0e900743b1e3844a73d383a42155b77e931efd8b82f9412a3.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1} + \\alpha_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 是特征向量， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 是特征值，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T(\\alpha_{1} + \\alpha_{2}) = \\lambda (\\alpha_{1} + \\alpha_{2})"
                    },
                    {
                        "type": "text",
                        "content": " ，得到"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(\\lambda - \\lambda_ {1}) \\alpha_ {1} + (\\lambda - \\lambda_ {2}) \\alpha_ {2} = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/d798c221b2ca60295de2a88040f8fad12467da35d88449649ece0704bc6203d8.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda - \\lambda_{1} = \\lambda - \\lambda_{2} = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，这与假设矛盾。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T\\alpha = \\lambda \\alpha"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = \\lambda T^{-1}(\\alpha)"
                    },
                    {
                        "type": "text",
                        "content": "，即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T^{-1}(\\alpha) = \\lambda^{-1}\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.4.2 特征值与特征向量进一步结论"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.22 (AB 与 BA 的特征值)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V, V'"
                    },
                    {
                        "type": "text",
                        "content": " 是 K 上的线性空间，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A} \\in \\mathrm{Hom}(V, V')"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{B} \\in \\mathrm{Hom}(V', V)"
                    },
                    {
                        "type": "text",
                        "content": " （矩阵同理），则\n- AB 与 BA 有相同非零特征值"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\xi"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB"
                    },
                    {
                        "type": "text",
                        "content": " 属于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_0"
                    },
                    {
                        "type": "text",
                        "content": " 的特征向量，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B\\xi"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "BA"
                    },
                    {
                        "type": "text",
                        "content": " 属于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_0"
                    },
                    {
                        "type": "text",
                        "content": " 的一个特征向量"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{0}"
                    },
                    {
                        "type": "text",
                        "content": "  是 AB 非零特征值，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists\\xi\\in V^{\\prime}"
                    },
                    {
                        "type": "text",
                        "content": "  使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\mathcal{AB})\\xi=\\lambda_{0}\\xi"
                    },
                    {
                        "type": "text",
                        "content": " ，两侧同时作用 B 得到："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\mathcal {B} (\\mathcal {A B}) \\xi = \\mathcal {B} (\\lambda_ {0} \\xi) \\Rightarrow (\\mathcal {B A}) (\\mathcal {B} \\xi) = \\lambda_ {0} (\\mathcal {B} \\xi)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/29c61bcc69c301c8fd7fe49714492dabaf5c9c5a9f720f31797fca54d15829e1.jpg"
                }
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.23 (Sylvester 公式)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A_{m\\times n},B_{n\\times m}"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{AB}(\\lambda),f_{BA}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB,BA"
                    },
                    {
                        "type": "text",
                        "content": " 的特征多项式，则"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\lambda^ {n} f _ {A B} (\\lambda) = \\lambda^ {m} f _ {B A} (\\lambda)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/6da2b14ba7cea52f39655b57e5de3f6abb9bdabe622d6590fc29f60cb52bfea4.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 证明：设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda \\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " ，根据"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c} E _ {n} & 0 \\\\ - A & E _ {m} \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {n} & \\frac {1}{\\lambda} B \\\\ A & E _ {m} \\end{array} \\right] = \\left[ \\begin{array}{c c} E _ {n} & \\frac {1}{\\lambda} B \\\\ O & E _ {m} - \\frac {1}{\\lambda} A B \\end{array} \\right] \\Rightarrow | H | = \\left| \\begin{array}{c c} E _ {n} & \\frac {1}{\\lambda} B \\\\ O & E _ {m} - \\frac {1}{\\lambda} A B \\end{array} \\right| = \\left(\\frac {1}{\\lambda}\\right) ^ {m} | \\lambda E _ {m} - A B |",
                "math_type": "latex",
                "image_source": {
                    "path": "images/97d8843adf9f0b88257fa09ed98154c2f241f49731cb8c36784a8c963d5776c2.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.4 特征值、相似对角化"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "54"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "同理根据"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c} E _ {n} & \\frac {1}{\\lambda} B \\\\ A & E _ {m} \\end{array} \\right] \\cdot \\left[ \\begin{array}{c c} E _ {n} & 0 \\\\ - A & E _ {m} \\end{array} \\right] = \\left[ \\begin{array}{c c} E _ {n} - \\frac {1}{\\lambda} A B & \\frac {1}{\\lambda} B \\\\ 0 & E _ {m} \\end{array} \\right] \\Rightarrow | H | = \\left(\\frac {1}{\\lambda}\\right) ^ {n} | \\lambda E _ {n} - B A |",
                "math_type": "latex",
                "image_source": {
                    "path": "images/ff72c6229869aeb82e9bd842ae525b749f091f8a018439478980aef5735ab0f0.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "根据   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|H|"
                    },
                    {
                        "type": "text",
                        "content": "  的值即可推出结论。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习5.20相关结论(1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 是两个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵，证明： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB + A"
                    },
                    {
                        "type": "text",
                        "content": " 和  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "BA + A"
                    },
                    {
                        "type": "text",
                        "content": " 有相同的特征多项式"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 计算下面矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "C = \\left[ \\begin{array}{c c c c} a _ {1} b _ {1} & a _ {1} b _ {2} & \\dots & a _ {1} b _ {n} \\\\ a _ {2} b _ {1} & a _ {2} b _ {2} & \\dots & a _ {2} b _ {n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a _ {n} b _ {1} & a _ {n} b _ {2} & \\dots & a _ {n} b _ {n} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/cebc7600c2ba43583fa4e43da3d53a192c870aacc569a9d573f2ad25b1a19810.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 特例：已知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = (a_{1}, \\cdots, a_{n})"
                    },
                    {
                        "type": "text",
                        "content": " ，证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{T}A"
                    },
                    {
                        "type": "text",
                        "content": "  的特征值为 0 和   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{1}^{2} + \\cdots + a_{n}^{2}"
                    },
                    {
                        "type": "text",
                        "content": " ，这里 0 是 n - 1 重根证明 (1) 由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB + A = A(B + E)"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "BA + A = (B + E)A"
                    },
                    {
                        "type": "text",
                        "content": " ，因此根据前面定理可知。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 令  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = (a_{1}, \\cdots, a_{n})^{T}"
                    },
                    {
                        "type": "text",
                        "content": ",  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B = (b_{1}, \\cdots, b_{n})"
                    },
                    {
                        "type": "text",
                        "content": ", 则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C = AB"
                    },
                    {
                        "type": "text",
                        "content": ", 而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "BA = \\sum_{i=1}^{n} a_{i} b_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 为一阶矩阵, 其特征值为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\sum_{i=1}^{n} a_{i} b_{i}"
                    },
                    {
                        "type": "text",
                        "content": ",  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB"
                    },
                    {
                        "type": "text",
                        "content": " 和  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "BA"
                    },
                    {
                        "type": "text",
                        "content": " 特征多项式相差  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda^{n-1}"
                    },
                    {
                        "type": "text",
                        "content": ", 因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C = AB"
                    },
                    {
                        "type": "text",
                        "content": " 的特征多项式为"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\lambda^ {n - 1} (\\lambda - \\sum_ {i = 1} ^ {n} a _ {i} b _ {i})",
                "math_type": "latex",
                "image_source": {
                    "path": "images/3eff4838651a014df7e047e4a336881743259dba04731ab41e6783a177b49969.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 即 (2) 的特例"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.24 (可逆的特征值表达)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 可逆当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值全不为0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 用   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\left|A\\right|=\\lambda_{1}\\cdots\\lambda_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  即可"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.25 (逆矩阵的特征值)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "n 阶可逆矩阵 A 的特征值为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1},\\cdots,\\lambda_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": "  的全部特征值为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1}^{-1},\\cdots,\\lambda_{n}^{-1}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 用 A 相似于 Jordan 阵 J，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J^{-1}"
                    },
                    {
                        "type": "text",
                        "content": "  为上三角且对角为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1}^{-1},\\cdots,\\lambda_{n}^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": "  的特征值也为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1}^{-1},\\cdots,\\lambda_{n}^{-1}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.4.3 相似的进一步研究"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.26 (相似的矩阵分解)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 相似当且仅当存在方阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P, Q"
                    },
                    {
                        "type": "text",
                        "content": "，使得"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = P Q, B = Q P",
                "math_type": "latex",
                "image_source": {
                    "path": "images/9ba9f29f5ce56f831937c6d2684762c0adcd374c88ca38d9b1a48b65e69a28f6.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "这里  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P, Q"
                    },
                    {
                        "type": "text",
                        "content": " 至少有一个可逆"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 相似，故存在可逆  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^{-1}AP = B"
                    },
                    {
                        "type": "text",
                        "content": " ，令  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q = P^{-1}A"
                    },
                    {
                        "type": "text",
                        "content": " ，则得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = PQ, B = QP"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.27 (一些相似结论)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵，则"
                    }
                ]
            },
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            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB"
                    },
                    {
                        "type": "text",
                        "content": " 与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "BA"
                    },
                    {
                        "type": "text",
                        "content": ": 若方阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 可逆, 则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB"
                    },
                    {
                        "type": "text",
                        "content": " 与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "BA"
                    },
                    {
                        "type": "text",
                        "content": " 相似"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B^{-1}"
                    },
                    {
                        "type": "text",
                        "content": ": 若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 可逆, 则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 相似"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 多项式：任一多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": "，有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(A), f(B)"
                    },
                    {
                        "type": "text",
                        "content": " 相似"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.4 特征值、相似对角化"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "55"
                    }
                ]
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        }
    ],
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 伴随矩阵：若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 相似，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{*}"
                    },
                    {
                        "type": "text",
                        "content": " 与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B^{*}"
                    },
                    {
                        "type": "text",
                        "content": " 相似"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "BA = (A^{-1}A)(BA) = A^{-1}(AB)A"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(P^{-1}AP)^{-1} = B^{-1}"
                    },
                    {
                        "type": "text",
                        "content": ", 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^{-1}A^{-1}P = B^{-1}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 显然"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(4) B ^ {*} = (P ^ {- 1} A P) ^ {*} = P ^ {*} A ^ {*} (P ^ {- 1}) ^ {*} = P ^ {*} A ^ {*} (P ^ {*}) ^ {- 1}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/801abe53157e00a904316e1a1eec29169a297ed77580e1b67aa47c85b885d3db.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "推论5.2(伴随矩阵的特征值)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若矩阵 A 的特征值为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1},\\cdots,\\lambda_{m}"
                    },
                    {
                        "type": "text",
                        "content": " ，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{*}"
                    },
                    {
                        "type": "text",
                        "content": "  的特征值为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\prod_{i\\neq j}\\lambda_{i}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 考虑  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上相似于 Jordan 阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J"
                    },
                    {
                        "type": "text",
                        "content": "，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J^{*}"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值易算，相似时特征值不变，因此可算出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{*}"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.4.4 相似对角化"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 5.21 (相似对角化)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 内的线性变换，如果存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 的一组基使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 在该基下的矩阵为对角阵，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 可对角化。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定义 5.22 (代数重数、几何重数)"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "代数重数指特征值  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 在特征多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 中的重数。几何重数指  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 对应特征子空间的维数。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理 5.2 (代数重数与几何重数关系)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对每个特征值  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": "，其代数重数大于等于几何重数。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理 5.3 (不同特征值的特征向量)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "A 属于不同特征值的特征向量线性无关。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1},\\cdots,\\lambda_{s}"
                    },
                    {
                        "type": "text",
                        "content": "  是 A 的 s 个不同的特征值，对应特征向量   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\xi_{1},\\cdots,\\xi_{s}"
                    },
                    {
                        "type": "text",
                        "content": " 。用归纳法，假设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\xi_{1},\\cdots,\\xi_{k-1}"
                    },
                    {
                        "type": "text",
                        "content": "  线性无关，若"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "c _ {1} \\xi_ {1} + \\dots + c _ {k} \\xi_ {k} = 0\\tag{5.1}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/6c845711128b98b326f7c740a925355644d5bcc6c06bf30a13dc38fa0cb36bc7.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "两侧作用 A 得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "c_{1}\\lambda_{1}\\xi_{1}+\\cdots+c_{k}\\lambda_{k}\\xi_{k}=0"
                    },
                    {
                        "type": "text",
                        "content": " ，用该式子减去   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{k}"
                    },
                    {
                        "type": "text",
                        "content": "  倍的 (5.1) 得到"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "c _ {1} \\left(\\lambda_ {1} - \\lambda_ {k}\\right) \\xi_ {1} + \\dots + c _ {k - 1} \\left(\\lambda_ {k - 1} - \\lambda_ {k}\\right) \\xi_ {k - 1} = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/ec3e4b651683bf39cc2581c230b577e3d703d99853acbc23f0c45591246d87b4.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  两两不相等，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "c_{1}=\\cdots=c_{k-1}=0"
                    },
                    {
                        "type": "text",
                        "content": " ，即特征向量彼此线性无关。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.28 (可对角化条件)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若 V 是 n 维线性空间，则 A 可对角化的条件按照递进关系如下排列："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 充要条件 1:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 个线性无关特征向量。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 充分条件：若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 个不同特征值，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 可对角化"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 充要条件2：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 是所有特征子空间的直和  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = \\bigoplus_{i=1}^{m} V_i"
                    },
                    {
                        "type": "text",
                        "content": " (特征子空间维数可能没有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_i"
                    },
                    {
                        "type": "text",
                        "content": " 重数高)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 充要条件3：所有特征值的代数重数等于几何重数"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 充要条件4：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\dim V = \\sum_{i=1}^{m} \\dim(V_i)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 充要条件 5(最小多项式): 最小多项式都是一次幂的乘积"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.4 特征值、相似对角化"
                    }
                ]
            },
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "56"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 左推右：由对角化形式，右侧乘开  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 个基向量均是特征向量，显然基向量都线性无关。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(A \\eta_ {1}, A \\eta_ {2}, \\dots , A \\eta_ {n}) = (\\eta_ {1}, \\eta_ {2}, \\dots , \\eta_ {n}) \\left[ \\begin{array}{c c c} \\lambda_ {1} & & \\\\ & \\lambda_ {2} & \\\\ & \\ddots & \\lambda_ {n} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/01aac27a1be8b5957b9289192e9b1ada52ff1b43ff58a7cc58f2a3e06c923a7b.jpg"
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            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "右推左：n 个线性无关特征向量当一组基即可。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 根据引理5.3可知"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "后面三个都根据第一个可以推出"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习5.21对角化计算(1)计算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\left( \\begin{array}{ll}3 & 4\\\\ 5 & 2 \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值，并计算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q^{-1}AQ"
                    },
                    {
                        "type": "text",
                        "content": " 是对角矩阵"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.29 (相关矩阵的可对角化性质)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 可对角化，则"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "1.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{*}"
                    },
                    {
                        "type": "text",
                        "content": " 可对角化"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "2.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 为多项式，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(A)"
                    },
                    {
                        "type": "text",
                        "content": " 可对角化"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "3. 若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 可逆，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 可对角化"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^{-1}AP = \\text{diag}\\{\\lambda_1, \\cdots, \\lambda_n\\}"
                    },
                    {
                        "type": "text",
                        "content": " ，两侧取伴随得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^*A^*(P^*)^{-1} = P^{-1}A^*P = \\text{diag}\\{\\prod_{i \\neq 1} \\lambda_i, \\prod_{i \\neq 2} \\lambda_i, \\cdots, \\prod_{i \\neq n} \\lambda_i\\}"
                    },
                    {
                        "type": "text",
                        "content": " 。其中伴随转逆用了   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1} = \\frac{1}{|A|}A^*"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2,3) 显然"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.4.5 相似上三角型"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.30 (相似上三角型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "任意 n 阶矩阵 A 都相似于一个上三角矩阵，且上三角对角为 A 的特征值。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 n=1 时显然成立。假设 n-1 时成立，则取   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1}"
                    },
                    {
                        "type": "text",
                        "content": "  为 A 对应   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1}"
                    },
                    {
                        "type": "text",
                        "content": "  的特征向量，构造矩阵   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P_{1}=[x_{1},\\cdots,x_{n}](x_{2}\\sim x_{n})"
                    },
                    {
                        "type": "text",
                        "content": "  无要求)，可以得到："
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A P _ {1} = (x _ {1}, \\dots , x _ {n}) \\left[ \\begin{array}{c c c c} \\lambda_ {1} & b _ {1 2} & \\dots & b _ {1 n} \\\\ 0 & b _ {2 2} & \\dots & b _ {2 n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ 0 & b _ {n 1} & \\dots & b _ {n n} \\end{array} \\right] \\Rightarrow P _ {1} ^ {- 1} A P _ {1} = \\left[ \\begin{array}{c c c c} \\lambda_ {1} & b _ {1 2} & \\dots & b _ {1 n} \\\\ 0 & b _ {2 2} & \\dots & b _ {2 n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ 0 & b _ {n 1} & \\dots & b _ {n n} \\end{array} \\right]. = \\left[ \\begin{array}{c c} \\lambda_ {1} & * \\\\ 0 & A _ {1} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/d71cedfd0e671442a26ad27ee4bd24139d2031d34d0601e247c8083f54dc0198.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "根据归纳假设可知对   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}"
                    },
                    {
                        "type": "text",
                        "content": "  有可逆矩阵 Q 使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q^{-1}A_{1}Q"
                    },
                    {
                        "type": "text",
                        "content": "  为上三角且对角为特征值，取   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P_{2}=\\left[\\begin{array}{cc}1 & \\mathbf{0}^{T}\\\\ \\mathbf{0} & Q\\end{array}\\right]"
                    },
                    {
                        "type": "text",
                        "content": " ，取 S=P₁P₂，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "S^{-1}AS"
                    },
                    {
                        "type": "text",
                        "content": "  即所求上三角"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.5 最小多项式"
                    }
                ]
            },
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "57"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.5 最小多项式"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.5.1 最小多项式与基本性质"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 5.23 (零化多项式与最小多项式)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是数域  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "K"
                    },
                    {
                        "type": "text",
                        "content": " 上的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "K"
                    },
                    {
                        "type": "text",
                        "content": " 上的多项式，"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 零化多项式：若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(A) = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的一个零化多项式。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 最小多项式：将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的次数最低首一零化多项式称为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的最小多项式"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.31 (最小多项式的性质)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "最小多项式有以下性质："
                    }
                ]
            },
            "bbox": [
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                290
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 唯一性：最小多项式是唯一的"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 整除性：最小多项式整除任意零化多项式"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 分块对角：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\text{diag}\\{A_1, \\cdots, A_s\\}"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m_A(\\lambda) = [m_1(\\lambda), \\cdots, m_s(\\lambda)]"
                    },
                    {
                        "type": "text",
                        "content": " (最小公倍数)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "相似：相似矩阵有相同最小多项式"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 假设有两个不同最小多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m_{1}(\\lambda), m_{2}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": "，则令  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "h(\\lambda) = m_{1}(\\lambda) - m_{2}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": "，有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "h(A) = m_{1}(A) - m_{2}(A) = 0"
                    },
                    {
                        "type": "text",
                        "content": "，显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "h(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 是最小多项式，矛盾。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 对任意零化多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": "，其可以表示为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = q(x)m(x) + r(x)"
                    },
                    {
                        "type": "text",
                        "content": "，代入  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = A"
                    },
                    {
                        "type": "text",
                        "content": " 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = 0"
                    },
                    {
                        "type": "text",
                        "content": "，其只能为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(x) = 0"
                    },
                    {
                        "type": "text",
                        "content": "，否则与最小多项式矛盾，因此整除。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m(A)=\\mathrm{diag}\\{m(A_{1}),\\cdots,m(A_{s})\\}"
                    },
                    {
                        "type": "text",
                        "content": " ，最大公因式即可满足所有分块均为 0。"
                    }
                ]
            },
            "bbox": [
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                722,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(4) 若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B = P^{-1}AP"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(B) = f(P^{-1}AP) = P^{-1}f(A)P"
                    },
                    {
                        "type": "text",
                        "content": "，因此可知最小多项式相同。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.5.2 Hamilton-Cayley 定理"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 5.32 (Hamilton-Cayley 定理)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "K"
                    },
                    {
                        "type": "text",
                        "content": " 上的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶矩阵， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(\\lambda) = |\\lambda E - A|"
                    },
                    {
                        "type": "text",
                        "content": " 即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征多项式，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的零化多项式。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": "  是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E - A"
                    },
                    {
                        "type": "text",
                        "content": "  的伴随矩阵，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B(\\lambda)(\\lambda E - A) = f(\\lambda)E"
                    },
                    {
                        "type": "text",
                        "content": " ，由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": "  元素含   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1 \\sim \\lambda^{n-1}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此可以写为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B(\\lambda) = \\lambda^{n-1}B_{0} + \\cdots + B_{n-1}"
                    },
                    {
                        "type": "text",
                        "content": " ，从而得到："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "B (\\lambda) (\\lambda E - A) = \\lambda^ {n} B _ {0} + \\lambda^ {n - 1} (B _ {1} - B _ {0} A) + \\dots + \\lambda (B _ {n - 1} - B _ {n - 2} A) - B _ {n - 1} A",
                "math_type": "latex",
                "image_source": {
                    "path": "images/b71919a787a8a10c4dbcb0d13f8e7e4e167cbff428983564e46cf104f26850e5.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "根据   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B(\\lambda)(\\lambda E - A) = f(\\lambda)E = \\lambda^{n}E + a_{1}\\lambda^{n-1}E + \\cdots + a_{n-1}\\lambda E + a_{n}E"
                    },
                    {
                        "type": "text",
                        "content": "  进行比较得到："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} B _ {0} = E \\\\ B _ {1} - B _ {0} A = a _ {1} E \\\\ \\vdots \\\\ B _ {n - 1} - B _ {n - 2} A = a _ {n - 1} E \\\\ - B _ {n - 1} A = a _ {n} E \\end{array} \\right. \\Rightarrow \\left\\{ \\begin{array}{l} B _ {0} A ^ {n} = A ^ {n} \\\\ B _ {1} A ^ {n - 1} - B _ {0} A ^ {n} = a _ {1} A ^ {n - 1} \\\\ \\vdots \\\\ B _ {n - 1} A - B _ {n - 2} A ^ {2} = a _ {n - 2} A \\\\ - B _ {n - 1} A = a _ {n} E \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/4a7d84bef1b97c72f201f0f15dcd1e436c00579ef217f8e1c23d476e89d417ba.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "两端分别相加，左侧为 0，因此右侧为 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 使用 Jordan 阵证明：分块看，显然每个   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  对应的块的阶数小于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  的代数重数，每个分块都是零，因此是零化多项式。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.5 最小多项式"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "58"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "推论 5.3 (Hamilton-Cayley 定理推论)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "Hamilton-Cayley 定理有以下推论:"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 最小多项式与特征多项式同根(重数可以不同)。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "5.5.3 最小多项式与相似对角化"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定理 5.33 (最小多项式与相似对角化)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 上的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 可对角化当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的最小多项式在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{P}"
                    },
                    {
                        "type": "text",
                        "content": " 上可分解为互素一次因子乘积。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 5.22 最小多项式判断可对角化 (1) 幂等矩阵一定可对角化"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 对合矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^2 = E"
                    },
                    {
                        "type": "text",
                        "content": " 一定可对角化"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 幂零矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^l = O"
                    },
                    {
                        "type": "text",
                        "content": " 不一定可对角化"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A(A-E)=O"
                    },
                    {
                        "type": "text",
                        "content": " ，一定为一次因子"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A + E)(A - E) = O"
                    },
                    {
                        "type": "text",
                        "content": " ，一定可对角化"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 例如 Jordan 块"
                    }
                ]
            },
            "bbox": [
                122,
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "5.5 最小多项式"
                    }
                ]
            },
            "bbox": [
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                55,
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                68
            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "59"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "第 6 章 Jordan 标准型"
                    }
                ],
                "level": 1
            },
            "bbox": [
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                650,
                111
            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.1 Jordan 标准形：空间分解法"
                    }
                ],
                "level": 2
            },
            "bbox": [
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                442,
                165
            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.1.1 空间第二分解定理：循环不变子空间分解"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 6.1 (幂零线性变换)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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                304,
                231
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "A 是 V 内的线性变换，若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists m"
                    },
                    {
                        "type": "text",
                        "content": "  使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{m}=0"
                    },
                    {
                        "type": "text",
                        "content": " ，则 A 是幂零线性变换。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理6.1(幂零基)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "A 是幂零变换，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall\\alpha\\in V"
                    },
                    {
                        "type": "text",
                        "content": "  若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{k-1}\\alpha\\neq0,A^{k}\\alpha=0"
                    },
                    {
                        "type": "text",
                        "content": " ，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha,A\\alpha,\\cdots,A^{k-1}\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "  线性无关。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{0}\\alpha + a_{1}A\\alpha + \\cdots + a_{k-1}A^{k-1}\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，若从左至右第一个非零的是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{i}"
                    },
                    {
                        "type": "text",
                        "content": " ，即   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{i}A^{i}\\alpha + \\cdots + a_{k}A^{k-1}\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，两侧同时作用   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{k-1-i}"
                    },
                    {
                        "type": "text",
                        "content": " ，上式变为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{i}A^{k-1}\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，而   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{k-1}\\alpha \\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " ，故   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{i} = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，与假设矛盾。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 6.2 (循环不变子空间)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall\\alpha\\in V,\\exists k"
                    },
                    {
                        "type": "text",
                        "content": "  使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{k}=0,A^{k-1}\\neq0"
                    },
                    {
                        "type": "text",
                        "content": "  ，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "I(\\alpha):=L(\\alpha,A\\alpha,\\cdots,A^{k-1}\\alpha)"
                    },
                    {
                        "type": "text",
                        "content": "  称为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "  生成的 A 的循环不变子空间。"
                    }
                ]
            },
            "bbox": [
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                432
            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.1 (循环不变子空间分解)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于 V 内的幂零变换 A，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists v_{1},\\cdots,v_{n}\\in V,\\ m_{1},\\cdots,m_{n}\\in N"
                    },
                    {
                        "type": "text",
                        "content": "  使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V=I(v_{1})\\oplus I(v_{2})\\oplus\\cdots\\oplus I(v_{n})"
                    },
                    {
                        "type": "text",
                        "content": " 。其中   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "I(v_{i})=L(v_{i},\\mathcal{A}v_{i}\\cdots,\\mathcal{A}^{m_{i}}v_{i})"
                    },
                    {
                        "type": "text",
                        "content": " ，且   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{m_{i}+1}v_{i}=0"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 采用数归与核像分解。(1) 若 dimV = 1，则显然可分解。(2) 若 dimV > 1，则用归纳法。由于 A 不是单射 (因为将不同的向量均映射到 0)，因此 A 不是满射 (因为线性映射是满射当且仅当是双射)，因此 A 的像维数低于 dimV。对 ImA 用归纳假设，得到 ImA 的一组基   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{m_{1}}v_{1}, \\cdots, Av_{1}, v_{1}, \\cdots, A^{m_{n}}v_{n}, \\cdots, v_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，由于上述向量组是 ImA 的一组基，对   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall j"
                    },
                    {
                        "type": "text",
                        "content": "  存在 V 中向量   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u_{j}"
                    },
                    {
                        "type": "text",
                        "content": "  使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "v_{j} = Au_{j}"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "下面说明下面的向量组线性无关:"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\mathcal {A} ^ {m _ {1} + 1} u _ {1}, \\dots , \\mathcal {A} u _ {1}, u _ {1}, \\dots , \\mathcal {A} ^ {m _ {n} + 1} u _ {n}, \\dots , u _ {n}\\tag{6.1}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/86d61f6da7d98fb478de1755847ae414d8a5f7155d183cfd753be0bc4435687d.jpg"
                }
            },
            "bbox": [
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                640
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "原因: 可设上述向量的一个线性组合为 0，两侧同时作用 A，则仅有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{m_{1}+1}u_{1},\\cdots,A^{m_{n}+1}u_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  的系数可以非零（因为它们被作用后为 0）。去除其他项后，相当于设上面这组向量的某个线性组合等于 0，由于它们也是 ImA 基的一部分，彼此线性无关，所以它们的系数也必须为零。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "于是将 (6.1) 扩成 V 的一组基   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{m_{1}+1}u_{1},\\cdots,u_{1},\\cdots,A^{m_{n}+1}u_{n},\\cdots,u_{n},w_{1},\\cdots,w_{p}"
                    },
                    {
                        "type": "text",
                        "content": " ，由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Aw_{j}\\in ImA"
                    },
                    {
                        "type": "text",
                        "content": " ，因此存在 (6.1) 张成空间中元素   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{j}"
                    },
                    {
                        "type": "text",
                        "content": " ，使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Aw_{j}=Ax_{j}"
                    },
                    {
                        "type": "text",
                        "content": " 。令   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u_{n+j}=w_{j}-x_{j}"
                    },
                    {
                        "type": "text",
                        "content": " ，则有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Au_{n+j}=0"
                    },
                    {
                        "type": "text",
                        "content": " ，从而 (6.1) 配上   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u_{n+1},\\cdots,u_{n+p}"
                    },
                    {
                        "type": "text",
                        "content": "  是所需 V 的一组基。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.1.2 核像升链与根子空间"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 上的线性变换，定义  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_0 = \\{0\\}, M_i = \\mathrm{Ker}\\mathcal{A}^i"
                    },
                    {
                        "type": "text",
                        "content": " ， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "N_0 = V, N_i = \\mathrm{Im}(\\mathcal{A}^i)"
                    },
                    {
                        "type": "text",
                        "content": "，下面来研究它们的性质。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理 6.2 (核像空间升链)"
                    }
                ],
                "level": 2
            },
            "bbox": [
                122,
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                304,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "上述空间有：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\{0\\}=M_{0}\\subseteq M_{1}\\subseteq M_{2}\\subseteq\\cdots,\\quad V=N_{0}\\supseteq N_{1}\\supseteq N_{2}\\supseteq\\cdots"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 对  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in M_i"
                    },
                    {
                        "type": "text",
                        "content": "，由定义有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}^i\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": "，显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}^{i+1}\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_i \\subseteq M_{i+1}"
                    },
                    {
                        "type": "text",
                        "content": "。"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 对于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\in N_i"
                    },
                    {
                        "type": "text",
                        "content": "，由定义  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists \\beta \\in V"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = \\mathcal{A}^i\\beta"
                    },
                    {
                        "type": "text",
                        "content": "，从而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = \\mathcal{A}^{i-1}(\\mathcal{A}\\beta) \\in N_{i-1}"
                    },
                    {
                        "type": "text",
                        "content": "。由  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha"
                    },
                    {
                        "type": "text",
                        "content": " 任意性可知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "N_i \\subseteq N_{i-1}"
                    },
                    {
                        "type": "text",
                        "content": "。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理 6.3 (核空间增长)"
                    }
                ],
                "level": 2
            },
            "bbox": [
                124,
                111,
                285,
                126
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "核空间   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  的增长满足以下性质："
                    }
                ]
            },
            "bbox": [
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                359,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{k} = M_{k + 1}"
                    },
                    {
                        "type": "text",
                        "content": " ，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{k} = M_{k + 1} = M_{k + 2} = \\dots"
                    }
                ]
            },
            "bbox": [
                139,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若 dimV = n，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{n} = M_{n+1} = M_{n+2} = \\cdots"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
                139,
                168,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 显然有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{k+1} \\subseteq M_{k+2}"
                    },
                    {
                        "type": "text",
                        "content": ", 只需要证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{k+2} \\subseteq M_{k+1}"
                    },
                    {
                        "type": "text",
                        "content": "。对于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in M_{k+2}"
                    },
                    {
                        "type": "text",
                        "content": ", 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}^{k+2} \\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": ", 即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}^{k+1}(\\mathcal{A} \\alpha) = 0"
                    },
                    {
                        "type": "text",
                        "content": ", 因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A} \\alpha \\in M_{k+1} = M_k"
                    },
                    {
                        "type": "text",
                        "content": ", 即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}^{k+1} \\alpha = 0 \\to \\alpha \\in M_{k+1}"
                    },
                    {
                        "type": "text",
                        "content": ", 由任意性可知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{k+2} \\subseteq M_{k+1}"
                    },
                    {
                        "type": "text",
                        "content": ", 故  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{k+1} = M_{k+2}"
                    },
                    {
                        "type": "text",
                        "content": ", 后续类似。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 只需要证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{n} = M_{n+1}"
                    },
                    {
                        "type": "text",
                        "content": " ，假设不然，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\{0\\} = M_{0} \\subset M_{1} \\subset \\cdots \\subset M_{n} \\subset M_{n+1}, dimM_{n+1} > n"
                    },
                    {
                        "type": "text",
                        "content": " ，与   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{n+1} \\subseteq V"
                    },
                    {
                        "type": "text",
                        "content": "  矛盾。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.2 (幂次变换核像分解)"
                    }
                ],
                "level": 2
            },
            "bbox": [
                122,
                303,
                336,
                318
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V = M_{n} \\oplus N_{n}"
                    },
                    {
                        "type": "text",
                        "content": ", 其中  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n = dimV"
                    },
                    {
                        "type": "text",
                        "content": "。"
                    }
                ]
            },
            "bbox": [
                115,
                323,
                361,
                338
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 显然由核像性质有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "dimM_{n} + dimN_{n} = n"
                    },
                    {
                        "type": "text",
                        "content": " ，只需证   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{n} \\cap N_{n} = \\{0\\}"
                    },
                    {
                        "type": "text",
                        "content": " 。  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall v \\in M_{n} \\cap N_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{n}v = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，且   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists u \\in V"
                    },
                    {
                        "type": "text",
                        "content": "  使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "v = A^{n}u"
                    },
                    {
                        "type": "text",
                        "content": " ，从而   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{2n}u = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，而   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{2n} = M_{2n-1} = \\cdots = M_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，故   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{n}u = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，从而 v = 0。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 6.3 (广义特征向量与根子空间 (广义特征子空间))"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 广义特征向量：线性变换  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 关于特征值  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 的广义特征向量定义为满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists j \\in \\mathbb{N}^{+}, (\\mathcal{A} - \\lambda E)^{j}v = 0"
                    },
                    {
                        "type": "text",
                        "content": " 的所有向量。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 根子空间：线性变换  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 关于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 的所有广义特征向量加上零向量组成的空间称为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 关于特征值  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 的根子空间，记作  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G(\\lambda, \\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": "。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.3 (根子空间的具体表达)"
                    }
                ],
                "level": 2
            },
            "bbox": [
                122,
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                354,
                533
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "根子空间   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G(\\lambda, \\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": "  满足   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G(\\lambda, \\mathcal{A}) = \\operatorname{Ker}(\\mathcal{A} - \\lambda E)^{dimV}"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
                114,
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                505,
                554
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 右推左：若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "v \\in Ker(\\mathcal{A} - \\lambda E)^{dimV}"
                    },
                    {
                        "type": "text",
                        "content": "，则由  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G(\\lambda, \\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": " 定义可知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "v \\in G(\\lambda, \\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": "。左推右： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "v \\in G(\\lambda, \\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": " 即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists j \\in \\mathbb{N}"
                    },
                    {
                        "type": "text",
                        "content": " s.t.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\mathcal{A} - \\lambda E)^j v = 0"
                    },
                    {
                        "type": "text",
                        "content": "，而前面已证明这样的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "v"
                    },
                    {
                        "type": "text",
                        "content": " 必然满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "v \\in Ker(\\mathcal{A} - \\lambda E)^{dimV}"
                    },
                    {
                        "type": "text",
                        "content": "。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 为了知识体系的完整，这里补充一下：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  根子空间的次数最低压缩为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  对应 Jordan 块的最高阶数，即最小多项式中的   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  的阶数。其他书也常用特征多项式的代数重数表示 (题目中会遇到)，上面引理的 dimV 明显放缩过大。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.1.3 空间第一分解定理：根子空间分解"
                    }
                ],
                "level": 2
            },
            "bbox": [
                92,
                686,
                453,
                703
            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.4 (根子空间的不变性)"
                    }
                ],
                "level": 2
            },
            "bbox": [
                121,
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                337,
                733
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "根子空间   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G(\\lambda, \\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": "  是 A 不变的。"
                    }
                ]
            },
            "bbox": [
                114,
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                349,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 由定理6.3可知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G(\\lambda, \\mathcal{A}) = Ker(\\mathcal{A} - \\lambda E)^n"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in G(\\lambda, \\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": " 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\mathcal{A} - \\lambda E)^n \\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": "，显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\mathcal{A} - \\lambda E)^n \\mathcal{A} \\alpha = \\mathcal{A}((\\mathcal{A} - \\lambda E)^n \\alpha) = 0"
                    },
                    {
                        "type": "text",
                        "content": "，故  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A} \\alpha \\in G(\\lambda, \\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": "。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.5 (空间第一分解定理：根子空间分解)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若 V 是复线性空间，A 是线性变换，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1},\\cdots,\\lambda_{m}"
                    },
                    {
                        "type": "text",
                        "content": "  是其特征值，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V=G(\\lambda_{1},\\mathcal{A})\\oplus\\cdots\\oplus G(\\lambda_{m},\\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 用归纳法，(1)n=1 时显然成立 (2) 假设对所有小于 n 维空间均成立，由定理6.2可知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V=G(\\lambda_{1},\\mathcal{A})\\oplus Im(\\mathcal{A}-\\lambda_{1}E)^{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "U=Im(\\mathcal{A}-\\lambda_{1}E)^{n}"
                    },
                    {
                        "type": "text",
                        "content": "  为空，则显然成立。其他情况首先显然 A 在 U 上是不变的 (A 与   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\mathcal{A}-\\lambda_{1}E)"
                    },
                    {
                        "type": "text",
                        "content": "  可交换)，去除   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1}"
                    },
                    {
                        "type": "text",
                        "content": "  所有对应特征向量后，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A|_{U}"
                    },
                    {
                        "type": "text",
                        "content": "  在 U 有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{2},\\cdots,\\lambda_{m}"
                    },
                    {
                        "type": "text",
                        "content": "  特征值，对 U 用归纳法，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "U=G(\\lambda_{2},\\mathcal{A}|_{U})\\oplus\\cdots\\oplus G(\\lambda_{m},\\mathcal{A}|_{U})"
                    },
                    {
                        "type": "text",
                        "content": " ，由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall j\\geq2,G(\\lambda_{j},\\mathcal{A}|_{U})=G(\\lambda_{j},\\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": " ，从而得证。"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "61"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "6.1 Jordan 标准形：空间分解法"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": []
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.1.4 Jordan 标准形：空间分解法"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.6 (幂零变换的 Jordan 标准型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "幂零变换 A 在循环不变子空间分解基下的矩阵如下。称为幂零变换的 Jordan 标准形。"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "J = \\left[ \\begin{array}{c c c c} J _ {1} & & & \\\\ & J _ {2} & & 0 \\\\ 0 & & \\ddots & \\\\ & & & J _ {s} \\end{array} \\right], \\quad J _ {i} = \\left[ \\begin{array}{c c c c} 0 & 1 & & 0 \\\\ & 0 & \\ddots & \\\\ & & \\ddots & 1 \\\\ 0 & & & 0 \\end{array} \\right] _ {n _ {i} \\times n _ {i}}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/e44ffcc13f224a94c4bbc4b8ad08ec68ce68ea82629f7431f82e6e056002ec52.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 代入计算显然可得。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.7 (核限制变换是幂零变换)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall"
                    },
                    {
                        "type": "text",
                        "content": " 线性变换  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": "，令  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{B} = \\mathcal{A} - \\lambda E"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{B}"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{k} = Ker\\mathcal{B}^{k}"
                    },
                    {
                        "type": "text",
                        "content": " 中的限制变换  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\left.\\mathcal{B}\\right|_{M_k}"
                    },
                    {
                        "type": "text",
                        "content": " 是幂零变换。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M_{k} = \\mathrm{Ker}\\mathcal{B}^{k}"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in M_{k}"
                    },
                    {
                        "type": "text",
                        "content": " 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{B}^{k}\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": "，故  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\mathcal{B}|_{M_k})^k = 0"
                    },
                    {
                        "type": "text",
                        "content": "。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.8 (Jordan 标准型：空间分解法)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上线性变换  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 而言，其有在先根子空间分解，根子空间再循环不变子空间分解的基下有以下形式。称为线性变换的 Jordan 标准形。"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "J = \\left[ \\begin{array}{c c c c} J _ {1} & & & 0 \\\\ & J _ {2} & & \\\\ & & \\ddots & \\\\ 0 & & & J _ {s} \\end{array} \\right], \\quad J _ {i} = \\left[ \\begin{array}{c c c c} \\lambda_ {i} & 1 & & 0 \\\\ & \\lambda_ {i} & \\ddots & \\\\ & & \\ddots & 1 \\\\ 0 & & & \\lambda_ {i} \\end{array} \\right] _ {n _ {i} \\times n _ {i}}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f32f65e3e09f824330400f4f96b090c8e2a8dfacec4d7bd682535e0f00dc008a.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 先由空间分解第一定理将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 分解为根子空间直和，由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A} - \\lambda E"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G(\\lambda, \\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": " 中的限制变换是幂零变换，因此再对  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "G(\\lambda, \\mathcal{A})"
                    },
                    {
                        "type": "text",
                        "content": " 进行循环不变子空间分解，再由幂零变换的 Jordan 标准形可证。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.2 空间分解法：零化多项式角度"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.2.1 根子空间分解：矩阵角度"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.9 (解空间分解)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), f_1(x), f_2(x) \\in P[x]"
                    },
                    {
                        "type": "text",
                        "content": "，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = f_1(x)f_2(x), (f_1(x), f_2(x)) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(A)X = 0, f_1(A)X = 0, f_2(A)X = 0"
                    },
                    {
                        "type": "text",
                        "content": " 的解空间分别为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V, V_1, V_2"
                    },
                    {
                        "type": "text",
                        "content": "，则"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "V = V _ {1} \\oplus V _ {2}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/ff9b52ea6fa85cae6685c037abcd84475ac8f3611bdc2e5e6852c92f463f9ef8.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 由于互素，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists u(x), v(x)"
                    },
                    {
                        "type": "text",
                        "content": "，使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "E = u(A)f_1(A) + v(A)f_2(A)"
                    },
                    {
                        "type": "text",
                        "content": "，从而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in V"
                    },
                    {
                        "type": "text",
                        "content": " 有"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\alpha = u (A) f _ {1} (A) \\alpha + v (A) f _ {2} (A) \\alpha := \\alpha_ {1} + \\alpha_ {2}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/c4129e9a25232b43e4fe7da59cabeaaf27166946396ecf118c604d80e8f1febe.jpg"
                }
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
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                        "type": "text",
                        "content": "62"
                    }
                ]
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "6.2 空间分解法：零化多项式角度"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "且满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(A)\\alpha = f_{1}(A)f_{2}(A)\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": " 。故考察  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{2}(A)\\alpha_{1}"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "f _ {2} (A) \\alpha_ {1} = f _ {2} (A) u (A) f _ {1} (A) \\alpha = u (A) f _ {1} (A) f _ {2} (A) = 0 \\Rightarrow \\alpha_ {1} \\in V _ {2}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/33facedce5ead61649d0d2247beba337afcc9a8efbde19947871e76e4035c06b.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "同理得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{1}(A)\\alpha_{2} = 0, \\alpha_{2} \\in V_{1}"
                    },
                    {
                        "type": "text",
                        "content": "，从而得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = V_{1} + V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "下证直和，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in V_1\\cap V_2"
                    },
                    {
                        "type": "text",
                        "content": " ，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{1}(A)\\alpha = f_{2}(A)\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，因此"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\alpha = u (A) f _ {1} (A) \\alpha + v (A) f _ {2} (A) \\alpha = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/d3ad743c3adc5392c5e8102512e19fa42d1591c488112266f8017e3d7bfec13d.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此结论成立。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.10 (多空间分解)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "A 是数域 P 上 n 阶方阵，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x), f_{1}(x), \\cdots, f_{s}(x) \\in P[x]"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = f_{1}(x)f_{2}(x)\\cdots f_{s}(x)"
                    },
                    {
                        "type": "text",
                        "content": " ，且   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{1}(x), \\cdots, f_{s}(x)"
                    },
                    {
                        "type": "text",
                        "content": "  两两互素，设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(A)X = 0, f_{1}(A)X = 0, \\cdots, f_{s}(A)X = 0"
                    },
                    {
                        "type": "text",
                        "content": "  的解空间为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V, V_{1}, \\cdots, V_{s}"
                    },
                    {
                        "type": "text",
                        "content": " ，则"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "V = V _ {1} \\oplus V _ {2} \\oplus \\dots \\oplus V _ {s}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/0486db34981225120e9706233cf12f467095707fc0fc77b00c01ee538ddc792a.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 用数学归纳法即可。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.11 (空间分解研究矩阵秩)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "考虑   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(A)=f_{1}(A)\\cdots f_{s}(A)"
                    },
                    {
                        "type": "text",
                        "content": " ，则"
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "n - \\mathrm{r} (f (A)) = n - \\mathrm{r} (f _ {1} (A)) + \\dots + n - \\mathrm{r} (f _ {s} (A))",
                "math_type": "latex",
                "image_source": {
                    "path": "images/b0e736db12f8050d8e8ce0ea91e59dd95f0d302b2d58c812247e296aaef9e0f0.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(A)X = 0, f_{1}(A)X = 0, \\cdots, f_{s}(A)X = 0"
                    },
                    {
                        "type": "text",
                        "content": "  的解空间为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V, V_{1}, \\cdots, V_{s}"
                    },
                    {
                        "type": "text",
                        "content": " ，根据解空间分解得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = V_{1} \\oplus \\cdots \\oplus V_{s}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此根据解空间的维数可得到结论。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 6.1 空间分解研究矩阵的秩 证明以下命题"
                    }
                ]
            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{2}=A"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A)+r(A-E)=n"
                    }
                ]
            },
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        {
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            "content": {
                "paragraph_content": [
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                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^2 = E"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A + E) + r(A - E) = n"
                    }
                ]
            },
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{3}=E"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A-E)+r(A^{2}+A+E)=n"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(A)=A^{2}-A=A(A-E)"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n-r(A^{2}-A)=n-r(A)+n-r(A-E)"
                    },
                    {
                        "type": "text",
                        "content": " ，左侧等于n，右侧等于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "2n-r(A)-r(A-E)"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A)+r(A-E)=n"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.2.2 根子空间分解：线性变换角度"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理 6.4 (零变换的核空间)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "V 是 P 上的线性空间，O 是零变换，则其核空间为整个空间："
                    }
                ]
            },
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            "content": {
                "math_content": "\\operatorname{Ker} (\\mathcal {O}) = V",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f7f33522a296d490218f95b72457a792b70cb37ed5525cd0a65eb86646f46350.jpg"
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        {
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            "content": {
                "title_content": [
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                        "type": "text",
                        "content": "定理 6.12 (核空间分解)"
                    }
                ],
                "level": 2
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                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 为线性变换，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = f_{1}(x)f_{2}(x)"
                    },
                    {
                        "type": "text",
                        "content": " 为多项式，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f_{1}(x),f_{2}(x)) = 1"
                    },
                    {
                        "type": "text",
                        "content": " ，则有如下核空间分解："
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\operatorname{Ker} (f (\\mathcal {A})) = \\operatorname{Ker} (f _ {1} (\\mathcal {A})) \\oplus \\operatorname{Ker} (f _ {2} (\\mathcal {A}))",
                "math_type": "latex",
                "image_source": {
                    "path": "images/26b045b7f17c0ad645e3f5fde7cc9707562bf469c5e5a3903a83b7ae10604980.jpg"
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(f_{1}(x), f_{2}(x)) = 1"
                    },
                    {
                        "type": "text",
                        "content": " 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists u(x), v(x)"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "u(x)f_{1}(x) + v(x)f_{2}(x) = 1"
                    },
                    {
                        "type": "text",
                        "content": "，因此"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\mathcal {E} = u (\\mathcal {A}) f _ {1} (\\mathcal {A}) + v (\\mathcal {A}) f _ {2} (\\mathcal {A})",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1d685b028c76f2c4fe5d727eca9376e45780755eedc046893834dec6c6ca6694.jpg"
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            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall\\alpha\\in\\mathrm{Ker}(f(\\mathcal{A}))"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\alpha = \\alpha_ {1} + \\alpha_ {2} := u (\\mathcal {A}) f _ {1} (\\mathcal {A}) \\alpha + v (\\mathcal {A}) f _ {2} (\\mathcal {A}) \\alpha",
                "math_type": "latex",
                "image_source": {
                    "path": "images/3e490564b2680462cf8213c910276247a1dee26006e130f8f93133f38b87f5f6.jpg"
                }
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        },
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            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "63"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "6.2 空间分解法：零化多项式角度"
                    }
                ]
            },
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{2}(\\mathcal{A})\\alpha_{1} = u(\\mathcal{A})f(\\mathcal{A})\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}\\in \\mathrm{Ker}(f_{1}(\\mathcal{A}))"
                    },
                    {
                        "type": "text",
                        "content": " ，同理有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{2}\\in \\mathrm{Ker}(f_{2}(\\mathcal{A}))"
                    },
                    {
                        "type": "text",
                        "content": " ，综上得到"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\operatorname{Ker} (f (\\mathcal {A})) = \\operatorname{Ker} (f _ {1} (\\mathcal {A})) + \\operatorname{Ker} (f _ {2} (\\mathcal {A}))",
                "math_type": "latex",
                "image_source": {
                    "path": "images/9d8589403b88e5250443349e1ba80435bf6d940e74f0bade701a064bde6afd38.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "下证直和：对  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in \\mathrm{Ker}(f_1(\\mathcal{A}))\\cap \\mathrm{Ker}(f_2(\\mathcal{A}))"
                    },
                    {
                        "type": "text",
                        "content": " ，有"
                    }
                ]
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\alpha = u (\\mathcal {A}) f _ {1} (\\mathcal {A}) \\alpha + v (\\mathcal {A}) f _ {2} (\\mathcal {A}) \\alpha = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/86ac6975c1151aef02ad3b6cdca1859f3bf72c99bf393b722d5ed37438fe13a8.jpg"
                }
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此为直和"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.13 (全空间分解：零化多项式角度)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "V 是 P 上的线性空间, A 是 V 上的线性变换,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = f_{1}(x)f_{2}(x)\\cdots f_{s}(x)"
                    },
                    {
                        "type": "text",
                        "content": "  为零化多项式, 且   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{1}(x), \\cdots, f_{s}(x)"
                    },
                    {
                        "type": "text",
                        "content": "  两两互素, 则"
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "V = \\operatorname{Ker} (f _ {1} (\\mathcal {A})) \\oplus \\dots \\oplus \\operatorname{Ker} (f _ {s} (\\mathcal {A}))",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1c2dce82ce501cfb0169ef764e643b9dcad8110b5b3756d2d5c0c0eb13debae7.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(A) = \\mathcal{O}"
                    },
                    {
                        "type": "text",
                        "content": "，根据零变换的核空间以及核空间分解定理可知。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理 6.5 (Hamilton-Cayley)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 是线性变换  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 的特征多项式，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 的零化多项式"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 该定理在最小多项式一章已经证明过了"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.14 (根子空间分解：零化多项式角度)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上线性变换  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 的特征多项式为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(\\lambda) = (\\lambda - \\lambda_1)^{k_1} \\cdots (\\lambda - \\lambda_s)^{k_s}"
                    },
                    {
                        "type": "text",
                        "content": "，最小多项式为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m(\\lambda) = (\\lambda - \\lambda_1)^{l_1} \\cdots (\\lambda - \\lambda_s)^{l_s}"
                    },
                    {
                        "type": "text",
                        "content": "，则有以下根子空间分解"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "V = \\operatorname{Ker} (\\mathcal {A} - \\lambda_ {1} \\mathcal {E}) ^ {k _ {1}} \\oplus \\dots \\oplus \\operatorname{Ker} (\\mathcal {A} - \\lambda_ {s} \\mathcal {E}) ^ {k _ {s}} = \\operatorname{Ker} (\\mathcal {A} - \\lambda_ {1} \\mathcal {E}) ^ {l _ {1}} \\oplus \\dots \\oplus \\operatorname{Ker} (\\mathcal {A} - \\lambda_ {s} \\mathcal {E}) ^ {l _ {s}}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/e0f0beca03352ee63fd9fe1eed22fc3e3bc4a7f61bb4691e636dbb4ee1640910.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 根据 Hamilton-Cayley 定理以及零化多项式全空间分解可知。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 6.2 根子空间分解：零化多项式角度 (1) ZJU2021.8: V 是线性空间，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\varphi"
                    },
                    {
                        "type": "text",
                        "content": " 是 V 上线性变换，特征多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(\\lambda)=\\left(\\lambda-2\\right)^{6}(\\lambda+2)^{4}"
                    },
                    {
                        "type": "text",
                        "content": " ，将 V 分解为两个非平凡不变子空间的直和，并证明结论。"
                    }
                ]
            },
            "bbox": [
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                570
            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.15 (根子空间分解与 Jordan 标准型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "根子空间分解将Jordan阵按照特征值  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 分为若干份，每份由  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 为对角的若干个Jordan块组成，对角  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 在Jordan阵中的出现次数等于其代数重数。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.2.3 Jordan 阵与循环不变子空间"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理 6.6 (核限制变换是幂零变换)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\mathcal{A} - \\lambda_i\\mathcal{E})\\big|_{\\mathrm{Ker}(\\mathcal{A} - \\lambda_i\\mathcal{E})^k}"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathrm{Ker}(\\mathcal{A} - \\lambda_i\\mathcal{E})^k"
                    },
                    {
                        "type": "text",
                        "content": " 中为幂零变换"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in \\mathrm{Ker}(\\mathcal{A} - \\lambda_i\\mathcal{E})^k"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\mathcal{A} - \\lambda_i\\mathcal{E})^k\\alpha = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，因此限制变换满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\mathcal{A} - \\lambda_i\\mathcal{E})^k\\big|_{\\mathrm{Ker}(\\mathcal{A} - \\lambda_i\\mathcal{E})^k} = \\mathcal{O}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.16 (循环不变子空间分解)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于 V 内的幂零变换 A，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists v_{1},\\cdots,v_{n}\\in V,\\ m_{1},\\cdots,m_{n}\\in N"
                    },
                    {
                        "type": "text",
                        "content": "  使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V=I(v_{1})\\oplus I(v_{2})\\oplus\\cdots\\oplus I(v_{n})"
                    },
                    {
                        "type": "text",
                        "content": " 。其中   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "I(v_{i})=L(v_{i},\\mathcal{A}v_{i}\\cdots,\\mathcal{A}^{m_{i}}v_{i})"
                    },
                    {
                        "type": "text",
                        "content": " ，且   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{m_{i}+1}v_{i}=0"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 在空间第二分解定理中已经证明过了。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "64"
                    }
                ]
            },
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "6.2 空间分解法：零化多项式角度"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.17 (Jordan 阵与循环不变子空间)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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                102
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "线性变换 A 在根子空间   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathrm{Ker}(\\mathcal{A}-\\lambda_{i}\\mathcal{E})^{k}"
                    },
                    {
                        "type": "text",
                        "content": "  下的限制变换   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A|_{\\mathrm{Ker}(\\mathcal{A}-\\lambda_{i}\\mathcal{E})^{k}}"
                    },
                    {
                        "type": "text",
                        "content": "  在一组循环不变基   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "((\\mathcal{A}-\\lambda_{i}\\mathcal{E})^{s}\\alpha, (\\mathcal{A}-\\lambda_{i}\\mathcal{E})^{s-1}\\alpha,\\cdots,\\alpha)"
                    },
                    {
                        "type": "text",
                        "content": "  下的矩阵为一个 Jordan 块："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "J _ {s} (\\lambda_ {i}) = \\left[ \\begin{array}{c c c c c} \\lambda_ {i} & 1 & & & \\\\ & \\lambda_ {i} & 1 & & \\\\ & & \\ddots & \\ddots & \\\\ & & & \\lambda_ {i} & 1 \\\\ & & & & \\lambda_ {i} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1e4ced9d1a39ba3adc875e17faae0a941d6fcc316cc186ecceaf6eccc264e964.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 直接验证即可。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.18 (Jordan 阵与几何重数)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "特征值  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 的代数重数是特征多项式  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda = \\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 根的重数，几何重数  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\lambda_{i}E - A = 0"
                    },
                    {
                        "type": "text",
                        "content": " 解空间维数) 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 对应以  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 为对角的 Jordan 块个数。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}E - A = 0"
                    },
                    {
                        "type": "text",
                        "content": " 的解空间维数等于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}E - J = 0"
                    },
                    {
                        "type": "text",
                        "content": " 解空间维数，而每有一个以  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 为对角的Jordan块， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}E - J"
                    },
                    {
                        "type": "text",
                        "content": " 上就有一行全为0，因此秩减1，从而块数等于几何重数。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 即以   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  为对角的 Jordan 块个数为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n - r(\\lambda E - A)"
                    },
                    {
                        "type": "text",
                        "content": " ，反过来写也要分得清。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.3 Jordan 标准形： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵法"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.3.1  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵与相抵"
                    }
                ],
                "level": 2
            },
            "bbox": [
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                480,
                268,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定义 6.4 (  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵)"
                    }
                ]
            },
            "bbox": [
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                250,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 是元素为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 多项式的矩阵，则将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 称为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵。"
                    }
                ]
            },
            "bbox": [
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                532,
                548
            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 6.5 (  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵初等变换)"
                    }
                ],
                "level": 2
            },
            "bbox": [
                122,
                573,
                317,
                588
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵有三种初等变换："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 互换两行/列"
                    }
                ]
            },
            "bbox": [
                139,
                613,
                250,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 用常数  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "c"
                    },
                    {
                        "type": "text",
                        "content": " 乘一行/列 (不能乘  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 多项式！)"
                    }
                ]
            },
            "bbox": [
                139,
                630,
                452,
                645
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 将一行/列乘以一个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 多项式加到另一行/列上"
                    }
                ]
            },
            "bbox": [
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                490,
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 6.6 (相抵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若可以用初等变换将一个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵转换为另一个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵，则称两个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵相抵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.19 (相抵标准形：法式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于数值矩阵 A，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E - A"
                    },
                    {
                        "type": "text",
                        "content": "  总是相抵于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathrm{diag}\\{1,\\cdots,1,d_{1}(\\lambda),\\cdots,d_{m}(\\lambda)\\}"
                    },
                    {
                        "type": "text",
                        "content": " ，其中   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d_{i}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": "  是首一多项式，且   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d_{i}(\\lambda)|d_{i+1}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理 6.7 (对角整除化)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 为非零  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 一定相抵于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": "，满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "b_{11}(\\lambda) \\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " 且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "b_{11}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 可整除  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 中的所有元素"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 先将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 中次数最低的元素放置于对角  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{11}(\\lambda)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "65"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "6.3 Jordan 标准形：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": "  矩阵法"
                    }
                ]
            },
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    ],
    [
        {
            "type": "image",
            "content": {
                "image_source": {
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                "image_footnote": []
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(1) 消去第一行、第一列：不妨设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{i1}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 不被  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{11}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 整除，则做带余除法  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{i1}(\\lambda) = q(\\lambda)a_{11}(\\lambda) + r(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": "，再将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 放置于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{11}"
                    },
                    {
                        "type": "text",
                        "content": "，直至全部整除，可消去第一列和第一行。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 考虑第一列、第一行外的其他元素，若存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{ij}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 不被  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{11}"
                    },
                    {
                        "type": "text",
                        "content": " 整除，则把第  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "j"
                    },
                    {
                        "type": "text",
                        "content": " 列加到第一列上，做带余除法，把余项放到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{11}"
                    },
                    {
                        "type": "text",
                        "content": "，以此类推。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.20 (法式的计算：初等变换法)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "每次把最低次移动到角上，尝试消除，消除过程按照以下步骤"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 通过初等变换将当前对角元素变为所有非零元素中次数最低的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 多项式"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 将矩阵对角整除化：具体见引理的过程"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 消去一列，再消去一行，重复对角整除化和消去"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.21 (相抵与相似)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "A 与 B 相似当且仅当   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E - A"
                    },
                    {
                        "type": "text",
                        "content": "  与   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E - B"
                    },
                    {
                        "type": "text",
                        "content": "  相抵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 左推右：若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B = P^{-1}AP"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E - B = P^{-1}(\\lambda E - A)P"
                    },
                    {
                        "type": "text",
                        "content": "，显然右侧两个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵相抵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 右推左：若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E - B = P(\\lambda)(\\lambda E - A)Q(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " ，设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q(\\lambda) = Q_k \\lambda^k + \\cdots + Q_1 \\lambda + Q_0"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q(\\lambda)^{-1} = R_m \\lambda^m + \\cdots + R_1 \\lambda + R_0"
                    },
                    {
                        "type": "text",
                        "content": " ，下记   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W = Q(B)"
                    },
                    {
                        "type": "text",
                        "content": " 。由   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q(\\lambda)Q^{-1}(\\lambda) = E"
                    },
                    {
                        "type": "text",
                        "content": "  得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "R_m Q(\\lambda) \\lambda^m + \\cdots + R_1 Q(\\lambda) \\lambda + R_0 Q(\\lambda) = E"
                    },
                    {
                        "type": "text",
                        "content": " ，代入 B 得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "R_m WB^m + \\cdots + R_1 WB + R_0 W = E"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P(\\lambda)^{-1}(\\lambda E - B) = (\\lambda E - A)Q(\\lambda) = Q(\\lambda)\\lambda - AQ(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": "，代入  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda = B"
                    },
                    {
                        "type": "text",
                        "content": " 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q(B)B = AQ(B)"
                    },
                    {
                        "type": "text",
                        "content": " (前面一式子左  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "= 0"
                    },
                    {
                        "type": "text",
                        "content": ")，即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "WB = AW"
                    },
                    {
                        "type": "text",
                        "content": "，从而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "WB^{l} = A^{l}W"
                    },
                    {
                        "type": "text",
                        "content": " (多作用一个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 相当于多右乘一个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B"
                    },
                    {
                        "type": "text",
                        "content": ")。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "再由前面的式子得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(R_{m}A^{m}+\\cdots+R_{0})W=E"
                    },
                    {
                        "type": "text",
                        "content": " ，则表明W可逆，从而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B=W^{-1}AW"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.3.2 行列式因子、不变因子"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定义 6.7 (行列式 k 阶子式)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "|A|"
                    },
                    {
                        "type": "text",
                        "content": " 的 k 阶子式是其任选 k 行、k 列组成的矩阵的行列式。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 6.8 (行列式因子)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵，若其所有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k"
                    },
                    {
                        "type": "text",
                        "content": " 阶子式(行列式)均为0，则定义其  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k"
                    },
                    {
                        "type": "text",
                        "content": " 阶行列式因子  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D_{k}(\\lambda) = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，否则定义为所有非零  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k"
                    },
                    {
                        "type": "text",
                        "content": " 阶子式的首一最大公因式。"
                    }
                ]
            },
            "bbox": [
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                627
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记一般行列式因子比初等变换方便，不过一定注意这里的子式是所有子式，而非主子式，甚至可以不连着，0的情况也得分开考虑。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理 6.8 (行列式因子的相除性)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D_{1}(\\lambda),\\cdots,D_{r}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": "  是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": "  非零行列式因子，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D_{i}(\\lambda)|D_{i+1}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{i+1}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 任意  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i + 1"
                    },
                    {
                        "type": "text",
                        "content": " 阶子式，这个行列式按某一行展开，则其每一展开项都是一个多项式与一个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i"
                    },
                    {
                        "type": "text",
                        "content": " 阶子式乘积，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D_i(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 是所有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i"
                    },
                    {
                        "type": "text",
                        "content": " 阶子式公因子，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D_i(\\lambda)|_{A_{i+1}(\\forall i)}"
                    },
                    {
                        "type": "text",
                        "content": " ，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D_{i+1}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 是所有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{i+1}"
                    },
                    {
                        "type": "text",
                        "content": " 的最大公因式，因此证毕。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 6.9 (不变因子)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定义  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d_{1}(\\lambda) = D_{1}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "d_{i}(\\lambda) = \\frac{D_{i}(\\lambda)}{D_{i - 1}(\\lambda)} (\\forall i\\neq 1)"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 的不变因子。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "66"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "6.3 Jordan 标准形： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵法"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.22 (行列式因子、不变因子的性质)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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                102
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 相抵矩阵的因子：相抵的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵有相同的行列式因子、不变因子 (初等变换不改变行列式、不变因子)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 法式与不变因子： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E - A"
                    },
                    {
                        "type": "text",
                        "content": " 法式的所有对角元素即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E - A"
                    },
                    {
                        "type": "text",
                        "content": " 的不变因子。"
                    }
                ]
            },
            "bbox": [
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                143,
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                158
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 只需证明行列式因子在三种初等变换不变即可。而不变因子由行列式因子唯一确定。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 由于不变因子在初等变化中不变，求出法式的不变因子 (即对角)，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E - A"
                    },
                    {
                        "type": "text",
                        "content": " 的不变因子即其法式对角元素。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.23 (不变因子与最小多项式)"
                    }
                ],
                "level": 2
            },
            "bbox": [
                122,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的最小多项式即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E - A"
                    },
                    {
                        "type": "text",
                        "content": " 的最后一个不变因子"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设 A 的互异特征值为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1},\\cdots,\\lambda_{s}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此可以列出所有的初等因子 (具体见下一节)，最小多项式即初等因子的最小公倍式，最后一个不变因子也是，因此结论成立"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.3.3 初等因子与 Jordan 标准型"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 6.10 (初等因子)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "将所有不变因子分解为不可约因子之积，这些不可约因子全体称为初等因子组。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.24 (对角化计算初等因子)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E-A"
                    },
                    {
                        "type": "text",
                        "content": "  经过初等变化变为对角阵 (不一定要法式)"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left( \\begin{array}{c c c c} f _ {1} (\\lambda) & & & \\\\ & f _ {2} (\\lambda) & & \\\\ & & \\ddots & \\\\ & & & f _ {n} (\\lambda) \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/fb62022c338d9daf6df0623e7fead6bdb13b395948673255bc9436323fbde5c7.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "其中  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{i}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 为非零首一多项式，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的初等因子组是所有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{i}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 做不可约分解得到的因式组合。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理 6.9 (Jordan 块、Jordan 阵的初等因子)"
                    }
                ],
                "level": 2
            },
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        },
        {
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "r 阶   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{0}"
                    },
                    {
                        "type": "text",
                        "content": "  对角的 Jordan 块的初等因子组为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\lambda - \\lambda_{0})^{r}"
                    },
                    {
                        "type": "text",
                        "content": " 。对角为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J_{1}, J_{2}, \\cdots, J_{k}"
                    },
                    {
                        "type": "text",
                        "content": "  的 Jordan 阵的初等因子组为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\lambda - \\lambda_{1})^{r_{1}}, \\cdots, (\\lambda - \\lambda_{k})^{r_{k}}"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) Jordan 块的初等因子组直接用行列式因子计算即可 (只有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r"
                    },
                    {
                        "type": "text",
                        "content": " 阶非 1，因此不变因子好算)"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)Jordan 阵的初等因子进行分块初等变换，将各个 Jordan 块分块变成法式，再用引理6.24即可。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.25 (使用初等因子计算 Jordan 标准型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若 K 上的矩阵 A 有初等因子组   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\lambda - \\lambda_{1})^{r_{1}}, \\cdots, (\\lambda - \\lambda_{k})^{r_{k}}"
                    },
                    {
                        "type": "text",
                        "content": " ，则 A 相似于 Jordan 阵   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J = \\text{diag}\\{J_{1}, \\cdots, J_{k}\\}"
                    },
                    {
                        "type": "text",
                        "content": " ，其中   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  是以   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  为对角的   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  阶 Jordan 块。"
                    }
                ]
            },
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.26 (Jordan 标准型存在充要条件)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "K 上矩阵 A 有 Jordan 标准型当且仅当其初等因子都是一次项的幂次乘积。"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "6.4  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵的进一步研究"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_number",
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                "page_number_content": [
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                        "type": "text",
                        "content": "67"
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                ]
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    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.4  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵的进一步研究"
                    }
                ],
                "level": 2
            },
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        },
        {
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            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.4.1 Jordan 标准型的直接计算"
                    }
                ],
                "level": 2
            },
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        {
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "一、初等变换方法：只需要将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E - A"
                    },
                    {
                        "type": "text",
                        "content": " 转换为对角矩阵，对角所有分解即初等因子组，直接构造Jordan标准型"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 6.3 无法直接整除的情况 (1) 计算下面矩阵 A 的法式:"
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c c} - 1 & 1 & 3 \\\\ 3 & 0 & - 4 \\\\ - 2 & 1 & 4 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/6b669966e31695f900cb11798f6ed8c74aae2ece4802e37bd4bd21008a298e1d.jpg"
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            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解(1)进行到下面步骤的时候无法整除, 即发现不整除列上的多项式, 要进行对角整除化"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[\\begin{array}{c c c}1&0&0\\\\0&\\lambda - 1&- \\lambda + 1\\\\0&\\lambda^ {2} + \\lambda - 3&- 3 \\lambda + 4\\end{array}\\right]\\rightarrow \\left[\\begin{array}{c c c}1&0&0\\\\0&\\lambda - 1&- \\lambda + 1\\\\0&(\\lambda^ {2} + \\lambda - 3) - (\\lambda - 1) (\\lambda + 2)&(- 3 \\lambda + 4) - (\\lambda + 2) (- \\lambda + 1)\\end{array}\\right] = \\left[\\begin{array}{c c c}1&0&\\\\0&\\lambda - 1&- \\lambda\\\\0&- 1&\\lambda^ {2} - \\lambda^ {2}\\end{array}\\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/6219e4b1ad56ad38a09ec76667f54ac96c21cbcfed20fddbe0a2301a820aba6f.jpg"
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            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此最终可以化简为："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c c c} 1 & & & \\\\ & 1 & & \\\\ & & (\\lambda - 1) ^ {3} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/3a4ac451e57b2225d63034daed799efc1a8dab9f5e1de14071ebcf0d6bf809aa.jpg"
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            "bbox": [
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        {
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习6.4初等变换方法计算Jordan标准型 计算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\left[ \\begin{array}{ccc}1 & -1 & 2\\\\ 3 & -3 & 6\\\\ 2 & -2 & 4 \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": " 的Jordan标准型解通过初等变换得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E - A"
                    },
                    {
                        "type": "text",
                        "content": " 的法式为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\left[ \\begin{array}{lll}1 & 0 & 0\\\\ 0 & \\lambda & 0\\\\ 0 & 0 & -\\lambda^2 +2\\lambda \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": " ，因此初等因子为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda ,\\lambda ,( \\lambda -2)"
                    },
                    {
                        "type": "text",
                        "content": " ，对应Jordan阵为： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J = \\left[ \\begin{array}{ccccc}0 &  &  &  \\\\  & 0 &  &  \\\\  &  & 2 \\end{array} \\right]"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "二、行列式因子方法：求一般矩阵的 Jordan 阵用行列式因子可能比较麻烦，但是对于抽象的 n 阶矩阵却有很大的作用，原因在于只要有两个子式互素，则行列式因子就为 1。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 6.5 经典而重要的题目   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a, b \\in C"
                    },
                    {
                        "type": "text",
                        "content": " ，求下面矩阵 A 的 Jordan 标准型"
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c c c c} a & b & \\dots & b & b \\\\ & a & \\dots & b & b \\\\ & & \\ddots & \\vdots & \\vdots \\\\ & & & a & b \\\\ & & & & a \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/eff90178876eb6eb7690f5cfa6f1fa23dded696e30b84c00c0f8f63bae9c90f3.jpg"
                }
            },
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        },
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            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "6.4  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵的进一步研究"
                    }
                ]
            },
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        },
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                        "content": "68"
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    [
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解 若 b=0，则 Jordan 标准型即 aE。若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "b\\neq0"
                    },
                    {
                        "type": "text",
                        "content": " ，则考虑   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E-A"
                    },
                    {
                        "type": "text",
                        "content": " ："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\lambda E - A = \\left[ \\begin{array}{c c c c c} \\lambda - a & - b & \\dots & - b & - b \\\\ & \\lambda - a & \\dots & - b & - b \\\\ & & \\ddots & \\vdots & \\vdots \\\\ & & & \\lambda - a & - b \\\\ & & & & \\lambda - a \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/7ecf787c657c582b41f82d5f8e7d3f98c60b59850f1d7dbb83c4dc279268c9c6.jpg"
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        {
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "考虑  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n - 1"
                    },
                    {
                        "type": "text",
                        "content": " 阶子式，右下子式行列式为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\lambda - a)^{n - 1}"
                    },
                    {
                        "type": "text",
                        "content": "，右上子式在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda = a"
                    },
                    {
                        "type": "text",
                        "content": " 时显然不是0(根据余数定理)，因此互素，即初等因子为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\lambda - a)^n"
                    },
                    {
                        "type": "text",
                        "content": "，因此Jordan阵为一个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶的以  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a"
                    },
                    {
                        "type": "text",
                        "content": " 为对角的Jordan块"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习6.6行列式因子应用相关练习 求不变因子:(1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A=\\left[\\begin{array}{cccc} & & 1 & \\lambda+2 \\\\ & 1 & \\lambda+2 \\\\ 1 & \\lambda+2 & & \\\\ \\lambda+2 & & & \\end{array}\\right]"
                    },
                    {
                        "type": "text",
                        "content": " (2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A=\\left[\\begin{array}{cccc} \\lambda & -1 & & \\\\ & \\lambda & -1 & \\\\ & & \\lambda & -1 \\\\ 5 & 4 & 3 & \\lambda+2 \\end{array}\\right]"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解(1)左上行列式因子为-1，因此不变因子为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1,1,1,(\\lambda+2)^{4}"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 右上行列式因子为 -1，因此不变因子 1,1,1,  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A|"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.27 (重要   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": "  矩阵)"
                    }
                ],
                "level": 2
            },
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "下面   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": "  矩阵 A 的不变因子为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1,1,\\cdots,\\lambda^{n}+a_{1}\\lambda^{n-1}+a_{2}\\lambda^{n-2}+\\cdots+a_{n-1}\\lambda+a_{n}"
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c c c c c} \\lambda & & & & & a _ {n} \\\\ - 1 & \\lambda & & & & a _ {n - 1} \\\\ & - 1 & \\lambda & & & a _ {n - 2} \\\\ & & \\ddots & \\ddots & & \\vdots \\\\ & & & - 1 & \\lambda & a _ {2} \\\\ & & & & - 1 & \\lambda + a _ {1} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f1a5e369a9386e29bd73058bd9a22bc18629381544d1069652a00394a02c4c1f.jpg"
                }
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 考虑左下方行列式因子，其值为 1，因此不变因子为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1, 1, \\cdots, 1, |A|"
                    },
                    {
                        "type": "text",
                        "content": " ，而计算出   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A| = \\lambda^{n} + a_{1}\\lambda^{n-1} + \\cdots + a_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此结论成立"
                    }
                ]
            },
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "推论6.1"
                    }
                ],
                "level": 2
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "下列矩阵 A 在 C 可对角化当且仅当   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda^{n} + a_{1}\\lambda^{n-1} + a_{2}\\lambda^{n-2} + \\cdots + a_{n-1}\\lambda + a_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  在 C 上无重根"
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c c c c c} 0 & & & & & a _ {n} \\\\ - 1 & 0 & & & & a _ {n - 1} \\\\ & - 1 & 0 & & & a _ {n - 2} \\\\ & & \\ddots & \\ddots & & \\vdots \\\\ & & & - 1 & 0 & a _ {2} \\\\ & & & & - 1 & a _ {1} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/5f4c896c796612347b03a45987f00908486991640d4c5d81b5fb5741c5dc8072.jpg"
                }
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习6.7一些用行列式因子计算方便的特殊矩阵 计算Jordan标准型(1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\left[ \\begin{array}{cccc}1 & 2 & 3 & 4\\\\ & 1 & 2 & 3\\\\ & & 1 & 2\\\\ & & & 1 \\end{array} \\right]"
                    },
                    {
                        "type": "text",
                        "content": " (2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\left[ \\begin{array}{cccc}a & & & \\\\ a & a & & \\\\ \\vdots & \\vdots & \\ddots & \\\\ a & a & \\cdots & a \\end{array} \\right]"
                    }
                ]
            },
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                        "type": "text",
                        "content": "69"
                    }
                ]
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "6.4  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵的进一步研究"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(3) A = \\left[ \\begin{array}{c c c c} a & & & \\\\ b _ {1} & a & & \\\\ & \\ddots & \\ddots & \\\\ & & b _ {n - 1} & a \\end{array} \\right] (4) A = \\left[ \\begin{array}{c c c c c} a & a _ {1 2} & a _ {1 3} & \\dots & a _ {1 n} \\\\ & a & a _ {2 3} & \\dots & a _ {2 n} \\\\ & & \\ddots & & \\vdots \\\\ & & & a & a _ {n - 1, n} \\\\ & & & & a \\end{array} \\right], a _ {1 2} a _ {2 3} \\dots a _ {n - 1, n} \\neq 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f09a93dbfa0835d8dc6d546b11a2038f879a56ddfcdb4d032502903fd383bf66.jpg"
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                    {
                        "type": "text",
                        "content": "解 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda E - A"
                    },
                    {
                        "type": "text",
                        "content": " 左上行列式为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\lambda - 1)^3"
                    },
                    {
                        "type": "text",
                        "content": "，右上在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda = 1"
                    },
                    {
                        "type": "text",
                        "content": " 是显然非零，因此不变因子  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1, 1, 1, |\\lambda E - A|"
                    }
                ]
            },
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                    {
                        "type": "text",
                        "content": "(2) 左上行列式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\lambda - a)^{n-1}"
                    },
                    {
                        "type": "text",
                        "content": " ，左下在   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda = a"
                    },
                    {
                        "type": "text",
                        "content": "  显然非零，因此不变因子   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1, 1, \\cdots, 1, |\\lambda E - A|"
                    }
                ]
            },
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                        "type": "text",
                        "content": "(3)(4) 显然"
                    }
                ]
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 如果对角相等，则用行列式因子计算可能会比较方便"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.4.2  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k"
                    },
                    {
                        "type": "text",
                        "content": " 阶Jordan块的个数"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
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                        "type": "text",
                        "content": "定理 6.28 (k 阶 Jordan 块个数)"
                    }
                ],
                "level": 2
            },
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                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶复矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的Jordan标准型为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J"
                    },
                    {
                        "type": "text",
                        "content": " 中对角元  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k"
                    },
                    {
                        "type": "text",
                        "content": " 阶Jordan块  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J_{k}(\\lambda)"
                    },
                    {
                        "type": "text",
                        "content": " 的个数如下。这里约定  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r((A - \\lambda E_n)^0) = n"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "r ((A - \\lambda E _ {n}) ^ {k - 1}) - 2 r ((A - \\lambda E _ {n}) ^ {k}) + r ((A - \\lambda E _ {n}) ^ {k + 1})",
                "math_type": "latex",
                "image_source": {
                    "path": "images/dc51ab9df8df1bcedd60145f01f9e60eb5f9d1ddc4fec03776b1999c74690112.jpg"
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            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 考虑零对角 Jordan 块的幂次:"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "r (J _ {r} (0) ^ {k}) = \\left\\{ \\begin{array}{l l} r - k, & 0 \\leq k <   r \\\\ 0, & k \\geq r \\end{array} \\right. \\quad r (J _ {r} (0) ^ {k}) - r (J _ {r} (0) ^ {k + 1}) = \\left\\{ \\begin{array}{l l} 1, & 0 \\leq k <   r \\\\ 0, & k \\geq r \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1ba3b87a3cba8d79484faf8752611e852d854a520ad84b5e416fd9a1d60ab36c.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "而非零对角 Jordan 块的幂次满足   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(J_{r}(a)^{k}) - r(J_{r}(a)^{k+1}) = 0"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "设 A 的 Jordan 标准型为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J = \\text{diag}\\{J_{r_1}(\\lambda_1), \\cdots, J_{r_s}(\\lambda_s)\\}"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A - \\lambda E)^k, (J - \\lambda E)^k"
                    },
                    {
                        "type": "text",
                        "content": "  相似。根据 J 是分块对角的性质可知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r((A - \\lambda E)^k) = r((J - \\lambda E)^k) = \\sum_{i=1}^{s} r(J_{r_i}(\\lambda_i - \\lambda)^k)"
                    },
                    {
                        "type": "text",
                        "content": " ，因此得到："
                    }
                ]
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        {
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            "content": {
                "math_content": "r ((A - \\lambda E) ^ {k}) - r ((A - \\lambda E) ^ {k + 1}) = \\sum_ {i = 1} ^ {s} [ r (J _ {r _ {i}} (\\lambda_ {i} - \\lambda) ^ {k}) - r (J _ {r _ {i}} (\\lambda_ {i} - \\lambda) ^ {k + 1}) ]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/959febf62e3308176f0abd50f40f99a21ba9c94b3870e4f0b53c7fce68a2a9f0.jpg"
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            },
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        {
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                    {
                        "type": "text",
                        "content": "根据前面的分析，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r((A-\\lambda E)^{k})-r((A-\\lambda E)^{k+1})"
                    },
                    {
                        "type": "text",
                        "content": " 等于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}=\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 的且阶数大于k那些块个数，同理  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r((A-\\lambda E)^{k-1})-r((A-\\lambda E)^{k})"
                    },
                    {
                        "type": "text",
                        "content": " 等于对角为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 的且阶数大于k-1的块个数，相减即可得到阶数为k的块个数。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 6.8 上述定理的应用 计算 A 的 Jordan 标准型"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c c c c} a & 0 & 1 & & \\\\ & a & 0 & \\ddots & \\\\ & & a & \\ddots & 1 \\\\ & & & \\ddots & 0 \\\\ & & & & a \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f3442f3ad1589da8de1e19dc280242acd0ac3e9ad1a5207bef621fac9fd17c90.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 6.9 A, B 是两个 n 阶矩阵，证明 A, B 相似的充要条件是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\lambda \\in C, k = 1, 2, \\cdots, n"
                    },
                    {
                        "type": "text",
                        "content": " ，均有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r((A - \\lambda E)^{k}) = r((B - \\lambda E)^{k})"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 直接用结论即可"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "6.4  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵的进一步研究"
                    }
                ]
            },
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        },
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                        "content": "70"
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    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.4.3 分块矩阵的 Jordan 阵"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
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                        "type": "text",
                        "content": "定理 6.29 (分块矩阵的 Jordan 标准型)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "P"
                    },
                    {
                        "type": "text",
                        "content": " 上  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 无公共复特征值，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 的 Jordan 标准型分别为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J_{1}, J_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，则对任意  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C \\in P^{m \\times n}"
                    },
                    {
                        "type": "text",
                        "content": "，则："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c} A & C \\\\ O & B \\end{array} \\right] \\sim \\operatorname{diag} \\{J _ {1}, J _ {2} \\}\\tag{#}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/04456a0d9fb9d3806b202f90954e7eadd4ba53c38ed234d9308e9d10bab21574.jpg"
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            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 无公共特征值时，根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX - XB"
                    },
                    {
                        "type": "text",
                        "content": " 的矩阵方程结论可知，对任意矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX - XB = C"
                    },
                    {
                        "type": "text",
                        "content": " 都有唯一解  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_0"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX_0 - X_0B = C"
                    },
                    {
                        "type": "text",
                        "content": "，因此："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c} E _ {m} & X _ {0} \\\\ O & E _ {n} \\end{array} \\right] \\left[ \\begin{array}{c c} A & C \\\\ O & B \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {m} & - X _ {0} \\\\ O & E _ {n} \\end{array} \\right] = \\left[ \\begin{array}{c c} A & C - (A X _ {0} - X _ {0} B) \\\\ O & B \\end{array} \\right] = \\left[ \\begin{array}{c c} A & O \\\\ O & B \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/fcd0b67710e50653b7383f8016e54e788427c38929e860ab41993ac754f7c59e.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此得到："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c} A & C \\\\ O & B \\end{array} \\right] \\sim \\left[ \\begin{array}{c c} A & O \\\\ O & B \\end{array} \\right] \\sim \\left[ \\begin{array}{c c} J _ {1} & O \\\\ O & J _ {2} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/37fdd75595dbd24e546f4afb485deeb1fd273e912341dcd097bed72b2dd751af.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 第二种证明：直接看 Jordan 标准型，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  为对角的 k 阶 Jordan 块结论直接往上套即可。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "推论6.2"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于分块上三角矩阵   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, A_{11}, A_{22}, \\cdots, A_{ss}"
                    },
                    {
                        "type": "text",
                        "content": "  两两无公共特征值，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{ii}"
                    },
                    {
                        "type": "text",
                        "content": "  的 Jordan 标准型为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "J_i"
                    },
                    {
                        "type": "text",
                        "content": " ，则 A 的 Jordan 标准型为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathrm{diag}\\{J_1, \\cdots, J_s\\}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c c c} A _ {1 1} & A _ {1 2} & \\dots & A _ {1 s} \\\\ & A _ {2 2} & \\dots & A _ {2 s} \\\\ & & \\ddots & \\vdots \\\\ & & & A _ {s s} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/956fddce58f0ac89c02641213fcb1de3bb45fc4f42db0e2082010fcbe102e063.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 6.10 分块矩阵应用 求 A 的 Jordan 标准型:"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c c c} 1 & 5 & - 1 & 2 \\\\ 0 & 2 & 4 & 3 \\\\ 0 & 0 & - 3 & 0 \\\\ 0 & 0 & 4 & - 3 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/a9f9a97bb317269ecdf47f3dd2f9a76fffb77e8eeb4ad6b1a4c01929dbdb8ef7.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathrm{diag}\\{1,2,J_2(-3)\\}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "6.4.4 过渡矩阵的计算"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 6.30 (过渡矩阵计算总结)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若已知 A 的 Jordan 标准型为 J，要计算 P 使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^{-1}AP = J"
                    },
                    {
                        "type": "text",
                        "content": " ，则可以按照以下步骤："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "1. 对于非一阶，以   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": "  为对角的 Jordan 块，设对应的 P 位置为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "[X_{1},\\cdots,X_{k}]"
                    },
                    {
                        "type": "text",
                        "content": " ，则满足"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A X _ {1} = \\lambda X _ {1}, A X _ {2} = \\lambda X _ {2} + X _ {1}, \\dots , A X _ {k} = \\lambda X _ {k} + X _ {k - 1}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/98aeabbf2e1d0754d36af92d380ce9b8e329603b1b4412f76c250aa223c88310.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "设右端项为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "b = [b_{1}, \\cdots, b_{n}]^{T}"
                    },
                    {
                        "type": "text",
                        "content": "  (未知量)，统一对增广矩阵做初等变换   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "[A - \\lambda E, b] \\to [A', b']"
                    },
                    {
                        "type": "text",
                        "content": " ，选取顺序为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{1}, \\cdots, X_{k}"
                    },
                    {
                        "type": "text",
                        "content": " 。  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{1}"
                    },
                    {
                        "type": "text",
                        "content": "  要保证不能选   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "[0, \\cdots, 0]^{T}"
                    },
                    {
                        "type": "text",
                        "content": " ，且选的   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{k}"
                    },
                    {
                        "type": "text",
                        "content": "  要保证   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{k+1}"
                    },
                    {
                        "type": "text",
                        "content": "  有解。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "2. 对于一阶：即   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX = \\lambda X"
                    },
                    {
                        "type": "text",
                        "content": "  特征向量，先求完非一阶再求，不要选   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "[0, \\cdots, 0]^{T}"
                    },
                    {
                        "type": "text",
                        "content": " ，保证与其他列向量线性无关。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "6.4  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵的进一步研究"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "71"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 6.11 已知矩阵 A 及其 Jordan 标准型 J，求过渡矩阵 P 使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^{-1}AP = J"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c c} 2 & 3 & 2 \\\\ 1 & 8 & 2 \\\\ - 2 & - 1 4 & - 3 \\end{array} \\right] \\quad J = \\left[ \\begin{array}{c c c} 1 & 0 & 0 \\\\ 0 & 3 & 1 \\\\ 0 & 0 & 3 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/5b2c2d28f1831c349a9ad019d829ead012bc06b59478052dc0d75edbcee15688.jpg"
                }
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P = (X_{1}, X_{2}, X_{3})"
                    },
                    {
                        "type": "text",
                        "content": "，根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^{-1}AP = J"
                    },
                    {
                        "type": "text",
                        "content": " 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(AX_{1}, AX_{2}, AX_{3}) = (X_{1}, X_{2}, X_{3})J"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX_{1} = X_{1}, AX_{2} = 3X_{2}, AX_{3} = X_{2} + 3X_{3}"
                    },
                    {
                        "type": "text",
                        "content": "，即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A - E)X_{1} = 0, (A - 3E)X_{2} = 0, (A - 3E)X_{3} = X_{2}"
                    },
                    {
                        "type": "text",
                        "content": "。先算后面非一阶，做初等变换得到："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "[ A - 3 E | b ] \\rightarrow \\left[\\begin{array}{c c c c}{- 1}&{3}&{2}&{b _ {1}}\\\\{0}&{8}&{4}&{b _ {1} + b _ {2}}\\\\{0}&{8}&{4}&{\\frac {1}{2} b _ {1} + \\frac {5}{2} b _ {2} + b _ {3}}\\end{array}\\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/1412603ee6ee1dec7b36555ee288edbf4c94a07868e74f045c3f8d7b013568c8.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "有解必须满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\frac{1}{2} b_1 + \\frac{5}{2} b_2 + b_3 = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX_{2} = 0"
                    },
                    {
                        "type": "text",
                        "content": " 等价于："
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c c} - 1 & 3 & 2 \\\\ 0 & 8 & 4 \\\\ \\frac {1}{2} & \\frac {5}{2} & 1 \\end{array} \\right] X _ {2} = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/860dfbf4c912a304beb19f1c6d2c802050e55d005fdd96dfd4229eee14318be9.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "而最后一个条件显然可以被前两行消掉，因此可以无视，随便取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{2} = [1, - 1,2]^{T}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{3}"
                    },
                    {
                        "type": "text",
                        "content": " 满足"
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c c} - 1 & 3 & 2 \\\\ 0 & 8 & 4 \\\\ 0 & 0 & 0 \\end{array} \\right] X _ {3} = \\left[ \\begin{array}{c} 1 \\\\ 0 \\\\ 0 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/fc6f47fb3ab73cc0a37cfc4c0f59767c81d043ac836341db77b5cfa55616233a.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "X_{3}"
                    },
                    {
                        "type": "text",
                        "content": " 取法无所谓，因此取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{3} = [-1,0,0]^{T}"
                    },
                    {
                        "type": "text",
                        "content": " 。而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 取线性无关即可，如  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{1} = [-2,0,1]^{T}"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 6.12 已知矩阵 A 及其 Jordan 标准型 J，求过渡矩阵 P 使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^{-1}AP = J"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{r r r} 0 & 1 & 0 \\\\ - 4 & 4 & 0 \\\\ - 2 & 1 & 2 \\end{array} \\right], J = \\left[ \\begin{array}{c c c} 2 & & \\\\ & 2 & 1 \\\\ & & 2 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/a26b831d8c43d0fdde81e02de42613beca6fb0ae81d98249527e68fbff511735.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P = [X_1, X_2, X_3]"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(AX_1, AX_2, AX_3) = (2X_1, 2X_2, X_2 + 2X_3)"
                    },
                    {
                        "type": "text",
                        "content": "，得到方程组  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(A - 2E)X_1 = 0, (A - 2E)X_2 = 0, (A - 2E)X_3 = X_2"
                    },
                    {
                        "type": "text",
                        "content": "。做初等变换后："
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "[ A | b ] \\to \\left[ \\begin{array}{c c c c} - 2 & 1 & 0 & b _ {1} \\\\ 0 & 0 & 0 & b _ {2} - 2 b _ {1} \\\\ 0 & 0 & 0 & b _ {3} - b _ {1} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/d77e9b7a17a77a41eed234a37650ed3313ccbb050fc6c515e153591028d00dba.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此要使  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AX_{3} = X_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 有解， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 必须满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "b_{3} = b_{1}"
                    },
                    {
                        "type": "text",
                        "content": "，先选取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{2} = [1,2,1]^{T}"
                    },
                    {
                        "type": "text",
                        "content": "，解出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{3} = \\left[-\\frac{1}{2},0,0\\right]^{T}"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 要线性无关，取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "[0,0,1]^{T}"
                    },
                    {
                        "type": "text",
                        "content": " 即可。"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "6.4  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵的进一步研究"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "72"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "第 7 章 双线性函数与二次型"
                    }
                ],
                "level": 1
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.1 双线性函数"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.1.1 双线性函数概念"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 7.1 (线性函数)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "K"
                    },
                    {
                        "type": "text",
                        "content": " 上的线性空间， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f: V \\to K"
                    },
                    {
                        "type": "text",
                        "content": " 且满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(\\alpha + \\beta) = f(\\alpha) + f(\\beta), f(k\\alpha) = kf(\\alpha)"
                    },
                    {
                        "type": "text",
                        "content": "，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 上的线性函数。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 7.2 (双线性函数)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "K"
                    },
                    {
                        "type": "text",
                        "content": " 上的线性空间， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f: V \\times V \\to K"
                    },
                    {
                        "type": "text",
                        "content": "，且满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(k_1\\alpha_1 + k_2\\alpha_2, \\beta) = k_1f(\\alpha_1, \\beta) + k_2f(\\alpha_2, \\beta), f(\\alpha, l_1\\beta_1 + l_2\\beta_2) = l_1f(\\alpha, \\beta_1) + l_2f(\\alpha, \\beta_2)"
                    },
                    {
                        "type": "text",
                        "content": "，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 上的双线性函数"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.1.2 双线性函数度量矩阵"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 7.3 (度量矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "V 是 K 上的 n 维线性空间，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1},\\cdots,\\epsilon_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  是 V 的一组基，f 为 V 上的双线性函数，则 f 的度量矩阵 G 定义如下："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "G = \\left[ \\begin{array}{c c c c} f (\\epsilon_ {1}, \\epsilon_ {1}) & f (\\epsilon_ {1}, \\epsilon_ {2}) & \\dots & f (\\epsilon_ {1}, \\epsilon_ {n}) \\\\ f (\\epsilon_ {2}, \\epsilon_ {1}) & f (\\epsilon_ {2}, \\epsilon_ {2}) & \\dots & f (\\epsilon_ {2}, \\epsilon_ {n}) \\\\ \\vdots & \\vdots & & \\vdots \\\\ f (\\epsilon_ {n}, \\epsilon_ {1}) & f (\\epsilon_ {n}, \\epsilon_ {2}) & \\dots & f (\\epsilon_ {n}, \\epsilon_ {n}) \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/d0a2c7e130706eb2319c74bf655219536b842cfa8371cc3f6a38cb15e700b46a.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.1 (双线性函数度量矩阵唯一确定)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x, y \\in V"
                    },
                    {
                        "type": "text",
                        "content": "  在   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1}, \\cdots, \\epsilon_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  下的坐标为 X, Y，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(\\alpha, \\beta) = X^{T}GY"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.1.3 双线性函数不同基下的矩阵：合同"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 7.4 (合同)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "K"
                    },
                    {
                        "type": "text",
                        "content": " 上的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵，若存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "K"
                    },
                    {
                        "type": "text",
                        "content": " 上的可逆  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶方阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B = T^T A T"
                    },
                    {
                        "type": "text",
                        "content": " ，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 合同。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 从矩阵角度来看，合同相当于 A 通过一系列相同的行与列变换变为 B"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.2 (不同基下度量矩阵关系)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 合同当且仅当它们是双线性函数两组基下的矩阵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.3 (分块矩阵合同)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "这里只考虑  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶实对称矩阵，希望对  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 整体或者一部分做合同变换："
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\left( \\begin{array}{cc}A_{1} & \\alpha \\\\ \\alpha^{T} & a_{nn} \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": "，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 可逆，则取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P = \\left( \\begin{array}{cc}E_{n - 1} & -A_{1}^{-1}\\alpha \\\\ 0 & 1 \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " (用列变换记忆)，得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "P ^ {T} A P = \\left( \\begin{array}{c c} E _ {n - 1} & 0 \\\\ 0 & a _ {n n} - \\alpha^ {T} A _ {1} ^ {- 1} \\alpha \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f94859c46aacb380aa2d60d0be33a1ee733c7a94f77268c1e5bebfcb0773408b.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\bullet A = \\left( \\begin{array}{cc}a_{11} & \\alpha^T\\\\ \\alpha & A_{n - 1} \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " ，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{11}\\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " ，取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P = \\left( \\begin{array}{cc}1 & -\\frac{1}{a_{11}}\\alpha^{T}\\\\ 0 & E_{n - 1} \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " ，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "P ^ {T} A P = \\left( \\begin{array}{c c} a _ {1 1} & 0 \\\\ 0 & A _ {n - 1} - \\frac {1}{a _ {1 1}} \\alpha \\alpha^ {T} \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/5dcf8cf96bc0766a422ddb20f109df0a12f608091ce09f96edc97dd5b2ecb800.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 若想对上面的右下角分块  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{n - 1} - \\frac{1}{a_{11}}\\alpha \\alpha^T"
                    },
                    {
                        "type": "text",
                        "content": " 做合同变换，变为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P_{1}^{T}\\left(A_{n - 1} - \\frac{1}{a_{11}}\\alpha \\alpha^{T}\\right)P_{1}"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q = P\\left( \\begin{array}{ll}1 & 0\\\\ 0 & P_{1} \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " (用列变换记忆)，得到"
                    }
                ]
            },
            "bbox": [
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                882,
                350
            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "Q ^ {T} A Q = \\left( \\begin{array}{c c} a _ {1 1} & 0 \\\\ 0 & P _ {1} ^ {T} \\left(A _ {n - 1} - \\frac {1}{a _ {1 1}} \\alpha \\alpha^ {T}\\right) P _ {1} \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/af586a97a28e2dad330762c46b3363603cb6005b7c65c2aaf8ccd2209e2a4355.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.2 二次型"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.2.1 二次型概念及其矩阵"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 7.5 (二次型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "定义多元二次多项式   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},x_{2},\\cdots,x_{n})=\\sum_{i=1}^{n}a_{ii}x_{i}^{2}+\\sum_{1\\leq i<j\\leq n}2a_{ij}x_{i}x_{j}"
                    },
                    {
                        "type": "text",
                        "content": "  为二次型"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 7.6 (二次型的矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于任意二次型   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},x_{2},\\cdots,x_{n})"
                    },
                    {
                        "type": "text",
                        "content": " ，总能找到一个对称矩阵 A，使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n})=x^{T}Ax"
                    },
                    {
                        "type": "text",
                        "content": " ，称矩阵 A 为二次型的矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 对于一个二次型   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n})"
                    },
                    {
                        "type": "text",
                        "content": " ，存在不止一个 A 使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n})=x^{T}Ax"
                    },
                    {
                        "type": "text",
                        "content": " ，但是二次型的矩阵特指那个对称矩阵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习7.1二次型的矩阵(1)给定二次型  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},x_{2},x_{3}) = 2x_{1}x_{2} + 2x_{1}x_{3} - 6x_{2}x_{3}"
                    },
                    {
                        "type": "text",
                        "content": " ，计算二次型的矩阵"
                    }
                ]
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                        "type": "text",
                        "content": "(2) 给定二次型   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(X)"
                    },
                    {
                        "type": "text",
                        "content": " ，计算二次型的矩阵："
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            "content": {
                "math_content": "f (x _ {1}, x _ {2}) = \\left( \\begin{array}{c c} x _ {1} & x _ {2} \\end{array} \\right) \\left( \\begin{array}{c c} 2 & 1 \\\\ 3 & 1 \\end{array} \\right) \\binom{x _ {1}}{x _ {2}}",
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                        "type": "text",
                        "content": "(3) 重点：求  "
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                    {
                        "type": "equation_inline",
                        "content": "f(x_{1}, \\cdots, x_{n}) = \\sum_{i=1}^{m} (a_{i1}x_{1} + \\cdots + a_{in}x_{n})^{2}"
                    },
                    {
                        "type": "text",
                        "content": " 的矩阵"
                    }
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                        "type": "text",
                        "content": "解(1)  "
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                    {
                        "type": "equation_inline",
                        "content": "A = \\left( \\begin{array}{ccc}0 & 1 & 1\\\\ 1 & 0 & -3\\\\ 1 & -3 & 0 \\end{array} \\right)"
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                ]
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                        "content": "7.2 二次型"
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                        "type": "text",
                        "content": "(2) 展开得到  "
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                        "content": "f(x_{1}, x_{2}) = 2x_{1}^{2} + 4x_{1}x_{2} + x_{2}^{2}"
                    },
                    {
                        "type": "text",
                        "content": "，因此矩阵为"
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                "math_content": "\\left( \\begin{array}{c c} 2 & 2 \\\\ 2 & 1 \\end{array} \\right)",
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                        "type": "text",
                        "content": "(3) 令  "
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                    {
                        "type": "equation_inline",
                        "content": "A = (a_{ij})"
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                    {
                        "type": "text",
                        "content": "，则"
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            "content": {
                "math_content": "f = \\sum_ {i = 1} ^ {m} (a _ {i 1} x _ {1} + \\dots + a _ {i n} x _ {n}) ^ {2} = \\sum_ {i = 1} ^ {m} \\left[ \\left( \\begin{array}{c c c} x _ {1} & \\dots & x _ {n} \\end{array} \\right) \\left( \\begin{array}{c} a _ {i 1} \\\\ \\vdots \\\\ a _ {i n} \\end{array} \\right) \\right] ^ {2}",
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            "content": {
                "math_content": "= \\sum_ {i = 1} ^ {m} \\left[ \\left( \\begin{array}{c c c} x _ {1} & \\dots & x _ {n} \\end{array} \\right) \\left( \\begin{array}{c} a _ {i 1} \\\\ \\vdots \\\\ a _ {i n} \\end{array} \\right) \\left( \\begin{array}{c c c} a _ {i 1} & \\dots & a _ {i n} \\end{array} \\right) \\left( \\begin{array}{c} x _ {1} \\\\ \\vdots \\\\ x _ {n} \\end{array} \\right) \\right] = X ^ {T} (A ^ {T} A) X",
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                        "content": "其中计算出  "
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                        "content": "A^T A"
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                        "content": " 的过程为"
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                "math_content": "A = \\left[ \\begin{array}{c} A _ {1} \\\\ A _ {2} \\\\ \\vdots \\\\ A _ {m} \\end{array} \\right] \\Rightarrow A ^ {T} A = \\left[ \\begin{array}{c c c c} A _ {1} ^ {T} & A _ {2} ^ {T} & \\dots & A _ {m} ^ {T} \\end{array} \\right] \\left[ \\begin{array}{c} A _ {1} \\\\ A _ {2} \\\\ \\vdots \\\\ A _ {m} \\end{array} \\right] = \\sum_ {i = 1} ^ {m} A _ {i} ^ {T} A _ {i}",
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                        "content": "定义 7.7 (二次型等价)"
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                        "content": "给定二次型  "
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                    {
                        "type": "equation_inline",
                        "content": "f = x^{T}Ax, g = y^{T}By"
                    },
                    {
                        "type": "text",
                        "content": "，若存在可逆矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T"
                    },
                    {
                        "type": "text",
                        "content": "，使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = Ty"
                    },
                    {
                        "type": "text",
                        "content": "，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f = y^{T}(T^{T}AT)y = y^{T}By = g"
                    },
                    {
                        "type": "text",
                        "content": "，则称二次型  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f, g"
                    },
                    {
                        "type": "text",
                        "content": " 等价（即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 合同）"
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                        "content": "定理 7.4 (合同的性质)"
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                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = C^T BC"
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                    {
                        "type": "text",
                        "content": " ，则"
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                        "type": "text",
                        "content": "- 秩不变：  "
                    },
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                        "content": "r(A)=r(B)"
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                        "type": "text",
                        "content": "-  "
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                    {
                        "type": "equation_inline",
                        "content": "C"
                    },
                    {
                        "type": "text",
                        "content": " 不一定唯一"
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                        "type": "text",
                        "content": "- 分块合同：若  "
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                    {
                        "type": "equation_inline",
                        "content": "A_{1}, A_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 分别与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B_{1}, B_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 合同，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\operatorname{diag}\\{A_{1}, A_{2}\\}"
                    },
                    {
                        "type": "text",
                        "content": " 与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\operatorname{diag}\\{B_{1}, B_{2}\\}"
                    },
                    {
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                        "content": " 合同"
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                ]
            },
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                        "type": "text",
                        "content": "证明 (1) 由于  "
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                    {
                        "type": "equation_inline",
                        "content": "C"
                    },
                    {
                        "type": "text",
                        "content": " 是满秩矩阵, 相当于一系列初等变换, 因此不改变秩"
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                ]
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                        "type": "text",
                        "content": "(2)例如  "
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                    {
                        "type": "equation_inline",
                        "content": "\\left( \\begin{array}{cc}1 & 0\\\\ 0 & -1 \\end{array} \\right),\\left( \\begin{array}{cc}1 & 0\\\\ 0 & 1 \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " 在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上合同，但是"
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                ]
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            "content": {
                "math_content": "\\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & - 1 \\end{array} \\right) = \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & i \\end{array} \\right) \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & 1 \\end{array} \\right) \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & i \\end{array} \\right) = \\left( \\begin{array}{c c} - 1 & 0 \\\\ 0 & i \\end{array} \\right) \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & 1 \\end{array} \\right) \\left( \\begin{array}{c c} - 1 & 0 \\\\ 0 & i \\end{array} \\right)",
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            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习7.2 反对称矩阵的二次型 证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是反对称矩阵当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall X, X^T A X = 0"
                    },
                    {
                        "type": "text",
                        "content": " 证明 左推右显然。右推左根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^T A X = X^T A^T X = -X^T A X = 0"
                    },
                    {
                        "type": "text",
                        "content": " 可知"
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                ]
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                        "type": "text",
                        "content": "7.2.2 二次型的标准型存在与配方法"
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                        "content": "定理 7.5 (二次型标准型的存在性)"
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                "level": 2
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            "content": {
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                        "type": "text",
                        "content": "对数域 P 上任意对称矩阵 A，其都合同于一个对角矩阵，且对角矩阵不唯一。"
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                        "type": "text",
                        "content": "证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n = 1"
                    },
                    {
                        "type": "text",
                        "content": " 则显然。设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n - 1"
                    },
                    {
                        "type": "text",
                        "content": " 时成立，则考虑  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\left( \\begin{array}{ll}a_{11} & \\alpha^T\\\\ \\alpha & A_1 \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " ，这里  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 对称。若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{11}\\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " ，则根据"
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                ]
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                        "content": "75"
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                        "content": "7.2 二次型"
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        }
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "前面分块矩阵合同结论，取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P = \\left( \\begin{array}{cc}1 & -\\frac{1}{a_{11}}\\alpha^T\\\\ 0 & E_{n - 1} \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " ，得到"
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                "math_content": "P ^ {T} A P = \\left( \\begin{array}{c c} a _ {1 1} & 0 \\\\ 0 & A _ {1} - \\frac {1}{a _ {1 1}} \\alpha \\alpha^ {T} \\end{array} \\right) := \\left( \\begin{array}{c c} a _ {1 1} & 0 \\\\ 0 & B _ {1} \\end{array} \\right)",
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        {
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                        "type": "text",
                        "content": "根据归纳假设，存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q_{1}^{T}B_{1}Q_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 为对角阵，则取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q = \\left( \\begin{array}{cc}1 & 0\\\\ 0 & Q_{1} \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " ，则"
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "Q ^ {T} P ^ {T} A P Q = \\left( \\begin{array}{c c} a _ {1 1} & \\\\ & Q _ {1} ^ {T} B _ {1} Q _ {1} \\end{array} \\right)",
                "math_type": "latex",
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                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{11} = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，但  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{kk}\\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " ，则取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P(1,k)"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "E_{n}"
                    },
                    {
                        "type": "text",
                        "content": " 第一列与第  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "k"
                    },
                    {
                        "type": "text",
                        "content": " 列互换的矩阵，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^T (1,k)AP(1,k)"
                    },
                    {
                        "type": "text",
                        "content": " 的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{11}^{\\prime}\\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " 且两个矩阵合同，转换为第一种情况。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若对角全为 0，讨论 A 第一行，若 A 第一行全为 0，则显然。否则不妨设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{12} \\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " ，则 A 如下，做相同行列变换变为："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left( \\begin{array}{c c c} 0 & a _ {1 2} & * \\\\ a _ {1 2} & 0 & * \\\\ * & * & * \\end{array} \\right) \\to \\left( \\begin{array}{c c c} 2 a _ {1 2} & a _ {1 2} & * \\\\ a _ {1 2} & a _ {2 2} & * \\\\ * & * & * \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/e253e290d3e26829aaf1994ad3543bb2273e5cd8b62281a7eaf8a809103ebac9.jpg"
                }
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "转换为第一种类型"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 要特别注意，如果对角均为零，则将同行的非零元素加到对角上！"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 7.8 (二次型标准型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于二次型  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f = x^{T}Ax"
                    },
                    {
                        "type": "text",
                        "content": "，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 与对角阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "D"
                    },
                    {
                        "type": "text",
                        "content": " 合同，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "g = z^{T}Dz"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f"
                    },
                    {
                        "type": "text",
                        "content": " 的标准型"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.6 (配方法)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "配方法按照以下步骤进行："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "1. 若无  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 的平方项，但有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1}x_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 交叉项，则用  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1} = y_{1} + y_{2}, x_{2} = y_{1} - y_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 换元。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "2. 把  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 的项放在一起，全部配方。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "3. 把凑成的   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(x_{1}+\\cdots)^{2}"
                    },
                    {
                        "type": "text",
                        "content": "  设为新变量"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "4. 以此类推"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "image",
            "content": {
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                "content": "",
                "image_caption": [],
                "image_footnote": []
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "笔记 配方法一般算过渡矩阵不太方便，不过算标准型还行"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 7.3 配方法 (1) 用配方法计算   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1}, x_{2}, x_{3}) = 2x_{1}x_{2} + 2x_{1}x_{3} - 6x_{2}x_{3}"
                    },
                    {
                        "type": "text",
                        "content": "  的标准型解 (1) 无   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1}^{2}"
                    },
                    {
                        "type": "text",
                        "content": "  项，因此令   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1} = y_{1} + y_{2}, x_{2} = y_{1} - y_{2}, x_{3} = y_{3}"
                    },
                    {
                        "type": "text",
                        "content": " ，得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "f = 2 (y _ {1} - y _ {3}) ^ {2} - 2 y _ {2} ^ {2} - 2 y _ {3} ^ {2} + 8 y _ {2} y _ {3}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/850425b6cd01327c9265d8958a6dee4169ef01c14e3fbc8954f58c873204e653.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "令  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1} = y_{1} - y_{3}, x_{2} = y_{2}, x_{3} = y_{3}"
                    },
                    {
                        "type": "text",
                        "content": " 得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f = 2x_{1}^{2} - 2(x_{2}^{2} - 4x_{2}x_{3} + 4x_{3}^{2}) + 6x_{3}^{2}"
                    },
                    {
                        "type": "text",
                        "content": ", 再做换元  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "y_{1} = x_{1}, y_{2} = x_{2} - 2x_{3}, y_{3} = x_{3}"
                    },
                    {
                        "type": "text",
                        "content": ", 得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "f = 2 y _ {1} ^ {2} - 2 y _ {2} ^ {2} + 6 y _ {3} ^ {2}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/96acf09d552f61d73e05417f5f75e2abe4a7cc702f3d8ff09d4723f6723eb1fb.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "7.2 二次型"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "76"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "image",
            "content": {
                "image_source": {
                    "path": "images/1fbb47a9c475507d8ef1e789ee4ed7772a02bb0019876a7edda07571148f915d.jpg"
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                "image_caption": [],
                "image_footnote": []
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.2.3 二次型的标准型：初等变换法"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.7 (标准型的计算：初等变换法)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对于对称矩阵 A 使用以下格式计算初等变换 (注意这不一定得出正交矩阵):"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\binom{{{{A}}}}{{{{E}}}} \\rightarrow \\binom{{{{C ^ {T} A C}}}}{{{{C}}}} \\quad \\text {对} \\mathbf {A} \\text {行变换和对应列变换，对} \\mathbf {I} \\text {只列变换}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/ac5913a9399af6b05c94d56a492a3fe7583a12938a4169f0254b7c548b4064d6.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "具体的步骤就是证明标准型存在性中的过程：若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 首对角为零，存在非零对角，则互换位置。若对角均为零，则找到第一行非零的元素加到首对角位置(下面的题目马上会用)"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 做同样的行列变换一定要小心，两种变换并非 “同时” 做的，而是先后做的，即做行变换时的矩阵已经是列变换后的，而非原本的矩阵！"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习7.4标准型计算：初等变换法(1)简单：计算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f = x_{1}^{2} + 2x_{1}x_{2} + 2x_{2}^{2} + 4x_{2}x_{3} + 4x_{3}^{2}"
                    },
                    {
                        "type": "text",
                        "content": " 的标准型"
                    }
                ]
            },
            "bbox": [
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            ]
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 对角均零：计算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1}, x_{2}, x_{3}) = 2x_{1}x_{2} + 2x_{1}x_{3} - 6x_{2}x_{3}"
                    },
                    {
                        "type": "text",
                        "content": "，计算其标准型，并求出过渡矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解(1)结果为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f = y_{1}^{2} + y_{2}^{2}"
                    },
                    {
                        "type": "text",
                        "content": " ，千万小心别行列变换错了"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 首先显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\left( \\begin{array}{ccc}0 & 1 & 1\\\\ 1 & 0 & -3\\\\ 1 & -3 & 0 \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " ，列出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\left( \\begin{array}{c}A\\\\ E \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " 并做行列初等变换，由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 对角均为0，因此将第二列和第二行加到首对角位置："
                    }
                ]
            },
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left(\\begin{array}{c c c}0&1&1\\\\1&0&- 3\\\\1&- 3&0\\\\1&0&0\\\\0&1&0\\\\0&0&1\\end{array}\\right)\\rightarrow \\left(\\begin{array}{c c c}2&1&- 2\\\\1&0&- 3\\\\- 2&- 3&0\\\\1&0&0\\\\1&1&0\\\\0&0&1\\end{array}\\right)\\rightarrow \\left(\\begin{array}{c c c}2&1&0\\\\1&0&- 2\\\\0&- 2&- 2\\\\1&0&1\\\\1&1&1\\\\0&0&1\\end{array}\\right)\\rightarrow \\dots",
                "math_type": "latex",
                "image_source": {
                    "path": "images/d80fefc0b66f9e3f3eec6b2e455304b19a2e46a609e6c7ef251b199ac012fdb2.jpg"
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            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "最终变为结果为:"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left( \\begin{array}{c c c} 2 & 0 & 0 \\\\ 0 & - \\frac {1}{2} & 0 \\\\ 0 & 0 & 6 \\\\ 1 & - \\frac {1}{2} & 3 \\\\ 1 & \\frac {1}{2} & - 1 \\\\ 0 & 0 & 1 \\end{array} \\right) = \\binom{C ^ {T} A C}{C}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/49d237973150b0594265f43a64e65af01c06aad66d5f96377f93ddfc265822a7.jpg"
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.2.4 二次型的标准型：正交变换法"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.8 (正交对角化的可行性)"
                    }
                ],
                "level": 2
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "任何实对称矩阵 A 一定正交相似于对角阵。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 做数学归纳法，n=1 显然，假设 n-1 成立，任取特征值   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda"
                    },
                    {
                        "type": "text",
                        "content": " ，对应单位特征向量   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}"
                    },
                    {
                        "type": "text",
                        "content": " ，将   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}"
                    },
                    {
                        "type": "text",
                        "content": "  扩充为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "R^{n}"
                    },
                    {
                        "type": "text",
                        "content": "  的一组标准正交基   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，令   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T=(\\alpha_{1},\\cdots,\\alpha_{n})"
                    },
                    {
                        "type": "text",
                        "content": " ，则"
                    }
                ]
            },
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        {
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            "content": {
                "math_content": "A T = T \\left( \\begin{array}{c c} \\lambda & \\alpha^ {T} \\\\ 0 & A _ {n - 1} \\end{array} \\right) \\Rightarrow T ^ {T} A T = \\left( \\begin{array}{c c} \\lambda & \\alpha^ {T} \\\\ 0 & A _ {n - 1} \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/e6fa7344d4600d47882459374c4414bd2cf03a798009d839fa1c15bf521f403e.jpg"
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T^T A T"
                    },
                    {
                        "type": "text",
                        "content": " 对称，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha^T = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，再根据归纳假设，存在正交矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P_{1}^{T}A_{n - 1}P_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 为对角阵，因此令  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P ="
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "7.2 二次型"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "77"
                    }
                ]
            },
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\mathrm{diag}(1, P_1)"
                    },
                    {
                        "type": "text",
                        "content": "，此时"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "P ^ {T} T ^ {T} A T P = \\left( \\begin{array}{c c} \\lambda & 0 \\\\ 0 & P ^ {T} A _ {n - 1} P \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/deeb0158441a760d1219164ffa0c6f1cf7d08c5b6be6da506f7944fc42528470.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "TP = Q"
                    },
                    {
                        "type": "text",
                        "content": " ，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q"
                    },
                    {
                        "type": "text",
                        "content": " 也是正交阵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.9 (标准型计算：正交变换)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "正交变换计算标准型过程如下："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 计算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_1, \\cdots, \\lambda_n"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 计算  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": " 对应的特征向量  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbf{v}_{i}"
                    },
                    {
                        "type": "text",
                        "content": "，不同特征值的特征向量必正交，同一特征值的特征向量需正交化"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 将特征向量单位化，组成  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T = [\\mathbf{v}_1, \\dots, \\mathbf{v}_n]"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 7.5 正交变换法 (1) 已知矩阵 A 如下，(a) 若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "tE + A"
                    },
                    {
                        "type": "text",
                        "content": "  为正定矩阵，求 t 的取值范围 (b) 求正交矩阵 T 使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T^{T}AT"
                    },
                    {
                        "type": "text",
                        "content": "  为对角矩阵"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{r r r} 2 & 2 & - 1 \\\\ 2 & 5 & - 4 \\\\ - 2 & - 4 & 5 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/e5ce20d1ab0f88a334c30ea686e9741d8e73344a3c39ec8cdd8e298b4b53203c.jpg"
                }
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|\\lambda E - A| = (\\lambda - 1)^{2} (\\lambda - 10)"
                    },
                    {
                        "type": "text",
                        "content": " ，特征值为 1, 10，正定要求 t > -1。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "先求  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(E - A)X = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，得到："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c c} 1 & 2 & - 2 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right] X = 0 \\Rightarrow X = k _ {1} \\left( \\begin{array}{c} - 2 \\\\ 1 \\\\ 0 \\end{array} \\right) + k _ {2} \\left( \\begin{array}{c} 2 \\\\ 0 \\\\ 1 \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/855d046d209813551ed424713af19b8b1cca285ad7c9f584fbb454fbe41b81ad.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "单位化得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\frac{1}{\\sqrt{5}} (-2,1,0)^T,\\frac{1}{\\sqrt{5}} (2,0,1)^T"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "再求  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(10E - A)X = 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c c c} 2 & 4 & 5 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 0 \\end{array} \\right] X = 0 \\Rightarrow X = k _ {1} \\left( \\begin{array}{c} 1 \\\\ 2 \\\\ - 2 \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/211884562d53ed88505b8b258c9562c324569a0c8942a2a84d8f208ed39d50f0.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "单位化得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\frac{1}{3} (1,2, - 2)"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "最终结果为:"
                    }
                ]
            },
            "bbox": [
                124,
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "T = \\left[ \\begin{array}{c c c} - \\frac {2}{\\sqrt {5}} & \\frac {2}{3 \\sqrt {5}} & \\frac {1}{3} \\\\ - \\frac {1}{\\sqrt {5}} & \\frac {4}{3 \\sqrt {5}} & \\frac {2}{3} \\\\ 0 & \\frac {5}{3 \\sqrt {5}} & - \\frac {2}{3} \\end{array} \\right], T ^ {T} A T = \\left[ \\begin{array}{c c c} 1 & & \\\\ & 1 & \\\\ & & 1 0 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/673d57f9854059917d5dffa58d775961f8f2de5b13b9c126cb66c8f207426ab3.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.10 (二次型与特征值)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "设 n 阶实对称矩阵 A 的全部特征值排序为：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1} \\geq \\lambda_{2} \\geq \\cdots \\geq \\lambda_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\max _ {X \\in \\mathbb {R} ^ {n}} \\frac {X ^ {T} A X}{X ^ {T} X} = \\lambda_ {1}, \\min _ {X \\in \\mathbb {R} ^ {n}} \\frac {X ^ {T} A X}{X ^ {T} X} = \\lambda_ {n}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/7c35c89a4dbf6e75a44dd94f4b84d4b7477a48e7117bfcf975d3b1673aba1bb0.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_1E - A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值均非负，故  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_1E - A"
                    },
                    {
                        "type": "text",
                        "content": " 半正定，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall X\\in \\mathbb{R}^n"
                    },
                    {
                        "type": "text",
                        "content": " 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^{T}(\\lambda E - A)X\\geq 0"
                    },
                    {
                        "type": "text",
                        "content": " ，得到 "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^{T}AX\\leq \\lambda_{1}X^{T}X"
                    },
                    {
                        "type": "text",
                        "content": " ，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_1E - A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值，故  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists X_0\\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_0^T (\\lambda_1E - A)X_0 = 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 7.6 二次型的特征值研究 (1) 重点: A, B 是 n 阶实对称矩阵, A 的特征值都大于 a, B 的特征值都大于 b,"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "78"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "7.2 二次型"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A + B"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值都大于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a + b"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) B 是 n 阶实矩阵，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B^{T}B"
                    },
                    {
                        "type": "text",
                        "content": "  的全部特征值按照大小顺序排为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{1} \\geq \\lambda_{2} \\geq \\cdots \\geq \\lambda_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，证明：B 的任意实特征值   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mu"
                    },
                    {
                        "type": "text",
                        "content": "  满足   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\sqrt{\\lambda_{n}} \\leq |\\mu| \\leq \\sqrt{\\lambda_{1}}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 根据前面定理可知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^{T}AX > aX^{T}X, X^{T}BX > bX^{T}X"
                    },
                    {
                        "type": "text",
                        "content": "，从而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^{T}(A + B)X > (a + b)X^{T}X"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A + B"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值大于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a + b"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B\\alpha = \\mu \\alpha"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha^T B^T = \\mu \\alpha^T"
                    },
                    {
                        "type": "text",
                        "content": "，得到"
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\alpha^ {T} B ^ {T} B \\alpha = \\mu^ {2} \\alpha^ {T} \\alpha \\in [ \\lambda_ {n} \\alpha^ {T} \\alpha , \\lambda_ {1} \\alpha^ {T} \\alpha ] \\Rightarrow \\lambda_ {n} \\leq \\mu^ {2} \\leq \\lambda_ {1}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/9b89f5f97f1a4c05205c06e032c587305b7a646e5df09c85d06b37bb4c6e482c.jpg"
                }
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.2.5 R, C 上的规范型"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "前面根据标准型对二次型进行分类，但是可以发现二次型的标准型是不唯一的，因此在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{R},\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上可以进一步划分，即每个规范型表示一个类。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.11 (C 上的规范型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上任一二次型  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f"
                    },
                    {
                        "type": "text",
                        "content": " 都等价于"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "g = u _ {1} ^ {2} + \\dots + u _ {r} ^ {2}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/2b6488daec2f43337c4b6dae3b6a676e4f55e512fb0bef9f738bfdf0d89a4bb9.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "推论 7.1 (C 上合同充要条件)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "C 上两个对称矩阵合同的充要条件为它们的秩相同。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.12 (R 上的规范型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{R}"
                    },
                    {
                        "type": "text",
                        "content": " 上每个二次型  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f"
                    },
                    {
                        "type": "text",
                        "content": " 都等价于一个如下二次型。将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "p"
                    },
                    {
                        "type": "text",
                        "content": " 称为正惯性指数， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r - p"
                    },
                    {
                        "type": "text",
                        "content": " 称为负惯性指数"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "g = u _ {1} ^ {2} + \\dots + u _ {p} ^ {2} - u _ {p + 1} ^ {2} - \\dots - u _ {r} ^ {2}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/be706adf4165ac46d4c1d4d0fc410fd811c3b8df9f74ed32fcf2934fb90834f2.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "推论 7.2 (  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{R}"
                    },
                    {
                        "type": "text",
                        "content": " 上合同充要条件)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "R 上两个对称矩阵充要条件是它们有相同的正负惯性指数。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习7.7规范型的应用(1)设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶实对称矩阵， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A| < 0"
                    },
                    {
                        "type": "text",
                        "content": "，证明： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists X \\in \\mathbb{R}^n"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^T A X < 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(X) = X^{T}AX"
                    },
                    {
                        "type": "text",
                        "content": " 为实二次型，若存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{1}, X_{2}"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_{1}^{T}AX_{1} > 0, X_{2}^{T}AX_{2} < 0"
                    },
                    {
                        "type": "text",
                        "content": " ，证明： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists X_0 \\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X_0^TAX_0 = 0"
                    }
                ]
            },
            "bbox": [
                89,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A| < 0"
                    },
                    {
                        "type": "text",
                        "content": ", 可知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 满秩且负惯性指数个数不为零, 做非退化线性替换  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X = CY"
                    },
                    {
                        "type": "text",
                        "content": " 即可得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Y^{T}C^{T}ACX"
                    },
                    {
                        "type": "text",
                        "content": ", 这里"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "C ^ {T} A C = \\operatorname{diag} \\{1, 1, \\dots , 1, - 1, - 1, \\dots , - 1 \\}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/a46d657f26114740ce3cbc1154fc95f0d723391cc631885ef17fdf4d716385cd.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Y = e_{n}"
                    },
                    {
                        "type": "text",
                        "content": " 时  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Y^{T}C^{T}ACY = -1"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 正负惯性指数均非零，因此可转换为规范型   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(X)=y_{1}^{2}+\\cdots+y_{p}^{2}-y_{p+1}^{2}-\\cdots-y_{n}^{2}"
                    },
                    {
                        "type": "text",
                        "content": " ，取   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Y=e_{1}+e_{p+1}"
                    },
                    {
                        "type": "text",
                        "content": "  即可"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.3 (半) 正定二次型的研究"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.3.1 正定二次型的概念与判别"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "正定二次型是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{R}"
                    },
                    {
                        "type": "text",
                        "content": " 中二次型中的特殊等价类，其代表为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "y^{T}Ey"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "79"
                    }
                ]
            },
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "7.3 (半) 正定二次型的研究"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 7.9 (正定二次型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "给定实数域上的二次型   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},x_{2},\\cdots,x_{n})"
                    },
                    {
                        "type": "text",
                        "content": " ，若对任意不全为 0 的   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1},\\cdots,x_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n})>0"
                    },
                    {
                        "type": "text",
                        "content": " ，则称   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n})"
                    },
                    {
                        "type": "text",
                        "content": "  为正定矩阵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.13 (正定二次型的规范型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "正定二次型的规范型是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "y^{T}Ey=y_{1}^{2}+y_{2}^{2}+\\cdots+y_{n}^{2}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "image",
            "content": {
                "image_source": {
                    "path": "images/c263772a56ac3460e77a1671c982bbce6d1734c60908b4b744d1e5be1797c031.jpg"
                },
                "content": "",
                "image_caption": [
                    {
                        "type": "text",
                        "content": "图 7.1: 二次型分类示意图"
                    }
                ],
                "image_footnote": []
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.14 (正定充要条件)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f = x^{T}Ax"
                    },
                    {
                        "type": "text",
                        "content": " 是正定二次型当且仅当："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 合同于矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "E"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 存在可逆矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C"
                    },
                    {
                        "type": "text",
                        "content": "，使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = C^T C"
                    },
                    {
                        "type": "text",
                        "content": "，也可以分解为非方阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\left( \\begin{array}{cc} C^T & O^T \\end{array} \\right) \\left( \\begin{array}{c} C \\\\ O \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": "."
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "所有特征值都大于0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的各阶顺序主子式/所有主子式行列式大于0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 等价于顺序主子式大于 0 需要证明：对 A 做如下分块："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\alpha^ {T} & a _ {n n} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/4db91d9f9c58fb099777b47141b22a79c4504f36f05a202aa8efe2c885333b22.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "正定推顺序主子式：若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 正定，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 正定，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 特征值均为正， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|A_1| > 0, |A| > 0"
                    },
                    {
                        "type": "text",
                        "content": "，根据归纳法成立。顺序主子式推正定： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n = 1"
                    },
                    {
                        "type": "text",
                        "content": " 时显然成立， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶矩阵时， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n - 1"
                    },
                    {
                        "type": "text",
                        "content": " 阶矩阵且顺序主子式均正，根据归纳假设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 正定，故存在  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P_{1}^{T} A_{1} P_{1} = E_{n - 1}"
                    },
                    {
                        "type": "text",
                        "content": "，因此令："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "P _ {2} = \\left[ \\begin{array}{c c} P _ {1} & \\\\ & 1 \\end{array} \\right], P _ {3} = \\left[ \\begin{array}{c c} E _ {n - 1} & - P _ {1} ^ {T} \\alpha \\\\ & 1 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/43d8d6269cfe30d3d550633fe965512534e860fbd054893fb01defafdeafa98c.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "得到："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "P _ {3} ^ {T} P _ {2} ^ {T} A P _ {2} P _ {3} = \\left[ \\begin{array}{c c} E _ {n - 1} & \\\\ & a _ {n n} - \\alpha^ {T} P _ {1} P _ {1} ^ {T} \\alpha \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/51e5534f85068538c6524b2de906ff61b641f6f2dfed390180d91a9839110ae0.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "而根据乘积矩阵的行列式性质得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "a _ {n n} - \\alpha^ {T} P _ {1} P _ {1} ^ {T} \\alpha = | A | | P _ {2} | ^ {2} | P _ {3} | ^ {2} > 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/89b5adb007d27da927662cd79cfbd0b97e4a8979b242fa67c05b4a0f14a26099.jpg"
                }
            },
            "bbox": [
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                636,
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "令  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P = P_{2}P_{3}\\mathrm{diag}\\{E_{n - 1},\\frac{1}{\\sqrt{a_{nn} - \\alpha^{T}P_{1}P_{1}^{T}\\alpha}}\\}"
                    },
                    {
                        "type": "text",
                        "content": " 得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "P ^ {T} A P = E",
                "math_type": "latex",
                "image_source": {
                    "path": "images/29be3799e15a612764760284ad94ee929eb20d1212317a57ca299c1400ef5748.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此正定"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 7.8 正定二次型的判定 (1)t 为何值时，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f = x_{1}^{2} + x_{2}^{2} + 5x_{3}^{2} + 2tx_{1}x_{2} - 2x_{1}x_{3} + 4x_{2}x_{3}"
                    },
                    {
                        "type": "text",
                        "content": "  为正定的？"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为实对称矩阵，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists M, \\forall t > M"
                    },
                    {
                        "type": "text",
                        "content": " 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "tE + A"
                    },
                    {
                        "type": "text",
                        "content": " 为正定矩阵"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "80"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "7.3 (半) 正定二次型的研究"
                    }
                ]
            },
            "bbox": [
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        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解(1)算顺序主子式即可，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "1-t^{2}>0,-5t^{2}-4t>0"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "-\\frac{4}{5}<t<0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 看特征值即可"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 7.9 正定矩阵性质的简单应用 (1) A 为 n 阶正定矩阵，证明：任意 n 阶实矩阵 B 有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(B^{T}AB) = r(B)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶正定矩阵， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n \\times m"
                    },
                    {
                        "type": "text",
                        "content": " 实矩阵，证明：若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(B) = m"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C = B^T AB"
                    },
                    {
                        "type": "text",
                        "content": " 为正定矩阵证明 (1) 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 正定，存在满秩矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = C^T C"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B^T AB = (CB)^T (CB)"
                    },
                    {
                        "type": "text",
                        "content": "，而根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T A"
                    },
                    {
                        "type": "text",
                        "content": " 的结论可知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r((CB)^T (CB)) = r(CB) = r(B)"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(B^T AB) = r(B)"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 正定，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B^T AB"
                    },
                    {
                        "type": "text",
                        "content": " 实对称， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall X, X^T CX = (BX)^T ABX"
                    },
                    {
                        "type": "text",
                        "content": "，根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的正定性可知其大于0，因此为正定矩阵"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.15 (正定矩阵对角性质)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶正定矩阵，则"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "1.   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{ii} > 0, \\forall i = 1, 2, \\cdots, n"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "2.   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "2|a_{ij}| < a_{ii} + a_{jj} (i \\neq j)"
                    }
                ]
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "3.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的所有元素中，绝对值最大的元素一定是在主对角线上，且一定是正数"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 不妨取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X = e_{i}"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^{T}AX = a_{ii} > 0"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X = e_i + e_j"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^TAX = a_{ii} + a_{jj} + 2a_{ij} > 0"
                    },
                    {
                        "type": "text",
                        "content": "，取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X = e_i - e_j"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^TAX = a_{ii} + a_{jj} - 2a_{ij} > 0"
                    },
                    {
                        "type": "text",
                        "content": "，综上得到结论成立。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 根据 (2) 可知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall i, j"
                    },
                    {
                        "type": "text",
                        "content": " ，有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "|a_{ij}| < \\max\\{a_{ii}, a_{jj}\\}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此结论成立"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.3.2 半正定二次型的概念与判别"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 7.10 (半正定二次型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若对任意   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1},\\cdots,x_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  均有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n})\\geq0"
                    },
                    {
                        "type": "text",
                        "content": " ，则称   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n})"
                    },
                    {
                        "type": "text",
                        "content": "  为半正定二次型，对应矩阵 A 为半正定矩阵。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.16 (半正定二次型的基础充要条件)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n})=X^{T}AX"
                    },
                    {
                        "type": "text",
                        "content": "  为半正定二次型当且仅当："
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 和  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\left( \\begin{array}{ll}E_r & O\\\\ O & O \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": " 合同， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的正惯性指数与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的秩相同"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的特征值都大于等于0"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "-  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的所有主子式 (不等价于顺序主子式) 大于等于 0"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 半正定不等价于顺序主子式大于等于 0，例如  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\left( \\begin{array}{rr} 0 & 0 \\\\ 0 & -1 \\end{array} \\right)"
                    },
                    {
                        "type": "text",
                        "content": "，顺序主子式都为 0，但是半负定。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.17 (半正定矩阵的   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C^{T}C"
                    },
                    {
                        "type": "text",
                        "content": "  分解)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n})=X^{T}AX"
                    },
                    {
                        "type": "text",
                        "content": "  为半正定二次型等价于下面两个命题"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 存在不列满秩的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n \\times n"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C"
                    },
                    {
                        "type": "text",
                        "content": ", 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = C^T C"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 存在行满秩 (列不满秩) 的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r \\times n (r < n)"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C"
                    },
                    {
                        "type": "text",
                        "content": "，使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = C^T C"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "7.3 (半) 正定二次型的研究"
                    }
                ]
            },
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "81"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = C^{T}\\left( \\begin{array}{cc}E_{r} & O\\\\ O & O \\end{array} \\right)C"
                    },
                    {
                        "type": "text",
                        "content": " 以及"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) = \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) = \\binom{E _ {r}}{O} \\left( \\begin{array}{c c} E _ {r} & O \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/257c41e9b4aae496a40455b3fd3f5c9a6d064b474901c54bfe3f0bf79ea11f33.jpg"
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            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "可以得到两种分解："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} C _ {1} = \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) C \\Rightarrow A = C _ {1} ^ {T} C _ {1} \\\\ C _ {2} = \\left( \\begin{array}{c c} E _ {r} & O \\end{array} \\right) C \\Rightarrow A = C _ {2} ^ {T} C _ {2} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/3de282387316e90c8cc44da658280b2875b9ec29bb58ad6515caca8e1a99b0fd.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "无论哪种分解列均不满秩。"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.18 (半正定矩阵的   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "CC^{T}"
                    },
                    {
                        "type": "text",
                        "content": "  分解)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\cdots,x_{n})=X^{T}AX"
                    },
                    {
                        "type": "text",
                        "content": "  为半正定二次型等价于下面两个命题"
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        {
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 存在不行满秩的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n \\times n"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C"
                    },
                    {
                        "type": "text",
                        "content": "，使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = CC^T"
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                ]
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 存在列满秩 (行不满秩) 的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n \\times r (r < n)"
                    },
                    {
                        "type": "text",
                        "content": " 矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C"
                    },
                    {
                        "type": "text",
                        "content": "，使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = CC^T"
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                ]
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 右推左：直接  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^{T}CC^{T}X = (C^{T}X)^{T}(C^{T}X)"
                    },
                    {
                        "type": "text",
                        "content": "，该多项式是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C^{T}X"
                    },
                    {
                        "type": "text",
                        "content": " 列向量中元素的平方和，因此大于等于0。左推右：合同的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = B^{T}\\mathrm{diag}\\{E_{r},O\\} B"
                    },
                    {
                        "type": "text",
                        "content": " 也可以写为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = C\\mathrm{diag}\\{E_r,O\\} C^T"
                    },
                    {
                        "type": "text",
                        "content": "，因此把中间的分解了即可。笔记 特别的可以取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = \\alpha^{T}\\alpha"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha"
                    },
                    {
                        "type": "text",
                        "content": " 为列向量(列满秩)，此时  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是半正定的。"
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                ]
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习7.10半正定矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "CC^T"
                    },
                    {
                        "type": "text",
                        "content": " 分解  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(1)\\alpha \\in \\mathbb{R}^n"
                    },
                    {
                        "type": "text",
                        "content": " ，证明：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\alpha^T"
                    },
                    {
                        "type": "text",
                        "content": " 一定是半正定矩阵"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 重点:  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的元素  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{ij} = (i \\times j)"
                    },
                    {
                        "type": "text",
                        "content": ",  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B"
                    },
                    {
                        "type": "text",
                        "content": " 的元素  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "b_{ij} = \\left(\\frac{1}{i \\times j}\\right)"
                    },
                    {
                        "type": "text",
                        "content": ", 则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 都是半正定矩阵"
                    }
                ]
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\alpha^T"
                    },
                    {
                        "type": "text",
                        "content": " 是实对称矩阵，从而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^T \\alpha \\alpha^T X = (\\alpha^T X)^T (\\alpha^T X) = (\\alpha^T X)^2 \\geq 0"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 可进行如下分解，根据 (1) 的结论可证是半正定矩阵"
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                ]
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        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{c} 1 \\\\ 2 \\\\ \\vdots \\\\ n \\end{array} \\right] \\left[ \\begin{array}{l l l l} 1 & 2 & \\dots & n \\end{array} \\right], \\quad B = \\left[ \\begin{array}{c} 1 \\\\ \\frac {1}{2} \\\\ \\vdots \\\\ \\frac {1}{n} \\end{array} \\right] \\left[ \\begin{array}{l l l l} 1 & \\frac {1}{2} & \\dots & \\frac {1}{n} \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/b3c483c4f3596be8b5af6cef0e871b78d3e1290c8eac14a4bf126e19a1e4b02b.jpg"
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        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.19 (半正定矩阵对角性质)"
                    }
                ],
                "level": 2
            },
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                        "type": "text",
                        "content": "半正定矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 有以下性质："
                    }
                ]
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "1. 对角元素非负"
                    }
                ]
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        {
            "type": "paragraph",
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                    {
                        "type": "text",
                        "content": "2.   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "2|a_{ij}| \\leq a_{ik} + a_{ki}"
                    }
                ]
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "3. 若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{ii} = 0"
                    },
                    {
                        "type": "text",
                        "content": "，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall k, a_{ik} = a_{ki} = 0"
                    }
                ]
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 只需要取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = e_i"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x^T Ax \\geq 0"
                    },
                    {
                        "type": "text",
                        "content": " 即可"
                    }
                ]
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        {
            "type": "paragraph",
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                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = e_i + e_j"
                    },
                    {
                        "type": "text",
                        "content": "，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{ii} + a_{jj} + 2a_{ij} \\geq 0"
                    },
                    {
                        "type": "text",
                        "content": "，同理取  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = e_i - e_j"
                    },
                    {
                        "type": "text",
                        "content": "，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "a_{ii} + a_{jj} - 2a_{ij} \\geq 0"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "2|a_{ij}| \\leq a_{ik} + a_{ki}"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 根据 (2) 可以得到"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.3.3  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C^T C"
                    },
                    {
                        "type": "text",
                        "content": " 的 (半) 正定性"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "C^{T}C"
                    },
                    {
                        "type": "text",
                        "content": "  对应着特殊的二次型   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(X)=\\sum_{i=1}^{n}(a_{i1}x_{1}+\\cdots+a_{in}x_{n})^{2}"
                    },
                    {
                        "type": "text",
                        "content": " ，且有着特殊的正定性，是正定二次型中特殊的一类。"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "7.3 (半) 正定二次型的研究"
                    }
                ]
            },
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        {
            "type": "page_number",
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                "page_number_content": [
                    {
                        "type": "text",
                        "content": "82"
                    }
                ]
            },
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        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "引理7.1  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(r(A^T A)"
                    },
                    {
                        "type": "text",
                        "content": " 与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A))"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为实数域上的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m \\times n"
                    },
                    {
                        "type": "text",
                        "content": " 的矩阵，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "r (A ^ {T} A) = r (A)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/31d9b14632efe608f13c7c049dba0ece045790f8236d550344a347cbee54fd6d.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "证明 在最小二乘法一节中已经给出了证明"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.20 (  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{T}A"
                    },
                    {
                        "type": "text",
                        "content": " 的正定性)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "考虑  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 为实数域上的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "m \\times n"
                    },
                    {
                        "type": "text",
                        "content": " 的矩阵，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "1.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T A, AA^T"
                    },
                    {
                        "type": "text",
                        "content": " 是半正定矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "2.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T A"
                    },
                    {
                        "type": "text",
                        "content": " 是正定矩阵当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 列满秩， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AA^T"
                    },
                    {
                        "type": "text",
                        "content": " 正定当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 行满秩"
                    }
                ]
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "且   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{T}A"
                    },
                    {
                        "type": "text",
                        "content": "  对应的二次型为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(X)=\\sum_{i=1}^{n}(a_{i1}x_{1}+a_{i2}x_{2}+\\cdots+a_{in}x_{n})^{2}"
                    }
                ]
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (半) 正定性：显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T A"
                    },
                    {
                        "type": "text",
                        "content": " 实对称，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^T A^T A X = (AX)^2 \\geq 0"
                    },
                    {
                        "type": "text",
                        "content": "，显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T A"
                    },
                    {
                        "type": "text",
                        "content": " 半正定。而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^T A"
                    },
                    {
                        "type": "text",
                        "content": " 正定当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A^T A) = n"
                    },
                    {
                        "type": "text",
                        "content": "，根据 (1) 当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r(A) = n"
                    },
                    {
                        "type": "text",
                        "content": "，这里  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 等于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的列数，因此等价于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 列满秩"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "二次型表达式：在二次型矩阵一节已经作为练习给出。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.3.4 正定矩阵同时对角化"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.21 (同时合同对角化)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶半正定矩阵，则存在一个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶可逆矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P"
                    },
                    {
                        "type": "text",
                        "content": "，使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^T AP, P^T BP"
                    },
                    {
                        "type": "text",
                        "content": " 同时为对角矩阵"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 由于 A, B 半正定，因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A + B"
                    },
                    {
                        "type": "text",
                        "content": "  半正定，存在可逆矩阵 M 使得"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "M ^ {T} (A + B) M = \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) = M ^ {T} A M + M ^ {T} B M",
                "math_type": "latex",
                "image_source": {
                    "path": "images/75bab922b01ae05cb8e32e9b80f23035abb78728029768f6ca43359c2f07a6b2.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M^T AM, M^T BM"
                    },
                    {
                        "type": "text",
                        "content": " 半正定，因此根据半正定矩阵的对角性质，当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r + 1 \\leq i \\leq n"
                    },
                    {
                        "type": "text",
                        "content": " ，其第  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i"
                    },
                    {
                        "type": "text",
                        "content": " 个对角元都非负，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r + 1 \\sim n"
                    },
                    {
                        "type": "text",
                        "content": " 的对角元都必须为0。再根据对角性质可知  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "r + 1 \\sim n"
                    },
                    {
                        "type": "text",
                        "content": " 行、列元素均为0，因此"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "M ^ {T} A M = \\left( \\begin{array}{c c} H _ {r} & O \\\\ O & O \\end{array} \\right), M ^ {T} B M = \\left( \\begin{array}{c c} E _ {r} - H _ {r} & O \\\\ O & O \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/b91b3fd3ef85811decd10405708406e942aeb2a7a76d596ea8e3abae33f5bd93.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "存在正交矩阵 T 使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T^{T}H_{r}T = \\Lambda = \\operatorname{diag}\\{\\lambda_{1}, \\cdots, \\lambda_{r}\\}"
                    },
                    {
                        "type": "text",
                        "content": " ，令   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P = M\\operatorname{diag}\\{T, E_{n-r}\\}"
                    },
                    {
                        "type": "text",
                        "content": " ，此时有"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "P ^ {T} A P = \\left( \\begin{array}{c c} \\Lambda & O \\\\ O & O \\end{array} \\right), P ^ {T} B P = \\left( \\begin{array}{c c} E _ {r} - \\Lambda & O \\\\ O & O \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/3e85f8eab00c544f636d0ad0ef7f8b1a7aab32a7faeaefd676e29a9b4868c879.jpg"
                }
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "同时为对角矩阵"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.22 (AB = BA 同时相似对角化)"
                    }
                ],
                "level": 2
            },
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 可相似对角化，且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB = BA"
                    },
                    {
                        "type": "text",
                        "content": " ，则存在可逆矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P"
                    },
                    {
                        "type": "text",
                        "content": " ，使得"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "P ^ {- 1} A P, P ^ {- 1} B P",
                "math_type": "latex",
                "image_source": {
                    "path": "images/bcf32d2eb69d6a067e3d56f52d471d7a2ef3a002132eb6f14462bbccb4a32101.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "同时为对角矩阵"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 由于 A 可相似对角化，因此存在 M，使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M^{-1}AM = \\text{diag}\\{\\lambda_{1}E, \\lambda_{2}E, \\cdots, \\lambda_{r}E\\}"
                    },
                    {
                        "type": "text",
                        "content": " ，而"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "M ^ {- 1} A B M = (M ^ {- 1} A M) (M ^ {- 1} B M) = M ^ {- 1} B A M = (M ^ {- 1} B M) (M ^ {- 1} A M)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/80ad04a2f342ecb8dbce4443ed8f83ca59b9cdb21f24cf2d639d52b92f4820f0.jpg"
                }
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M^{-1}BM"
                    },
                    {
                        "type": "text",
                        "content": " 以同样的形式进行分块，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "M^{-1}BM = (B_{ij})_{r\\times r}"
                    },
                    {
                        "type": "text",
                        "content": "，对于每个  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "B_{ii}"
                    },
                    {
                        "type": "text",
                        "content": "， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists Q_i"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "Q_i^{-1}B_{ii}Q_i = \\Lambda_i"
                    },
                    {
                        "type": "text",
                        "content": "，因此令"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "T = M \\operatorname{diag} \\left\\{Q _ {1}, \\dots , Q _ {r} \\right\\}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/7669633dfd14788e639f94766af6103691b784ff8b73a34ccc91cd21b4f8659e.jpg"
                }
            },
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            "content": {
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                        "type": "text",
                        "content": "83"
                    }
                ]
            },
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "7.3 (半) 正定二次型的研究"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "image",
            "content": {
                "image_source": {
                    "path": "images/18f1321b9a28cd1e7a912d0030429bba3abaea7c68f82d6ab7aab02fb1299de2.jpg"
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                "content": "",
                "image_caption": [],
                "image_footnote": []
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            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "可以得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "T ^ {- 1} A T = \\operatorname{diag} \\left\\{\\lambda_ {1} E, \\dots , \\lambda_ {r} E \\right\\}, T ^ {- 1} B T = \\operatorname{diag} \\left\\{\\Lambda_ {1}, \\dots , \\Lambda_ {r} \\right\\}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/529b874246d6221d5acc31299f91fcd07dfa3fe67bbc01774981d6dd44ca2ff3.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "同时为对角矩阵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.3.5 正定矩阵的四则运算性质"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.23 (正定的四则运算)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 是正定矩阵，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}, A^{k}, A + B, A^{*}"
                    },
                    {
                        "type": "text",
                        "content": " 都是正定矩阵，但  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB"
                    },
                    {
                        "type": "text",
                        "content": " 不一定正定 (原因在于不一定对称)，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB"
                    },
                    {
                        "type": "text",
                        "content": " 实对称，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB"
                    },
                    {
                        "type": "text",
                        "content": " 是正定矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}, A^{k}, A^{*}"
                    },
                    {
                        "type": "text",
                        "content": " : 由于 A 正定, 因此 A 的特征值   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  均大于 0,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{-1}"
                    },
                    {
                        "type": "text",
                        "content": "  的特征值   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}^{-1} > 0"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{*}"
                    },
                    {
                        "type": "text",
                        "content": "  的特征值   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\prod_{j \\neq i} \\lambda_{j} > 0"
                    },
                    {
                        "type": "text",
                        "content": " ,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{k}"
                    },
                    {
                        "type": "text",
                        "content": "  的特征值   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda_{i}^{k} > 0"
                    },
                    {
                        "type": "text",
                        "content": " , 因此均正定。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A + B"
                    },
                    {
                        "type": "text",
                        "content": " 不能用特征值，只能用合同，显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A + B"
                    },
                    {
                        "type": "text",
                        "content": " 实对称，而任意非零  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x \\in \\mathbb{R}^n"
                    },
                    {
                        "type": "text",
                        "content": " ，有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "X^T (A + B)X = X^T AX + X^T BX > 0"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A + B"
                    },
                    {
                        "type": "text",
                        "content": " 实对称。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB"
                    },
                    {
                        "type": "text",
                        "content": " 正定：由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "AB"
                    },
                    {
                        "type": "text",
                        "content": " 实对称，此时  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(AB)^T = B^TA^T = BA"
                    },
                    {
                        "type": "text",
                        "content": " 。由于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A, B"
                    },
                    {
                        "type": "text",
                        "content": " 正定，故存在可逆  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P, Q"
                    },
                    {
                        "type": "text",
                        "content": " 使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = P^TP, B = Q^TQ, AB = P^TPQ^TQ"
                    },
                    {
                        "type": "text",
                        "content": " ，而"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "Q A B Q ^ {- 1} = Q P ^ {T} P Q ^ {T} (Q Q ^ {- 1}) = (P Q ^ {T}) ^ {T} (P Q ^ {T})",
                "math_type": "latex",
                "image_source": {
                    "path": "images/eea8e295557eae8f4341cf94ba6a263d2aeb0ffc783345a8e55eb1a1e18316ea.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此 AB 与   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(PQ^{T})^{T}(PQ^{T})"
                    },
                    {
                        "type": "text",
                        "content": "  相似，而后者正定（因为列满秩），因此 AB 为正定矩阵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.3.6 正定矩阵分解进阶"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.24 (正定矩阵平方分解)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶正定矩阵，则存在实对称矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C_1"
                    },
                    {
                        "type": "text",
                        "content": " ，使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = C_1^2"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 由于   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\exists P"
                    },
                    {
                        "type": "text",
                        "content": "  使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^{T}AP = \\Lambda = \\text{diag}\\{\\lambda_{1}, \\cdots, \\lambda_{n}\\}"
                    },
                    {
                        "type": "text",
                        "content": " ，根据正定，所有特征值都为正，因此取"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "C _ {1} = P ^ {T} \\sqrt {\\Lambda} P",
                "math_type": "latex",
                "image_source": {
                    "path": "images/806d77cef21d23c8280529f128756692603790a97aa9f21ec8e68c10342eb326.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "即可得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = C_1^2"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "笔记 上述   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = C_{1}^{2}"
                    },
                    {
                        "type": "text",
                        "content": "  也是正定矩阵分解！且是唯一的。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.25 (正定矩阵三角分解)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 阶正定矩阵，则存在可逆上三角阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "R"
                    },
                    {
                        "type": "text",
                        "content": " ，使得  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = R^{T}R"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 首先  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = C^T C"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C"
                    },
                    {
                        "type": "text",
                        "content": " 为可逆矩阵，再对  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C"
                    },
                    {
                        "type": "text",
                        "content": " 做  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "QR"
                    },
                    {
                        "type": "text",
                        "content": " 分解得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "C = QR"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = R^T Q^T QR = R^TR"
                    },
                    {
                        "type": "text",
                        "content": " ，即所求。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.26 (极分解定理)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "对任一可逆矩阵 A，一定存在一个正交矩阵 T 和两个正定矩阵   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "S_{1}, S_{2}"
                    },
                    {
                        "type": "text",
                        "content": "  使得"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = T S _ {1} = S _ {2} T",
                "math_type": "latex",
                "image_source": {
                    "path": "images/bd9d76478b78775e3acf6d5e9857512f48b68cbf03200fa2f2f7a85162866659.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "且这种分解是唯一的。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 存在性: 由于 A 可逆, 因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{T}A"
                    },
                    {
                        "type": "text",
                        "content": "  正定, 根据前面结论存在正定矩阵   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "S_{1}"
                    },
                    {
                        "type": "text",
                        "content": " , 满足   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A^{T}A = S_{1}^{2}"
                    },
                    {
                        "type": "text",
                        "content": " , 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "T = (A^{T})^{-1}S_{1}"
                    },
                    {
                        "type": "text",
                        "content": " , 则"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = (A ^ {T}) ^ {- 1} S _ {1} ^ {2} = T S _ {1}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f3bf48c3101834e56323e785ed742288aecb32fccac407772bc43fe7da373b19.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "经过验证可发现  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "TT^{T} = (A^{T})^{-1}S_{1}S_{1}^{T}A^{-1} = (A^{T})^{-1}S_{1}^{2}A^{-1} = (A^{T})^{-1}A^{T}AA^{-1} = E"
                    },
                    {
                        "type": "text",
                        "content": " ，因此为正交矩阵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "7.3 (半) 正定二次型的研究"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "84"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "另一方面，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = TS_{1}T^{-1}T"
                    },
                    {
                        "type": "text",
                        "content": " ，记   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "S_{2} = TS_{1}T^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " ，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = S_{2}T"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "S_{2}"
                    },
                    {
                        "type": "text",
                        "content": "  正定。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 唯一性：设  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = TS_{1} = \\tilde{T}\\tilde{S}_{1}"
                    },
                    {
                        "type": "text",
                        "content": "，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} A ^ {T} A = S _ {1} ^ {2} \\\\ A ^ {T} A = \\tilde {S} _ {1} ^ {2} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/4983228d096b882181c6700d07e73c975fa1497f1ca9a6bec0bea3d1e9e15140.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "根据前面正定矩阵分解唯一性可知唯一。"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "7.3.7 负定二次型"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 7.11 (负定二次型)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若对于任意不全为零的  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{1},\\dots ,x_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\dots ,x_{n}) < 0"
                    },
                    {
                        "type": "text",
                        "content": " ，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x_{1},\\dots ,x_{n})"
                    },
                    {
                        "type": "text",
                        "content": " 为负定二次型"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 7.27 (负定二次型的充要条件)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 负定当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "-A"
                    },
                    {
                        "type": "text",
                        "content": " 正定，因此充要条件为：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A"
                    },
                    {
                        "type": "text",
                        "content": " 的所有奇数阶顺序主子式小于0，偶数阶顺序主子式大于0。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 用特征值判断即可。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "7.3 (半) 正定二次型的研究"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "85"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "第8章 内积空间"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "8.1 Euclid 空间"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "8.1.1 Euclid 空间与正交"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 8.1 (欧式空间)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{R}"
                    },
                    {
                        "type": "text",
                        "content": " 上线性空间，定义内积  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\alpha, \\beta)"
                    },
                    {
                        "type": "text",
                        "content": " 满足："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 线性性： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(k_{1}\\alpha_{1} + k_{2}\\alpha_{2}, \\beta) = k_{1}(\\alpha_{1}, \\beta) + k_{2}(\\alpha_{2}, \\beta)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 对称性：  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\alpha ,\\beta) = (\\alpha ,\\beta)"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "- 正定性： "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\alpha, \\alpha) = 0"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 8.2 (度量矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "给定 Euclid 空间 V 的一组基   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1},\\cdots,\\epsilon_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，则定义这组基对应的度量矩阵为"
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{c c c c} (\\epsilon_ {1}, \\epsilon_ {1}) & (\\epsilon_ {1}, \\epsilon_ {2}) & \\dots & (\\epsilon_ {1}, \\epsilon_ {n}) \\\\ (\\epsilon_ {2}, \\epsilon_ {1}) & (\\epsilon_ {2}, \\epsilon_ {2}) & \\dots & (\\epsilon_ {2}, \\epsilon_ {n}) \\\\ \\vdots & \\vdots & & \\vdots \\\\ (\\epsilon_ {n}, \\epsilon_ {1}) & (\\epsilon_ {n}, \\epsilon_ {2}) & \\dots & (\\epsilon_ {n}, \\epsilon_ {n}) \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/fcd00d1a4a2d4925cfd981e1d1e66950deb25949e46c0eb700eda88c059a787d.jpg"
                }
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 8.1 (内积的度量矩阵表示)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若 Euclid 空间中的向量   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha, \\beta"
                    },
                    {
                        "type": "text",
                        "content": "  在基   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\epsilon_{1}, \\cdots, \\epsilon_{n})"
                    },
                    {
                        "type": "text",
                        "content": "  中的坐标分别为 X, Y,   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\epsilon_{1}, \\cdots, \\epsilon_{n})"
                    },
                    {
                        "type": "text",
                        "content": "  对应的度量矩阵为 A，则"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(\\alpha , \\beta) = X ^ {T} A Y",
                "math_type": "latex",
                "image_source": {
                    "path": "images/c1c4250909a3b0bd7e3ee4de046119a9881f7071a880168a727c1283b2cb31f1.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 8.1 度量矩阵 已知 Euclid 空间 V 的一组基为:"
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\epsilon_ {1} = e _ {1} + e _ {2}, \\epsilon_ {2} = e _ {1} + e _ {3}, \\epsilon_ {3} = e _ {4} - e _ {1}, \\epsilon_ {4} = e _ {1} - e _ {2} - e _ {3} + e _ {4}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/197e1658986f9f3abdc5b8b7eea1270c7737e70b4fe2342f3d255e721d5e3f47.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "向量   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha,\\beta"
                    },
                    {
                        "type": "text",
                        "content": "  在   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\epsilon_{1},\\cdots,\\epsilon_{4})"
                    },
                    {
                        "type": "text",
                        "content": "  下的坐标分别为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(1,2,3,4)^{T},(2,0,1,0)^{T}"
                    },
                    {
                        "type": "text",
                        "content": " ，计算   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\epsilon_{1},\\cdots,\\epsilon_{4})"
                    },
                    {
                        "type": "text",
                        "content": "  的度量矩阵 A，以及   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\alpha,\\beta)"
                    },
                    {
                        "type": "text",
                        "content": "  解 把   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1},\\cdots,\\epsilon_{4}"
                    },
                    {
                        "type": "text",
                        "content": "  看成   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "R^{n}"
                    },
                    {
                        "type": "text",
                        "content": "  中的坐标："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} \\epsilon_ {1} = (1, 1, 0, 0) \\\\ \\epsilon_ {2} = (1, 0, 1, 0) \\\\ \\epsilon_ {3} = (- 1, 0, 0, 1) \\\\ \\epsilon_ {4} = (1, - 1, - 1, 1) \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/23fd981a525e005f48a94f962711eebf5acef0e72ee0ca4f4aec928089c3871f.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此根据向量的坐标内积公式可以得到度量矩阵为："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{r r r r} 2 & 1 & - 1 & 0 \\\\ 1 & 2 & - 1 & 0 \\\\ - 1 & - 1 & 2 & 0 \\\\ 0 & 0 & 0 & 4 \\end{array} \\right], (\\alpha , \\beta) = 5",
                "math_type": "latex",
                "image_source": {
                    "path": "images/341204480109b8428b2379588309bb2bfbcb8502915d1d711cb71405620ebef1.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 8.2 (正交向量线性无关)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若 Eulid 空间 V 中的向量   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{s}"
                    },
                    {
                        "type": "text",
                        "content": "  彼此正交，则它们线性无关。"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 不妨设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "c_{1}\\alpha_{1}+\\cdots+c_{s}\\alpha_{s}=0"
                    },
                    {
                        "type": "text",
                        "content": " ，则两侧与   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  做内积得到："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left(c _ {1} \\alpha_ {1} + \\dots + c _ {s} \\alpha_ {s}, \\alpha_ {i}\\right) = (0, \\alpha_ {i}) = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/d34cfaf345277b8f5292801b4321062ca3202cfed0e55d9e56049c9732675ffa.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "根据正交定义可知左侧等于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "c_{i}(\\alpha_{i},\\alpha_{i})"
                    },
                    {
                        "type": "text",
                        "content": " ，而  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\alpha_{i},\\alpha_{i})\\neq 0"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "c_{i} = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "i"
                    },
                    {
                        "type": "text",
                        "content": " 的任意性可知线性无关。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "8.1.2 标准正交基"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "一、标准正交基的概念与存在性"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 8.3 (标准正交基)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 维Eulid空间中  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 个两两正交的单位向量称为标准正交基"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "二、标准正交基之间的过渡矩阵"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 8.3 (标准正交基的过渡矩阵)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1},\\cdots,\\epsilon_{n}"
                    },
                    {
                        "type": "text",
                        "content": " 是 n 维 Euclid 空间 V 的一组基，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\eta_{1},\\cdots,\\eta_{n}"
                    },
                    {
                        "type": "text",
                        "content": " 下述等式，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\eta_{1},\\cdots,\\eta_{n}"
                    },
                    {
                        "type": "text",
                        "content": " 是标准正交基当且仅当 T 是正交矩阵。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(\\eta_ {1}, \\dots , \\eta_ {n}) = (\\epsilon_ {1}, \\dots , \\epsilon_ {n}) T",
                "math_type": "latex",
                "image_source": {
                    "path": "images/4902c94d7aa81c7a7817f041f4cf5e758f1e0abec3f2ff236f4508feaba0b439.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "三、标准正交基的计算：Schmidt 正交化"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 8.4 (Schmidt 正交化)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "给定一组线性无关的向量   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{s}"
                    },
                    {
                        "type": "text",
                        "content": " ，要求做出一组相互正交的向量 (单位化单独进行)  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1},\\cdots,\\epsilon_{s}"
                    },
                    {
                        "type": "text",
                        "content": " ，可以按照以下步骤："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\epsilon_ {1} = \\alpha_ {1}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/776329851f0d747d8aee3fd86dfbd79fd4470210bde8e297ceafd951cd43bacf.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\epsilon_ {2} = \\alpha_ {2} - \\frac {\\left(\\alpha_ {2} , \\epsilon_ {1}\\right)}{\\left(\\epsilon_ {1} , \\epsilon_ {1}\\right)} \\epsilon_ {1}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/474279af377668dc2b1f4ad1b94491e232514dbc7bb80807cbfbea15da2cbdd2.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\epsilon_ {s} = \\alpha_ {s} - \\sum_ {i = 1} ^ {s - 1} \\frac {(\\alpha_ {s} , \\epsilon_ {i})}{(\\epsilon_ {i} , \\epsilon_ {i})} \\epsilon_ {i}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/fddeeaa39583a1223d0c1a7eba331ed2d14631fd9432736ce11786db06c2d4f8.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 8.2Schmidt 正交化 已知   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1}=(1,2,-1)^{T},\\epsilon_{2}=(-1,3,1)^{T},\\epsilon_{3}=(4,-1,0)^{T}"
                    },
                    {
                        "type": "text",
                        "content": " ，用 Schmidt 正交化将其化为标准正交基"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解 最终结果为  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\eta_{1} = \\frac{1}{\\sqrt{6}} (1,2, - 1)^{T},\\eta_{2} = \\frac{1}{\\sqrt{3}} (-1,1,1)^{T},\\eta_{3} = \\frac{1}{\\sqrt{2}} (1,0,1)^{T}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 8.5 (QR 分解)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 假设原本的矩阵 A 的列向量为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1},\\cdots,\\alpha_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，经过 Schmidt 正交化后得到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{1},\\cdots,\\beta_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，则满足："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} \\alpha_ {1} = \\beta_ {1} \\\\ \\alpha_ {2} = a _ {1 2} \\beta_ {1} + \\beta_ {2} \\\\ \\quad \\vdots \\\\ \\alpha_ {n} = a _ {1 n} \\beta_ {1} + a _ {2 n} \\beta_ {2} + \\dots + \\beta_ {n} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/475b6010c15d11e2f853eafcc6eb2d5c3b6af8f366a06a1bde859559a9793670.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "87"
                    }
                ]
            },
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "8.1 Euclid 空间"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "再将  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta"
                    },
                    {
                        "type": "text",
                        "content": " 进行单位化得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\eta_{i} = \\frac{\\beta_{i}}{|\\beta_{i}|}"
                    },
                    {
                        "type": "text",
                        "content": "，此时写为"
                    }
                ]
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(\\alpha_ {1}, \\dots , \\alpha_ {n}) = (\\eta_ {1}, \\dots , \\eta_ {n}) \\left( \\begin{array}{c c c c} b _ {1 1} & b _ {1 2} & \\dots & b _ {1 n} \\\\ & b _ {2 2} & \\dots & b _ {2 n} \\\\ & & \\ddots & \\vdots \\\\ & & & b _ {n n} \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/8eca09523e3936cd96d20980532d698ff9349a70a2a9329b9971ea42cdf8a86e.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "此时对角   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "b_{11}=|\\beta_{1}|\\cdots,b_{nn}=|\\beta_{n}|"
                    },
                    {
                        "type": "text",
                        "content": " ，而   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\eta_{1},\\cdots,\\eta_{n})"
                    },
                    {
                        "type": "text",
                        "content": "  是标准正交基。因此是正交矩阵乘上三角阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习 8.3QR 分解 将下面的矩阵 A 分解为正交矩阵 T 与对角都是正数的上三角矩阵 B 的乘积"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A = \\left[ \\begin{array}{l l l} 1 & 1 & 1 \\\\ 1 & 0 & 1 \\\\ 1 & 2 & 0 \\end{array} \\right]",
                "math_type": "latex",
                "image_source": {
                    "path": "images/31fef9e1a27a623a5366777ee1c96acbc8c98cc8f208f3390e9838d7e68953cb.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解 先进行   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1}, \\alpha_{2}, \\alpha_{3}"
                    },
                    {
                        "type": "text",
                        "content": "  正交化到   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta_{1}, \\beta_{2}, \\beta_{3}"
                    },
                    {
                        "type": "text",
                        "content": " :"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} \\beta_ {1} = \\alpha_ {1} = (1, 1, 1) ^ {T}, | \\beta_ {1} | = \\sqrt {3} \\\\ \\beta_ {2} = \\alpha_ {2} - \\frac {3}{3} (1, 1, 1) ^ {T} = (0, - 1, 1) ^ {T}, | \\beta_ {2} | = \\sqrt {2} \\\\ \\beta_ {3} = \\alpha_ {3} + \\frac {1}{2} \\beta_ {2} - \\frac {2}{3} \\beta_ {1} = (\\frac {1}{3}, - \\frac {1}{6}, - \\frac {1}{6}), | \\beta_ {3} | = \\sqrt {\\frac {1}{6}} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/dba7ff1122fcf0c9efa4a00ee8ea03b5751d8acf2218e2d4065c76a6a2e11e26.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "进行单位化得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} \\eta_ {1} = (\\frac {\\sqrt {3}}{3}, \\frac {\\sqrt {3}}{3}, \\frac {\\sqrt {3}}{3}) \\\\ \\eta_ {2} = (0, - \\frac {\\sqrt {2}}{2}, \\frac {\\sqrt {2}}{2}) \\\\ \\eta_ {3} = (\\frac {\\sqrt {6}}{3}, - \\frac {\\sqrt {6}}{6}, - \\frac {\\sqrt {6}}{6}) \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/942239fed00a544765f676c049f138043f4a577f49a8a36ab4e8987f629da5b1.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "反向得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left\\{ \\begin{array}{l} \\alpha_ {1} = \\beta_ {1} \\\\ \\alpha_ {2} = \\beta_ {1} + \\beta_ {2} \\\\ \\alpha_ {3} = \\frac {2}{3} \\beta_ {1} - \\frac {1}{2} \\beta_ {2} + \\beta_ {3} \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/7a2e0ba7f7c795c4c24593ca589b7645a69a15ef46d3d5fe38fd6906d41a671e.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}) = (\\beta_ {1}, \\beta_ {2}, \\beta_ {3}) \\left( \\begin{array}{c c c} 1 & 1 & \\frac {2}{3} \\\\ 0 & 1 & - \\frac {1}{2} \\\\ 0 & 0 & 1 \\end{array} \\right) = (\\eta_ {1}, \\eta_ {2}, \\eta_ {3}) \\left( \\begin{array}{c c c} | \\beta_ {1} | & & \\\\ & | \\beta_ {2} | & \\\\ & & | \\beta_ {3} | \\end{array} \\right) \\left( \\begin{array}{c c c} 1 & 1 & \\frac {2}{3} \\\\ 0 & 1 & - \\frac {1}{2} \\\\ 0 & 0 & 1 \\end{array} \\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/302fc0280904603aa019f7250d50eaf9ab8efcf91747b2e8b5c9f79758c718c9.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "8.1.3 正交和与正交补"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 8.4 (正交和、正交补)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V_{1},\\cdots,V_{s}"
                    },
                    {
                        "type": "text",
                        "content": "  为一列子空间，若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall i,j"
                    },
                    {
                        "type": "text",
                        "content": "  有   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{i}\\perp V_{j}"
                    },
                    {
                        "type": "text",
                        "content": " ，则称   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}+\\cdots+V_{s}"
                    },
                    {
                        "type": "text",
                        "content": "  为正交和。若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V=V_{1}+V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "  且为正交和，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1},V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "  互为正交补。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 8.6 (正交和必为直和)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1} + \\cdots + V_{s}"
                    },
                    {
                        "type": "text",
                        "content": "  为正交和，则其必为直和。"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "0 = \\alpha_{1} + \\cdots + \\alpha_{s}"
                    },
                    {
                        "type": "text",
                        "content": " ，两侧与   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  做内积可知"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "0 = (\\alpha_ {i}, \\alpha_ {i})",
                "math_type": "latex",
                "image_source": {
                    "path": "images/96b29782a3602ce8c89a81b9f914831183ba38affb04d5496c7e8de25c662830.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "page_header",
            "content": {
                "page_header_content": [
                    {
                        "type": "text",
                        "content": "8.1 Euclid 空间"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "page_number",
            "content": {
                "page_number_content": [
                    {
                        "type": "text",
                        "content": "88"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{i}=0"
                    },
                    {
                        "type": "text",
                        "content": " ，故零元素分解唯一，为直和。"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 8.7 (正交补的唯一性)"
                    }
                ],
                "level": 2
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "Euclid 空间 V 的任一子空间   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}"
                    },
                    {
                        "type": "text",
                        "content": "  有唯一的正交补   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W^{\\perp}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}=\\{0\\}"
                    },
                    {
                        "type": "text",
                        "content": "  时，即   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}^{\\perp}=V"
                    },
                    {
                        "type": "text",
                        "content": " 。否则设   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}=L(\\epsilon_{1},\\cdots,\\epsilon_{s})"
                    },
                    {
                        "type": "text",
                        "content": "  为标准正交基，将其扩充为 V 的标准正交基："
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "V = L \\left(\\epsilon_ {1}, \\dots , \\epsilon_ {n}\\right)",
                "math_type": "latex",
                "image_source": {
                    "path": "images/77ce258df5dd9bceea2e0e99e15aec8001f774f857bfe4309ed3765b83c888cf.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}^{\\perp}=L(\\epsilon_{s+1},\\cdots,\\epsilon_{n})"
                    },
                    {
                        "type": "text",
                        "content": "  为正交补"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "唯一性: 若   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}"
                    },
                    {
                        "type": "text",
                        "content": "  有正交补   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{2}, V_{3}"
                    },
                    {
                        "type": "text",
                        "content": " ，则   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in V_{2} \\subset V_{1} \\oplus V_{3}, \\exists \\alpha_{1} \\in V_{1}, \\alpha_{3} \\in V_{3}"
                    },
                    {
                        "type": "text",
                        "content": "  使得   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = \\alpha_{1} + \\alpha_{3}"
                    },
                    {
                        "type": "text",
                        "content": " ，而   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\perp \\alpha_{1}, \\alpha_{1} \\perp \\alpha_{3}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "0 = (\\alpha , \\alpha_ {1}) = (\\alpha_ {1}, \\alpha_ {1}) + (\\alpha_ {3}, \\alpha_ {1}) = (\\alpha_ {1}, \\alpha_ {1})",
                "math_type": "latex",
                "image_source": {
                    "path": "images/7e57f2c83f4bfaea8d7dcadd24353e64b95756076b326d763ebc1d7dafc8a636.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1} = 0, \\alpha = \\alpha_{3} \\in V_{3}"
                    },
                    {
                        "type": "text",
                        "content": "，得到  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{2} \\subseteq V_{3}, V_{3} \\subseteq V_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{2} = V_{3}"
                    }
                ]
            },
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        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 8.8 (正交补的具体表达式)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 是线性空间  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 的子空间，则  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}^{\\perp}"
                    },
                    {
                        "type": "text",
                        "content": " 恰由所有与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_{1}"
                    },
                    {
                        "type": "text",
                        "content": " 正交的向量组成，即"
                    }
                ]
            },
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "V _ {1} ^ {\\perp} = \\{\\alpha \\in V, \\alpha \\perp V _ {1} \\}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/d14d57b49c8892cfa811e3baa4e091848b10db6a204608f0d5932e70af03f5b1.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 令  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "W = \\{\\alpha \\in V : \\alpha \\perp V_1\\}"
                    },
                    {
                        "type": "text",
                        "content": "，下面证明  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_1^\\perp = W"
                    },
                    {
                        "type": "text",
                        "content": "。首先  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_1 \\perp V_1^\\perp"
                    },
                    {
                        "type": "text",
                        "content": "，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V_1^\\perp \\subseteq W"
                    },
                    {
                        "type": "text",
                        "content": "。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "反之， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in W"
                    },
                    {
                        "type": "text",
                        "content": "，根据  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V = V_{1} \\oplus V_{1}^{\\perp}"
                    },
                    {
                        "type": "text",
                        "content": "，因此可分解  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha = \\alpha_{1} + \\alpha_{2}"
                    },
                    {
                        "type": "text",
                        "content": "，得到"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left(\\alpha_ {1}, \\alpha_ {1}\\right) = \\left(\\alpha - \\alpha_ {2}, \\alpha_ {1}\\right) = \\left(\\alpha , \\alpha_ {1}\\right) - \\left(\\alpha_ {2}, \\alpha_ {1}\\right) = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/e347fd4c0c96719d8b7572dd347333353d1c939c8a53904a93f3d21011f05479.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha_{1} = 0"
                    },
                    {
                        "type": "text",
                        "content": " ，即  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\in V_1^\\perp"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定理 8.9 (正交补的性质)"
                    }
                ],
                "level": 2
            },
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(1) (U ^ {\\perp}) ^ {\\perp} = U (2) (U _ {1} + U _ {2}) ^ {\\perp} = U _ {1} ^ {\\perp} \\cap U _ {2} ^ {\\perp} (3) (U _ {1} \\cap U _ {2}) ^ {\\perp} = U _ {1} ^ {\\perp} + U _ {2} ^ {\\perp}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/cafe621d79731811a09154907b7ce913887e6b77e94690ab68ef62a053e87eb4.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "证明 (1) 根据正交补的具体表达式一下得出"
                    }
                ]
            },
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(2) 显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(U_{1} + U_{2})^{\\perp}\\subseteq U_{1}^{\\perp}"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(U_{1} + U_{2})^{\\perp}\\subseteq U_{2}^{\\perp}"
                    },
                    {
                        "type": "text",
                        "content": " ，因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(U_{1} + U_{2})^{\\perp}\\subseteq U_{1}^{\\perp}\\cap U_{2}^{\\perp}"
                    },
                    {
                        "type": "text",
                        "content": " 。"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "另一方面， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in U_1^\\perp \\cap U_2^\\perp, \\beta \\in U_1 + U_2"
                    },
                    {
                        "type": "text",
                        "content": "，有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\beta = \\beta_1 + \\beta_2"
                    },
                    {
                        "type": "text",
                        "content": "，因此"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "(\\alpha , \\beta) = (\\alpha , \\beta_ {1} + \\beta_ {2}) = (\\alpha , \\beta_ {1}) + (\\alpha , \\beta_ {2}) = 0",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f9dded78208f57483106b8b49a2ba1d01cdea97560f3ae2c9ec29c2fcdb06367.jpg"
                }
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "U_{1}^{\\perp}\\cap U_{2}^{\\perp}\\subseteq (U_{1} + U_{2})^{\\perp}"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "(3) 可转换为 (2)"
                    }
                ]
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "8.1.4 正交变换"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "定义 8.5 (正交变换)"
                    }
                ],
                "level": 2
            },
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n"
                    },
                    {
                        "type": "text",
                        "content": " 维Eulid空间，  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 上线性变换，若  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha ,\\beta \\in V"
                    },
                    {
                        "type": "text",
                        "content": " 有  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\mathcal{A}\\alpha ,\\mathcal{A}\\beta) = (\\alpha ,\\beta)"
                    },
                    {
                        "type": "text",
                        "content": " ，则称  "
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                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
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                        "content": " 为正交变换"
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                        "content": "定理 8.10 (正交变换充要条件)"
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                        "content": "-  "
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                    {
                        "type": "text",
                        "content": " 在标准正交基下矩阵为正交矩阵"
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                        "content": "- 标准正交基在  "
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                        "content": " 的作用下仍为标准正交基"
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                        "content": "-  "
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                        "content": "\\mathcal{A}"
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                        "type": "text",
                        "content": " 保持向量长度不变"
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                        "content": "若  "
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                        "type": "equation_inline",
                        "content": "\\epsilon_1, \\dots, \\epsilon_n"
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                        "type": "text",
                        "content": " 是  "
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                        "type": "equation_inline",
                        "content": "V"
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                    {
                        "type": "text",
                        "content": " 的标准正交基，则  "
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                        "type": "equation_inline",
                        "content": "\\mathcal{A}(\\epsilon_1), \\dots, \\mathcal{A}(\\epsilon_n)"
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                    {
                        "type": "text",
                        "content": " 也是  "
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                        "type": "equation_inline",
                        "content": "V"
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                        "content": " 的标准正交基"
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                "page_header_content": [
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                        "type": "text",
                        "content": "8.1 Euclid 空间"
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                        "content": "89"
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                        "content": "8.1.5 镜面反射"
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                        "content": "若对  "
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                    {
                        "type": "equation_inline",
                        "content": "\\forall \\alpha \\in V"
                    },
                    {
                        "type": "text",
                        "content": " ，  "
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                        "type": "equation_inline",
                        "content": "\\mathcal{A}:V\\to V"
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                        "content": " 满足"
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                "math_content": "\\mathcal {A} \\alpha = \\alpha - 2 (\\eta , \\alpha) \\eta",
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                    "path": "images/5db0e4860cac3ed6e535f0127f15e1b4fb058af1d3932ae4e4fd19c5b7278856.jpg"
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                        "content": "其中  "
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                        "content": "\\eta"
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                        "content": " 是单位向量"
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                        "content": "引理 8.1 (镜面反射是正交变换)"
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                        "content": "镜面反射是正交变换"
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                        "content": "证明"
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                ]
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                        "content": "定理 8.11 (镜面反射的特征)"
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                        "content": "8.2.1 Unitary 空间概念"
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                        "content": "V"
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                    {
                        "type": "text",
                        "content": " 是  "
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                    {
                        "type": "equation_inline",
                        "content": "\\mathbb{C}"
                    },
                    {
                        "type": "text",
                        "content": " 上内积空间，其上定义内积  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\alpha, \\beta)"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\forall k_1, k_2 \\in \\mathbb{C}"
                    }
                ]
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                "math_content": "\\bullet \\left(k _ {1} \\alpha_ {1} + k _ {2} \\alpha_ {2}, \\beta\\right) = k _ {1} (\\alpha_ {1}, \\beta) + k _ {2} (\\alpha_ {2}, \\beta)",
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                "math_content": "\\bullet (\\alpha , \\beta) = \\overline {{(\\beta , \\alpha)}}",
                "math_type": "latex",
                "image_source": {
                    "path": "images/54776cb7dd4b3ae5e6cf41def958fb85335f83c5258eed1af8bceb6b69d84700.jpg"
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                        "type": "equation_inline",
                        "content": "(\\alpha, \\alpha) \\geq 0"
                    },
                    {
                        "type": "text",
                        "content": " 且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\alpha, \\alpha) = 0"
                    },
                    {
                        "type": "text",
                        "content": " 当且仅当  "
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                        "content": "定义 8.8 (酉矩阵)"
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                    },
                    {
                        "type": "text",
                        "content": " 上矩阵  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "U^{-1} = \\overline{U}^T"
                    },
                    {
                        "type": "text",
                        "content": " ，则称  "
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                        "type": "equation_inline",
                        "content": "U"
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                        "type": "text",
                        "content": " 为酉矩阵。"
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                        "type": "equation_inline",
                        "content": "\\mathcal{U}"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\mathcal{U}\\alpha, \\mathcal{U}\\beta) = (\\alpha, \\beta)"
                    },
                    {
                        "type": "text",
                        "content": "，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{U}"
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                    {
                        "type": "text",
                        "content": " 为酉变换"
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                "level": 2
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        },
        {
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                "paragraph_content": [
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                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 是酉空间  "
                    },
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                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 的线性变换，若  "
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                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}^*"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "(\\mathcal{A}\\alpha, \\beta) = (\\alpha, \\mathcal{A}^*\\beta)"
                    },
                    {
                        "type": "text",
                        "content": "，则称  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}^*"
                    },
                    {
                        "type": "text",
                        "content": " 是  "
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                    {
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                        "content": "\\mathcal{A}"
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                    {
                        "type": "text",
                        "content": " 的共轭变换"
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                ]
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                "level": 2
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                        "content": "若酉空间  "
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                    },
                    {
                        "type": "text",
                        "content": " 的线性变换  "
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                        "type": "equation_inline",
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                    {
                        "type": "text",
                        "content": " 与其共轭变换  "
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                    {
                        "type": "text",
                        "content": " 可交换，则称  "
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                        "content": " 为正规变换。"
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                        "content": "若酉空间  "
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                    },
                    {
                        "type": "text",
                        "content": " 的线性变换  "
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                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
                    },
                    {
                        "type": "text",
                        "content": " 满足  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A} = \\mathcal{A}^*"
                    },
                    {
                        "type": "text",
                        "content": "，则称  "
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                    {
                        "type": "equation_inline",
                        "content": "\\mathcal{A}"
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                    {
                        "type": "text",
                        "content": " 为Hermite变换"
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                        "content": "90"
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                "title_content": [
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                        "content": "第9章 对偶空间"
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                    {
                        "type": "text",
                        "content": " 是  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "K"
                    },
                    {
                        "type": "text",
                        "content": " 上的线性空间，将定义于  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V"
                    },
                    {
                        "type": "text",
                        "content": " 上的所有有界线性泛函组成的空间称为对偶空间  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V^{*}"
                    },
                    {
                        "type": "text",
                        "content": " 。"
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                ]
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        {
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                        "content": "给定 V 的一组基   "
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                        "content": "\\epsilon_{1},\\cdots,\\epsilon_{n}"
                    },
                    {
                        "type": "text",
                        "content": " ，定义线性泛函   "
                    },
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                        "type": "equation_inline",
                        "content": "f_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  满足："
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                ]
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            "type": "equation_interline",
            "content": {
                "math_content": "f _ {i} (\\epsilon_ {j}) = \\left\\{ \\begin{array}{l l} 1, & i = j \\\\ 0, & i \\neq j \\end{array} \\right.",
                "math_type": "latex",
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                        "content": "则   "
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                    {
                        "type": "equation_inline",
                        "content": "\\{f_{1},\\cdots,f_{n}\\}"
                    },
                    {
                        "type": "text",
                        "content": "  是   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "V^{*}"
                    },
                    {
                        "type": "text",
                        "content": "  的一组基，称其为   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1},\\cdots,\\epsilon_{n}"
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                        "content": "  的对偶基。"
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                ]
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        },
        {
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            "content": {
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                        "content": "笔记 计算对偶基时，将   "
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                    {
                        "type": "equation_inline",
                        "content": "f_{i}"
                    },
                    {
                        "type": "text",
                        "content": "  视为一个向量，即   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{i}(\\alpha)=f_{i1}x_{1}+\\cdots+f_{in}x_{n}=f_{i}^{T}\\alpha"
                    },
                    {
                        "type": "text",
                        "content": " 。计算   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1},\\cdots,\\epsilon_{n}"
                    },
                    {
                        "type": "text",
                        "content": "  的对偶基只需要满足："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "\\left[ \\begin{array}{c} f _ {1} ^ {T} \\\\ f _ {2} ^ {T} \\\\ \\vdots \\\\ f _ {n} ^ {T} \\end{array} \\right] \\left[ \\begin{array}{c c c c} \\epsilon_ {1} & \\epsilon_ {2} & \\dots & \\epsilon_ {n} \\end{array} \\right] = E",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f35f81960fd5383d1df0beb460ab7ba62f63a3fda3cfa1f10e9e043460f21fb6.jpg"
                }
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "练习9.1给定  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\epsilon_{1} = [1,1,0]^{T},\\epsilon_{2} = [0,1,1]^{T},\\epsilon_{3} = [0,1, - 1]^{T}"
                    },
                    {
                        "type": "text",
                        "content": " ，求其对偶基"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "解显然  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f_{1}, f_{2}, f_{3}"
                    },
                    {
                        "type": "text",
                        "content": " 都是三个系数组成的线性函数，不妨记  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = [f_{1}, f_{2}, f_{3}]^{T}, P = [\\epsilon_{1}, \\epsilon_{2}, \\epsilon_{3}]"
                    },
                    {
                        "type": "text",
                        "content": "，则"
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "A P = \\left[ \\begin{array}{l} f _ {1} ^ {T} \\\\ f _ {2} ^ {T} \\\\ f _ {3} ^ {T} \\end{array} \\right] [ \\epsilon_ {1}, \\epsilon_ {2}, \\epsilon_ {3} ] = E",
                "math_type": "latex",
                "image_source": {
                    "path": "images/f9a4e63e8e98ee85aa41e550b39070d4e9b6adc4402bef03955c18c62894fee3.jpg"
                }
            },
            "bbox": [
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        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "因此  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "A = P^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " ，只需求出  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "P^{-1}"
                    },
                    {
                        "type": "text",
                        "content": " 即可。最终求出："
                    }
                ]
            },
            "bbox": [
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        },
        {
            "type": "equation_interline",
            "content": {
                "math_content": "P ^ {- 1} = \\left[ \\begin{array}{c c c} 1 & 0 & 0 \\\\ - \\frac {1}{2} & \\frac {1}{2} & \\frac {1}{2} \\\\ - \\frac {1}{2} & \\frac {1}{2} & - \\frac {1}{2} \\end{array} \\right] \\Rightarrow \\left\\{ \\begin{array}{l} f _ {1} ^ {T} = (1, 0, 0) \\\\ f _ {2} ^ {T} = (- \\frac {1}{2}, \\frac {1}{2}, \\frac {1}{2}) \\\\ f _ {3} ^ {T} = (- \\frac {1}{2}, \\frac {1}{2}, - \\frac {1}{2}) \\end{array} \\right.",
                "math_type": "latex",
                "image_source": {
                    "path": "images/eaca715ea38f03fef6ad25e59aa5a3b8726381cba8855e6af095b72dbc2bf3e9.jpg"
                }
            },
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        }
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]