[
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    {
        "type": "text",
        "text": "高等代数笔记",
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    {
        "type": "text",
        "text": "作者：Reichtum",
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    {
        "type": "text",
        "text": "组织：ZheJiang University",
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    },
    {
        "type": "text",
        "text": "时间：September 20, 2022",
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    },
    {
        "type": "text",
        "text": "版本：0.1",
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    },
    {
        "type": "text",
        "text": "There was no royal road to geometry. ——Euclid",
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    },
    {
        "type": "image",
        "img_path": "images/d840dcbb2187119fefcbbad478a371deb4b6159c9ef6f1e6f1baa4fef154e71b.jpg",
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    {
        "type": "text",
        "text": "目录",
        "text_level": 2,
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        ],
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    },
    {
        "type": "text",
        "text": "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": "text",
        "text": "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 $A^T A$ 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": "header",
        "text": "目录",
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    },
    {
        "type": "page_number",
        "text": "ii",
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    {
        "type": "text",
        "text": "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： $\\lambda$ 矩阵法 65  \n6.3.1 $\\lambda$ 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 $\\lambda$ 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 $k$ 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": "header",
        "text": "目录",
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    },
    {
        "type": "footer",
        "text": "第7章 双线性函数与二次型",
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    {
        "type": "page_number",
        "text": "73",
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    },
    {
        "type": "page_number",
        "text": "iii",
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    },
    {
        "type": "text",
        "text": "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 $C^T C$ 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": "header",
        "text": "目录",
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    },
    {
        "type": "page_number",
        "text": "iv",
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    {
        "type": "text",
        "text": "第1章 多项式",
        "text_level": 2,
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    {
        "type": "text",
        "text": "1.1 一元多项式",
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    {
        "type": "text",
        "text": "1.1.1 整除理论",
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    {
        "type": "text",
        "text": "定义 1.1 (整除)",
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        "text": "$g(x), f(x)$ 是 $\\mathbb{P}$ 上的多项式，若 $\\exists h(x)$ 使得 $f(x) = g(x)h(x)$ ，则称 $g(x)$ 整除 $f(x)$ ，记为 $g(x)|f(x)$ 。称 $g(x)$ 是 $f(x)$ 的因式， $f(x)$ 是 $g(x)$ 的倍式。",
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    {
        "type": "text",
        "text": "定理 1.1 (整除的数域不变性)",
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    {
        "type": "text",
        "text": "$\\mathbb{P}, \\mathbb{F}$ 为两个数域， $\\mathrm{P} \\subseteq \\mathbb{F}$ ， $f(x), g(x) \\in \\mathbb{P}[x]$ (在小数域中)，则 $\\mathbb{P}$ 中 $g(x)|f(x)$ 当且仅当 $\\mathbb{F}$ 中 $g(x)|f(x)$",
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    {
        "type": "text",
        "text": "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": "text",
        "text": "定理 1.2 (带余除法)",
        "text_level": 2,
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    {
        "type": "text",
        "text": "对于 $\\forall f(x), g(x) \\in \\mathbb{P}[x]$ ，存在唯一 $q(x), r(x) \\in \\mathbb{P}[x]$ 使得 $f(x) = q(x)g(x) + r(x)$ 。这里 $r(x) = 0$ 或者 $\\partial(r(x)) < \\partial(g(x))$",
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    {
        "type": "text",
        "text": "练习1.1带余除法的计算 $(1)f(x) = x^{4} + 2x^{3} - 5x + 7, g(x) = x^{2} - 3x + 1$ ，求 $g(x)$ 除 $f(x)$ 的商式与余式解(1)商为 $x^{2} + 5x + 14$ ，余式为 $32x - 7$",
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        "type": "text",
        "text": "练习 1.2 带余除法的应用 (1) 设 $g(x)=ax+b$ ， $f(x)\\in\\mathbb{P}[x]$ ，证明： $g(x)|f^{2}(x)$ 当且仅当 $g(x)|f(x)$",
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        "text": "(2) $f(x), h(x), g(x)$ 为三个非零多项式，证明： $h(x)|(f(x)-g(x))$ 当且仅当 $f(x), g(x)$ 除以 $h(x)$ 的余式相同",
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    },
    {
        "type": "equation",
        "text": "$$\nf (x), g (x)\n$$",
        "text_format": "latex",
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    {
        "type": "equation",
        "text": "$$\ng ^ {m} (x) | f ^ {m} (x)\n$$",
        "text_format": "latex",
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    {
        "type": "equation",
        "text": "$$\ng (x) | f (x)\n$$",
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        "text": "证明 (1) 右推左显然。左推右: $f^2(x) = g(x)h(x)$ ，若 $f(x) = q(x)g(x) + r(x)$ ，则 $f^2(x) = q^2(x)g^2(x) + 2q(x)g(x)r(x) + r^2(x)$ ，根据 $g(x)|f(x)$ 得到 $g(x)|r^2(x)$ ，显然 $r(x) = 0$",
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        "type": "text",
        "text": "(2) 设 $f(x)=q_{1}(x)h(x)+r_{1}(x), g(x)=q_{1}(x)h(x)+r_{2}(x)$ ，相减得到",
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    },
    {
        "type": "equation",
        "text": "$$\nf (x) - g (x) = h (x) \\left[ q _ {1} (x) - q _ {2} (x) \\right] + r _ {1} (x) - r _ {2} (x)\n$$",
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        "type": "text",
        "text": "显然 $\\partial (r_1(x) - r_2(x)) < \\partial (h(x))$ ，而 $h(x)|(r_1(x) - r_2(x))$ ，因此当且仅当 $r_1(x) = r_2(x)$",
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    {
        "type": "text",
        "text": "(3) 右推左：用 $f(x)=h(x)g(x)$ ，两侧取 m 次方即可。",
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        "type": "text",
        "text": "左推右：设 $f(x)=c_{1}p_{1}^{r_{1}}(x)\\cdots p_{s}^{r_{s}}(x)$ ， $g(x)=c_{2}p_{1}^{t_{1}}(x)\\cdots p_{s}^{t_{s}}(x)$ ，其中 $p_{1}(x)\\sim p_{s}(x)$ 两两不可约且互素， $r_{i}$ 可以为 0 或非负整数。根据条件可知 $mt_{i}\\leq mr_{i}$ ，因此 $t_{i}\\leq r_{i}$ ，因此 $g(x)|f(x)$",
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    {
        "type": "text",
        "text": "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": "text",
        "text": "1.1.2 最大公因式",
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    {
        "type": "text",
        "text": "定义 1.2 (最大公因式)",
        "text_level": 2,
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        "type": "text",
        "text": "$f(x), g(x), d(x) \\in \\mathbb{P}[x]$ ，若 $d(x)$ 同时整除 $f(x), g(x)$ ，则称其为公因式。若没有可以整除 $d(x)$ 的公因式存在，则称其为最大公因式，记为",
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    {
        "type": "equation",
        "text": "$$\nd (x) = (f (x), g (x))\n$$",
        "text_format": "latex",
        "bbox": [
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    {
        "type": "text",
        "text": "定理 1.3 (最大公因式表示定理)",
        "text_level": 2,
        "bbox": [
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        "page_idx": 6
    },
    {
        "type": "text",
        "text": "$\\forall f(x), g(x) \\in \\mathbb{P}[x]$ ，均存在 $d(x) = (f(x), g(x)) \\in \\mathbb{P}[x]$ ，且 $\\exists u(x), v(x) \\in \\mathbb{P}[x]$ 满足 $d(x) = u(x)f(x) + v(x)g(x)$ (反之不可，互素时才可)",
        "bbox": [
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        "page_idx": 6
    },
    {
        "type": "text",
        "text": "证明 假设通过辗转相除法得到:",
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    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} f (x) = q _ {1} (x) g (x) + r _ {1} (x) \\\\ g (x) = q _ {2} (x) r _ {1} (x) + r _ {2} (x) \\\\ r _ {1} (x) = q _ {3} (x) r _ {2} (x) + r _ {3} (x) \\\\ \\vdots \\\\ r _ {s - 3} (x) = q _ {s - 1} (x) r _ {s - 2} (x) + r _ {s - 1} (x) \\\\ r _ {s - 2} (x) = q _ {s} (x) r _ {s - 1} (x) + r _ {s} (x) \\\\ r _ {s - 1} (x) = q _ {s + 1} (x) r _ {s} (x) + 0 \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "则此时可以得到 $r_s(x)$ 是 $f(x), g(x)$ 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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        "page_idx": 6
    },
    {
        "type": "equation",
        "text": "$$\n\\begin{array}{r l} r _ {s} (x) & = r _ {s - 2} (x) - q _ {s} (x) r _ {s - 1} (x) \\\\ & = r _ {s - 2} (x) - q _ {s} (x) (r _ {s - 3} (x) - q _ {s - 1} (x) r _ {s - 2} (x)) \\\\ & = \\dots \\dots \\\\ & = u (x) f (x) + v (x) g (x) \\end{array}\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 6
    },
    {
        "type": "text",
        "text": "即逐个将前面的等式代入方程，逐个消去 $r_{s-2}(x), \\cdots, r_{2}(x), r_{1}(x)$ ，最终只剩下 $g(x), f(x)$",
        "bbox": [
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        "page_idx": 6
    },
    {
        "type": "text",
        "text": "笔记 最大公因式 $d(x) = u(x)f(x) + v(x)g(x)$ 的 $u(x)$ 和 $v(x)$ 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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        "page_idx": 6
    },
    {
        "type": "text",
        "text": "练习1.3辗转相除法 $(1)f(x) = x^{4} + 3x^{3} - x^{2} - 4x - 3, g(x) = 3x^{3} + 10x^{2} + 2x - 3$ ，求 $(f(x), g(x))$ ，并求 $u(x), v(x)$ 使得 $u(x)f(x) + v(x)g(x) = (f(x), g(x))$",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "(2) 已知 $f(x) = x^4 + x^3 - 3x^2 - 4x - 1$ ， $g(x) = x^3 + x^2 - x - 1$ ，求满足 $u(x)f(x) + v(x)g(x) = x + 1$ 的 $u(x), v(x)$",
        "bbox": [
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        "page_idx": 6
    },
    {
        "type": "text",
        "text": "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",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "定理 1.4 (保最大公因式)",
        "text_level": 2,
        "bbox": [
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        "page_idx": 6
    },
    {
        "type": "text",
        "text": "$f(x), g(x) \\in P[x]$ ，若 $a, b, c, d \\in P$ 满足 $ad - bc \\neq 0$ ，则",
        "bbox": [
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        ],
        "page_idx": 6
    },
    {
        "type": "equation",
        "text": "$$\n(f (x), g (x)) = (a f (x) + b g (x), c f (x) + d g (x))\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 6
    },
    {
        "type": "text",
        "text": "证明 假设左侧为 $d(x)$ ，右侧为 $d_{1}(x)$ ，下面要证 $d(x)=d_{1}(x)$ ，不妨证明 $d(x)|d_{1}(x), d_{1}(x)|d(x)$ 。",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "(1) $d(x)|d_{1}(x)$ ： $d(x)$ 是左侧最大公因式，其是右侧每个部分的最大公因式，因此 $d(x)|d_{1}(x)$",
        "bbox": [
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        "page_idx": 6
    },
    {
        "type": "text",
        "text": "(2) $d_{1}(x)|d(x)$ ：由于 $ad - bc \\neq 0$ ，从方程组角度 $f(x), g(x)$ 也可以用 $af(x) + bg(x), cf(x) + dg(x)$ 线性表出，",
        "bbox": [
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    },
    {
        "type": "header",
        "text": "1.1 一元多项式",
        "bbox": [
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        "page_idx": 6
    },
    {
        "type": "page_number",
        "text": "2",
        "bbox": [
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    },
    {
        "type": "equation",
        "text": "$$\n\\begin{array}{c} \\text {   ①   } \\\\ q _ {1} (x) = x - \\frac {1}{3} \\left| \\begin{array}{l l} f (x) & \\frac {1}{3} g (x) \\\\ x ^ {4} + 3 x ^ {3} - x ^ {2} - 4 x - 3 & x ^ {3} + \\frac {1 0}{3} x ^ {2} + \\frac {2}{3} x - 1 \\\\ x ^ {4} + \\frac {1 0}{3} x ^ {3} + \\frac {2}{3} x ^ {2} - x & x ^ {3} + 5 x ^ {2} + 6 x \\\\ \\hline - \\frac {1}{3} x ^ {3} - \\frac {5}{3} x ^ {2} - 3 x - 3 & - \\frac {5}{3} x ^ {2} - \\frac {1 6}{3} x - 1 \\\\ - \\frac {1}{3} x ^ {3} - \\frac {1 0}{9} x ^ {2} - \\frac {9}{9} x + \\frac {1}{3} & - \\frac {5}{3} x ^ {2} - \\frac {2 5}{3} x - 1 0 \\\\ \\hline r _ {1} (x) = - \\frac {5}{9} x ^ {2} - \\frac {2 5}{9} x - \\frac {1 0}{3} & r _ {2} (x) = 3 x + 9 \\\\ - \\frac {9}{5} r _ {1} (x) = x ^ {2} + 5 x + 6 & \\frac {1}{3} r _ {2} (x) = x + 3. \\end{array} \\right| \\quad \\text {   ②   } \\\\ q _ {2} (x) = x - \\frac {5}{3} \\end{array}\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 7
    },
    {
        "type": "equation",
        "text": "$$\n\\begin{array}{r l} & {\\mathrm{因此}: ① f (x) = \\frac {1}{3} g _ {1} (x) g (x) + r _ {1} (x) \\quad ② g (x) = g _ {2} (x) (- \\frac {9}{5} r _ {1} (x)) + r _ {2} (x)} \\\\ & {③ - \\frac {9}{5} r _ {1} (x) = (x + 2) \\frac {1}{3} r _ {2} (x) \\qquad r _ {2} (x) \\mathrm{即大公因式}} \\\\ & {\\Rightarrow ① ^ {\\prime}: r _ {1} (x) = f (x) - \\frac {1}{3} g _ {1} (x) g (x), ② ^ {\\prime} r _ {2} (x) = g (x) + \\frac {9}{5} g _ {2} (x) r _ {1} (x)} \\\\ & {\\mathrm{将} r _ {1} (x) \\mathrm{代入} ② ^ {\\prime} \\mathrm{中即可}} \\end{array}\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 7
    },
    {
        "type": "text",
        "text": "具体用 cramer 法则求出即可，如 $f(x)=\\frac{df_{1}(x)-bg_{1}(x)}{ad-bc}, g(x)=\\frac{ag_{1}(x)-cf_{1}(x)}{ad-bc}$ ，因此与 (1) 同理",
        "bbox": [
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        "page_idx": 7
    },
    {
        "type": "text",
        "text": "推论 1.1 (加减保最大公因式)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 7
    },
    {
        "type": "equation",
        "text": "$$\n(f (x), f (x) - g (x)) = (f (x), f (x) + g (x)) = (f (x), g (x))\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 7
    },
    {
        "type": "text",
        "text": "定理 1.5 (最小公倍式的表示)",
        "text_level": 2,
        "bbox": [
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        "page_idx": 7
    },
    {
        "type": "text",
        "text": "$f(x), g(x)$ 的首 1 最小公倍式 $[f(x), g(x)] = \\frac{f(x)g(x)}{(f(x), g(x))}$",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "证明 设 $(f(x), g(x)) = d(x), f(x) = d(x)f_1(x), g(x) = d(x)g_1(x)$ ，且 $(f_1(x), g_1(x)) = 1$ ，记 $s(x) = \\frac{f(x)g(x)}{d(x)} = d(x)f_1(x)g_1(x) = f(x)g_1(x) = f_1(x)g(x)$ ，下面说明 $s(x)$ 是最小公倍式：(1)是公倍式：显然(2)最小：设 $h(x)$ 也是公倍式，设 $h(x) = f(x)m(x)$ ，根据 $g(x)|h(x)$ 得到 $d(x)g_1(x)|d(x)f_1(x)m(x)$ ，根据 $g_1(x)$ 与 $f_1(x)$ 互素，得到 $g_1(x)|m(x)$ ，因此 $g_1(x)f(x)|f(x)m(x)$ ，得到 $s(x)|h(x)$ ，因此 $s(x)$ 是最小的。",
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    },
    {
        "type": "text",
        "text": "1.1.3 互素",
        "text_level": 2,
        "bbox": [
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    },
    {
        "type": "text",
        "text": "定义 1.3 (互素)",
        "text_level": 2,
        "bbox": [
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        "page_idx": 7
    },
    {
        "type": "text",
        "text": "若 $f(x), g(x) \\in \\mathbb{P}[x]$ 的最大公因式为1，则称它们互素。",
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        "page_idx": 7
    },
    {
        "type": "text",
        "text": "定理 1.6 (互素充要条件)",
        "text_level": 2,
        "bbox": [
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        "page_idx": 7
    },
    {
        "type": "text",
        "text": "$f(x), g(x) \\in \\mathbb{P}[x]$ 互素当且仅当 $\\exists u(x), v(x) \\in \\mathbb{P}[x]$ 使得 $u(x)f(x) + v(x)g(x) = 1$",
        "bbox": [
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        "page_idx": 7
    },
    {
        "type": "text",
        "text": "证明 左推右显然 (根据最大公因式定理)。右推左：设 $(f(x), g(x)) = d(x)$ ，则 $d(x)|f(x), d(x)|g(x)$ ，设 $f(x) = d(x)f_1(x), g(x) = d(x)g_1(x)$ ，从而",
        "bbox": [
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        "page_idx": 7
    },
    {
        "type": "equation",
        "text": "$$\nd (x) | [ u (x) f _ {1} (x) + v (x) g _ {1} (x) ] = 1\n$$",
        "text_format": "latex",
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    },
    {
        "type": "header",
        "text": "1.1 一元多项式",
        "bbox": [
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        ],
        "page_idx": 7
    },
    {
        "type": "page_number",
        "text": "3",
        "bbox": [
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        "page_idx": 7
    },
    {
        "type": "text",
        "text": "因此 $d(x), u(x)f_1(x) + v(x)g_1(x)$ 均为常数，故 $f(x), g(x)$ 互素笔记上述 $u(x), v(x)$ 在无条件限制下也是不一定唯一的。",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "text",
        "text": "练习1.4互素相关应用(1)若 $f(x),g(x)$ 互素，证明： $f(x^{m}),g(x^{m})$ 互素",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "text",
        "text": "(2) 重点：若 $(f(x), g(x)) = d(x)$ ，证明： $(f(x^m), g(x^m)) = d(x^m)$",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "text",
        "text": "(3) 设 $(f(x), g(x)) = 1, (f(x), h(x)) = 1$ ，证明： $(f(x), g(x)h(x)) = 1$",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "text",
        "text": "证明 (1) 由于 $f(x), g(x)$ 互素，得到 $u(x)f(x) + v(x)g(x) = 1$ ，从而 $u(x^{m})f(x^{m}) + v(x^{m})g(x^{m}) = 1$ ，由于是互素充要条件，因此 $f(x^{m}), g(x^{m})$ 互素",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "text",
        "text": "(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 $f(x) = d(x)m(x), g(x) = d(x)n(x)$ ，则 $(m(x), n(x)) = 1$ ，因此",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "equation",
        "text": "$$\nu (x ^ {k}) m (x ^ {k}) + v (x ^ {k}) n (x ^ {k}) = 1\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "text",
        "text": "得到 $m(x^{k}), n(x^{k})$ 互素，而 $f(x^{k}) = d(x^{k})m(x^{k}), g(x^{k}) = d(x^{k})n(x^{k})$ ，因此 $(f(x^{k}), g(x^{k})) = d(x^{k})$",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "text",
        "text": "(3) 根据互素，有 $u_{1}(x)f(x) + v_{1}(x)g(x) = 1, u_{2}(x)f(x) + v_{2}(x)h(x) = 1$ ，两式相乘得到",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ u _ {1} (x) u _ {2} (x) f (x) + v _ {1} (x) g (x) + v _ {2} (x) h (x) \\right] f (x) + v _ {1} (x) v _ {2} (x) g (x) h (x) = 1\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "text",
        "text": "因此根据充要条件得到 $(f(x), g(x)h(x)) = 1$",
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        ],
        "page_idx": 8
    },
    {
        "type": "text",
        "text": "练习 1.5 互素充要条件推广 $f(x)$ , $g(x)$ 是次数大于 0 的互素多项式，则存在唯一的 $u(x)$ , $v(x)$ 使得 $u(x)f(x) + v(x)g(x) = 1$ ，这里 $\\partial(u(x)) < \\partial(g(x))$ , $\\partial(v(x)) < \\partial(f(x))$",
        "bbox": [
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        ],
        "page_idx": 8
    },
    {
        "type": "text",
        "text": "证明 (1) 存在性：由于 $f(x), g(x)$ 互素，故存在 $a(x), b(x)$ 使得 $a(x)f(x) + b(x)g(x) = 1$ ，显然 $a(x), g(x)$ 也互素，作带余除法 $a(x) = q(x)g(x) + u(x)$ ，代入公式得到",
        "bbox": [
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        ],
        "page_idx": 8
    },
    {
        "type": "equation",
        "text": "$$\n(q (x) g (x) + u (x)) f (x) + b (x) g (x) = 1 \\Rightarrow u (x) f (x) + (b (x) + q (x) f (x)) g (x) = 1\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "text",
        "text": "记 $v(x) = b(x) + q(x)f(x)$ ，下面证明 $\\partial(v(x)) < \\partial(f(x))$ 。反设 $\\partial(v(x)) \\geq \\partial(f(x))$ ，根据 $\\partial(u(x)) < \\partial(g(x))$ 可知 $\\partial(v(x)g(x)) > \\partial(u(x)f(x))$ ，显然它们的和不为 1，矛盾。",
        "bbox": [
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        ],
        "page_idx": 8
    },
    {
        "type": "text",
        "text": "(2) 唯一性：假设还有 $u_{1}(x)f(x) + v_{1}(x)g(x) = 1$ ，那么有 $(u(x) - u_{1}(x))f(x) + (v(x) - v_{1}(x))g(x) = 0$ ，即",
        "bbox": [
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        ],
        "page_idx": 8
    },
    {
        "type": "equation",
        "text": "$$\n(u (x) - u _ {1} (x)) f (x) = (v _ {1} (x) - v (x)) g (x)\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "text",
        "text": "由于 $f(x)$ , $g(x)$ 互素，得到 $f(x)\\left|(v_{1}(x)-v(x)\\right)$ ，而后者次数低于前者，只能 $v_{1}(x)=v(x)$",
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        "page_idx": 8
    },
    {
        "type": "text",
        "text": "1.1.4 不可约多项式与因式分解",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 8
    },
    {
        "type": "text",
        "text": "定义 1.4 (不可约多项式)",
        "text_level": 2,
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "text",
        "text": "设 $p(x) \\in \\mathbb{P}[x]$ 且 $\\partial(p(x)) \\geq 1$ ，若 $p(x)$ 不能表示成 $\\mathbb{P}[x]$ 中两个次数小于 $p(x)$ 的多项式的乘积，则称 $p(x)$ 为 $\\mathbb{P}[x]$ 中的不可约多项式。",
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        "page_idx": 8
    },
    {
        "type": "text",
        "text": "定理 1.7 (不可约多项式的性质)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 8
    },
    {
        "type": "text",
        "text": "若 $p(x)\\in \\mathbb{P}[x]$ 是不可约多项式，则",
        "bbox": [
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        ],
        "page_idx": 8
    },
    {
        "type": "text",
        "text": "\\- $p(x)$ 与 $\\forall f(x) \\in \\mathbb{P}[x]$ 要么互素，要么 $p(x) | f(x)$",
        "bbox": [
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        ],
        "page_idx": 8
    },
    {
        "type": "text",
        "text": "若 $p(x)|f(x)g(x)$ ，则要么 $p(x)|f(x)$ 要么 $p(x)|g(x)$",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "image",
        "img_path": "images/94c21327ea1ce13f97d4196886e4a3c8357fc7b23d1e4155e67153e99c609ca1.jpg",
        "image_caption": [],
        "image_footnote": [],
        "content": "",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "笔记 不可约多项式经常用于证明乘积形式 $(h(x), f(x)g(x)) = 1$ 的命题。例如反设 $(h(x), f(x)g(x)) = d(x)$ ，不可约多项式 $p(x)|d(x)$ ，则 $p(x)$ 要么整除 $f(x)$ ，要么整除 $g(x)$",
        "bbox": [
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        ],
        "page_idx": 8
    },
    {
        "type": "text",
        "text": "练习 1.6 利用不可约多项式性质 + 反证法 (1) 重点: 已知 $(f(x), g(x)) = 1$ ，证明 $(f(x)g(x), f(x) + g(x)) = 1$",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "page_number",
        "text": "4",
        "bbox": [
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        "page_idx": 8
    },
    {
        "type": "header",
        "text": "1.1 一元多项式",
        "bbox": [
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        ],
        "page_idx": 8
    },
    {
        "type": "text",
        "text": "(2) 已知 $f_{1}(x)$ , $f_{2}(x)$ , $\\cdots$ , $f_{s}(x)$ 两两互素，记 $F_{i}(x) = f_{1}(x) \\cdots f_{i-1}(x) f_{i+1}(x) \\cdots f_{s}(x)$ ，证明 $F_{1}(x)$ , $F_{2}(x)$ , $\\cdots$ , $F_{s}(x)$ 互素",
        "bbox": [
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        "page_idx": 9
    },
    {
        "type": "text",
        "text": "(3) 稍加变化: 复数域上的多项式 $f(x)$ 没有重因式, 证明 $(f(x) + 2f'(x), 3f^2(x)) = 1$",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "证明 (1) 反设有不可约多项式 $q(x)$ 作为公因式，则 $q(x)$ 同时是 $f(x)g(x), f(x) + g(x)$ 因式。根据前者得到 $q(x)$ 至少整除 $f(x), g(x)$ 之一，不妨设 $q(x)|f(x)$ ，根据第二个式子得到 $q(x)$ 也一定整除 $g(x)$ ，否则无法得到多项式。因此矛盾。",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "(2) 同种方法，随便挑 $F_{i}(x), F_{j}(x)$ 说一说就行",
        "bbox": [
            122,
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            210
        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "(3) 同样设不可约 $q(x)$ ，根据 $3f^{2}(x)$ 可知 $q(x)|f(x)$ ，再根据 $q(x)|f(x)+2f'(x)$ 可知 $q(x)|f'(x)$ （否则不是多项式），根据重因式充要条件可知。",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "练习 1.7 证明不可约多项式 (1) $p(x)$ 是次数大于 0 的多项式，若对任意多项式 $f(x), g(x)$ ，根据 $p(x)|f(x)g(x)$ 可以得到 $p(x)|f(x)$ 或者 $p(x)|g(x)$ ，则 $p(x)$ 是不可约多项式。",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "证明 (1) 反设 $p(x)$ 不是不可约多项式，则存在 $f(x), g(x)$ 使得 $p(x) = f(x)g(x)$ ，此时 $p(x)|f(x)g(x)$ ，但是 $p(x) \\nmid f(x), p(x) \\nmid g(x)$",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "定理 1.8 (因式分解唯一性)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "$f(x)$ 是 $\\mathbb{P}$ 上次数大于等于1的多项式，则 $f(x)$ 可唯一分解为 $\\mathbb{P}$ 上有限个不可约多项式乘积：",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "equation",
        "text": "$$\nf (x) = c p _ {1} ^ {r _ {1}} (x) p _ {2} ^ {r _ {2}} (x) \\dots p _ {s} ^ {r _ {s}} (x)\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "1.1.5 余数定理及其应用",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "定理 1.9 (余数定理)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "$f(x)\\in \\mathbb{P}[x]$ ，对 $\\forall \\alpha \\in \\mathbb{P}$ ，总有",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "equation",
        "text": "$$\nf (x) = (x - \\alpha) q (x) + f (\\alpha)\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "证明 用带余除法得到 $f(x)=(x-\\alpha)q(x)+c$ ，而代入 $x=\\alpha$ 得到 $f(\\alpha)=0+c$ ，即 $c=f(\\alpha)$ ，因此",
        "bbox": [
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        "page_idx": 9
    },
    {
        "type": "equation",
        "text": "$$\nf (x) = (x - \\alpha) q (x) + f (\\alpha)\n$$",
        "text_format": "latex",
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        "page_idx": 9
    },
    {
        "type": "text",
        "text": "推论 1.2",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "$\\alpha \\in \\mathbb{P}$ 是多项式 $f(x)$ 的根当且仅当 $(x - \\alpha) \\mid f(x)$",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "笔记 余数定理的主要价值在于 $\\mathbb{P}$ 上的因式等价于 $\\mathbb{P}$ 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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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "练习 1.8 余数定理的应用 (1) 若 $(x-1)\\mid f(x^{m})$ ，证明 $(x^{m}-1)\\mid f(x^{m})$",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "(2) 证明 $x^{2} + x + 1 \\mid x^{3m} + x^{3n + 1} + x^{3p + 2}$",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "(3) 已知 $x^{2} + x + 1 \\mid f_{1}(x^{3}) + xf_{2}(x^{3})$ ，证明 $(x - 1) \\mid f_{1}(x)$ 且 $(x - 1) \\mid f_{2}(x)$",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "(4) 已知 $x^{2} + 1 \\mid f_{1}(x^{4}) + xf_{2}(x^{4})$ ，证明 1 是 $f_{1}(x), f_{2}(x)$ 的公共根。",
        "bbox": [
            122,
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "证明 (1) 由于 $(x - 1) \\mid f(x^m)$ ，因此 $x = 1$ 是 $f(x^m)$ 的根，故 $x = 1$ 是 $f(x)$ 的根，因此 $(x - 1) \\mid f(x)$ ，故 $(x^m - 1) \\mid f(x^m)$",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "(2) 放到 $\\mathbb{C}$ 上考虑: $\\omega = e^{i\\frac{2\\pi}{3}}$ , 则 $x^{2} + x + 1 = (x - \\omega)(x - \\omega^{2})$ , 代入 $\\omega$ 则 $\\omega^{3m} + \\omega^{3n+1} + \\omega^{3p+2} = 1 + \\omega + \\omega^{2} = 0$ , 代入 $\\omega^{2}$ 则 $\\omega^{6m} + \\omega^{6n+2} + \\omega^{6p+4} = 1 + \\omega^{2} + \\omega^{4} = 1 + \\omega^{2} + \\omega = 0$ , 根据余数定理可知成立",
        "bbox": [
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        "page_idx": 9
    },
    {
        "type": "text",
        "text": "(4) $x^{2}+1$ 复数根为 x=i，根据 $(x^{2}+1)\\mid f_{1}(x^{4})+xf_{2}(x^{4})$ 可知 $f_{1}(1)+if_{2}(1)=0$ ，从而为公共根。",
        "bbox": [
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        "page_idx": 9
    },
    {
        "type": "page_number",
        "text": "5",
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        "page_idx": 9
    },
    {
        "type": "header",
        "text": "1.1 一元多项式",
        "bbox": [
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        ],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "1.1.6 重因式与重根",
        "text_level": 2,
        "bbox": [
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        "page_idx": 10
    },
    {
        "type": "text",
        "text": "定理 1.10 (重因式性质汇总)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "$f(x)$ 是次数大于0的多项式， $p(x)$ 是不可约多项式，则",
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "\\- 核心：若 $p(x)$ 是 $f(x)$ 的 $k$ 重因式，则其是 $f'(x)$ 的 $k - 1$ 重因式。",
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "\\- 无重因式充要： $f(x)$ 没有重因式当且仅当 $f(x), f'(x)$ 互素。",
        "bbox": [
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        "page_idx": 10
    },
    {
        "type": "text",
        "text": "\\- 不可约因式： $\\frac{f(x)}{(f(x), f'(x))}$ 是 $f(x)$ 所有不可约因式 (重数均为 1) 乘积的常数倍",
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "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 $(x^{2} + 1)^{2}$ ，其在 $\\mathbb{C},\\mathbb{R},\\mathbb{Q}$ 上都有重因式，但是只在 $\\mathbb{C}$ 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。",
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "练习1.9用根判断因式(1)重点：设 $h(x), f(x), g(x)$ 为实系数多项式满足以下等式，证明 $x^{2} + 1|f(x), x^{2} + 1|g(x)$",
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} (x ^ {2} + 1) h (x) + (x + 1) f (x) + (x - 2) g (x) = 0 \\\\ (x ^ {2} + 1) k (x) + (x - 1) f (x) + (x + 2) g (x) = 0 \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 10
    },
    {
        "type": "text",
        "text": "证明 (1) 要证明 $\\mathbb{R}$ 上 $(x^2 + 1)|f(x)$ ，只需要证明 $\\mathbb{C}$ 上 $(x - i)|f(x), (x + i)|f(x)$ ，等价于 $x = \\pm i$ 在 $\\mathbb{C}$ 上为 $f(x)$ 的根。代入 $x = i$ 得到：",
        "bbox": [
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        "page_idx": 10
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} (i + 1) f (i) + (i - 2) g (i) = 0 \\\\ (i - 1) f (i) + (i + 2) g (i) = 0 \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 10
    },
    {
        "type": "text",
        "text": "解得 $f(i) = g(i) = 0$ 。同理可证 $f(-i) = g(-i) = 0$",
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "定理 1.11 (重根性质汇总)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "$f(x)$ 是 $\\mathbb{P}[x]$ 上次数大于0的多项式，则",
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "\\- 重根： $x_0$ 是 $f(x)$ 的 $k$ 重根当且仅当 $f(x_0) = f'(x_0) = \\cdots = f^{(k-1)}(x_0) = 0$ ，而 $f^{(k)}(x_0) \\neq 0$",
        "bbox": [
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        "page_idx": 10
    },
    {
        "type": "text",
        "text": "\\- 重根判定定理： $p(x)$ 是 $f(x)$ 重因式当且仅当其是 $f(x), f'(x)$ 公因式。",
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "笔记 $\\mathbb{C}$ 上的重根问题等价于 $\\mathbb{C}$ 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 $\\mathbb{C}$ 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。",
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "练习 1.10 判定重根存在性 (1) 证明 $f(x)=1+x+\\frac{x^{2}}{2!}+\\cdots+\\frac{x^{n}}{n!}$ 没有重根",
        "bbox": [
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        "page_idx": 10
    },
    {
        "type": "text",
        "text": "(2) 证明 $f(x)=x^{n}+nx^{n-1}+n(n-1)x^{n-2}+\\cdots+n!$ 无重根证明 (1) 直接证重根不好证，转换为重因式，若在 $\\mathbb{C}$ 上无重因式则肯定无重根。根据 $f'(x) = 1 + x + \\frac{x^2}{2!} + \\cdots + \\frac{x^{n-1}}{(n-1)!}$ 得到",
        "bbox": [
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        "page_idx": 10
    },
    {
        "type": "text",
        "text": "",
        "bbox": [
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        "page_idx": 10
    },
    {
        "type": "equation",
        "text": "$$\n(f (x), f ^ {\\prime} (x)) = (f (x), f (x) - f ^ {\\prime} (x)) = (f (x), \\frac {x ^ {n}}{n !}) = (f (x), x ^ {n})\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "由于 $x^n$ 项的原因， $(f(x), f'(x)) = x^k, k \\geq 0$ 。而 $f(0) = 1$ ，这说明 $x$ 不是 $f(x)$ 的因式，从而 $k = 0$ ，因此 $f(x), f'(x)$ 互素，无重根。",
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "(2) 也用无重因式证明",
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "练习 1.11 重根、重因式相互转化 (1) 求 $x^{5} + x^{4} - 6x^{3} - 14x^{2} - 11x - 3$ 的重根和重数",
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "(2) 重点: 求 $A, B$ 使得 $(x - 1)^{2} | Ax^{4} + Bx^{2} + 1$",
        "bbox": [
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        ],
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "(3) 重点：求 $t$ 使得 $f(x) = x^{3} - 3x^{2} + tx - 1$ 有重根",
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    },
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        "type": "text",
        "text": "(4) 重点 ZJU2018.2: 证明 $x^{3} + px + q$ 有重根的充要条件 $4p^{3} + 27q^{2} = 0$",
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    {
        "type": "header",
        "text": "1.1 一元多项式",
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    {
        "type": "page_number",
        "text": "6",
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    },
    {
        "type": "text",
        "text": "(5)ZJU2021: 求 $t$ 为何值时, $f(x) = x^3 + 6x^2 + tx + 8$ 有重根, 并求重根解 (1) 可以尝试猜。一般用辗转相除法算 $f(x), f'(x)$ 的最大公因式, 最终答案为 $(x + 1)^4(x - 3)$",
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    {
        "type": "text",
        "text": "(2) 根据重根与重因式关系，在 $\\mathbb{C}$ 上必有 $f(1) = 0, f'(1) = 0$ ，因此得出 $A + B + 1 = 0, 4A + 2B = 0$ ，得到 $A = -2, B = 1$",
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    },
    {
        "type": "text",
        "text": "(3) 同理考虑 $\\mathbb{C}$ 上的因式分解， $f'(x) = 3x^2 - 6x + t$ ，因此考虑 $f(x)$ 与 $\\frac{1}{3} f'(x) = x^2 - 2x + \\frac{t}{3}$ 辗转相除法，第一步得出 $f(x) = \\frac{1}{3} f'(x)(x - 1) + (\\frac{2}{3} t - 2)x + (\\frac{t}{3} - 1)$ ，如果 $t = 3$ 则直接整除有重根，如果 $t \\neq 3$ ，则接着算， $\\frac{1}{3} f' = (\\frac{1}{\\frac{2}{3} t - 2})r_1(x)(x - \\frac{5}{2}) + \\frac{t}{3} + \\frac{5}{4}$ ，则需要 $\\frac{t}{3} + \\frac{5}{4} = 0$ ，得到 $t = -\\frac{15}{4}$",
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    },
    {
        "type": "text",
        "text": "(4) 在 $\\mathbb{C}$ 上考虑，有重根即有重因式， $f'(x) = 3x^2 + p$ ，先有 $f(x) = \\frac{1}{3} f'(x)x + \\frac{2}{3} px + q$ ，若 $p = q = 0$ ，则成立，下面考虑 $p \\neq 0$ 。得到 $r_1(x) = \\frac{3}{2p} r_1(x)(x - \\frac{3q}{2p}) + \\frac{p}{3} + \\frac{9q^2}{4p^2}$ ，因此得到 $4p^3 + 27q^2 = 0$",
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    },
    {
        "type": "text",
        "text": "(5)t=12 时有 3 重 x=-2 根，t=-15 时有 2 重 x=1 根。",
        "bbox": [
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        "page_idx": 11
    },
    {
        "type": "text",
        "text": "练习 1.12 导数重根反推函数重根？ $x_{0}$ 是 $f'(x)$ 的 k 重根，举例说明其不能推出 $x_{0}$ 是 $f(x)$ 的 $k+1$ 重根，该命题需要 $x_{0}$ 是 $f(x)$ 的根。",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "解例如 $f(x) = x^{k + 1} + 1$ ， $x_0 = 0$",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "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 $f(x)$ 本身的根。",
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    },
    {
        "type": "text",
        "text": "1.1.7 整数与多项式",
        "text_level": 2,
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    },
    {
        "type": "text",
        "text": "定理 1.12 (整数与多项式)",
        "text_level": 2,
        "bbox": [
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        "page_idx": 11
    },
    {
        "type": "text",
        "text": "定理1.12（整数与多项式） $m,n$ 为正整数， $(m,n) = d$ 的充要条件是 $(x^{m} - 1, x^{n} - 1) = x^{d} - 1$",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "证明 (1) 左推右是公因式：设 m = kd，则 $x^{m} - 1 = (x^{d})^{k} - 1 = (x^{d} - 1)(1 + x^{d} + \\cdots + x^{(k-1)d})$ ，这说明 $x^{d} - 1|x^{m} - 1$ ，另一侧同理，因此是公因式",
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    },
    {
        "type": "text",
        "text": "(2) 最大: 设 $p(x)$ 是 $x^n - 1, x^m - 1$ 的公因式, 下面证明 $p(x)|x^d - 1$ 。由于 $x^n - 1$ 在 $\\mathbb{C}$ 上无重根 (因为和导数互素), 因此 $p(x)$ 也没有重根, 从而 $p(x) = a(x - x_1) \\cdots (x - x_s)$ , 这里 $x_1 \\sim x_s$ 是互异复数, 由于 $x_i^n = x_i^m = 1$ , 得到 $x_i^d = x_i^{um + vn} = (x_i^m)^u(x_i^n)^v = 1$ , 因此 $x - x_i|x^d - 1$ , $(x - x_1) \\cdots (x - x_s)|x^d - 1$ , 因此 $p(x)|x^d - 1$",
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    },
    {
        "type": "text",
        "text": "(3) 右推左：若 $(m, n) = d_{1}$ ，则根据前面的结论 $(x^{m} - 1, x^{n} - 1) = x^{d_{1}} - 1 = x^{d} - 1$ ，因此 $d_{1} = d$",
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    },
    {
        "type": "text",
        "text": "练习1.13整除与多项式 $(1)d, n \\in \\mathbb{N}^{+}$ ，证明 $x^{d} - 1|x^{n} - 1$ 的充要条件为 $d|n$",
        "text_level": 2,
        "bbox": [
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    {
        "type": "text",
        "text": "(2)m,n 是大于 1 的自然数，证明：m,n 互素的充要条件是 $1 + x + x^{2} + \\cdots + x^{m-1}$ 与 $1 + x + x^{2} + \\cdots + x^{n-1}$ 互素",
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    },
    {
        "type": "text",
        "text": "(3) $g(x)=1+x+x^{2}+\\cdots+x^{n},f(x)=g(x^{k}),(k,n+1)=1$ ，证明： $g(x)|f(x)$",
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    },
    {
        "type": "text",
        "text": "证明 (1) 本题即前面题目的特例。",
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    {
        "type": "text",
        "text": "右推左：设 $x^{d}-1=(x-x_{1})\\cdots(x-x_{d})(\\mathbb{C}$ 上)，因此 $x_{i}^{d}=1$ ，由于 n=dk，因此 $x_{i}^{n}=(x_{i}^{d})^{k}=1$ ，因此 $x^{d}-1|x^{n}-1$ 。",
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    {
        "type": "text",
        "text": "左推右：设 $x^{d}-1=(x-x_{1})\\cdots(x-x_{d})$ ，由于 $x^{d}-1|x^{n}-1,\\quad x_{i}^{d}=x_{i}^{n}=1$ ，假设 $n=kd+r, r\\neq0$ ，则 $x_{i}^{n}=x_{i}^{kd+r}=x_{i}^{r}$ ，由于 $x_{i}=1$ 只有一重，其他 $x_{i}\\neq1$ ，对于这些 $x_{i}$ 有 $x_{i}^{r}\\neq1$ ，因此矛盾。",
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    },
    {
        "type": "text",
        "text": "(2)m,n 互素当且仅当 $(x^{m}-1,x^{n}-1)=x-1$ ，等价于 $(1+x+x^{2}+\\cdots+x^{m-1},1+x+\\cdots+x^{n-1})=1$ (3) 根据 $g(x)=\\frac{x^{n+1}-1}{x-1}$ , $f(x)=\\frac{x^{k(n+1)}-1}{x^{k}-1}$ ，此时 $g(x)|f(x)$ 等价于 $(x^{k}-1)(x^{n+1}-1)|(x-1)(x^{k(n+1)}-1)$ ，由于 $(k,n+1)=1$ ，因此 $x^{k}-1$ 与 $x^{n+1}$ 的最大公因为 $(x-1)$ ，而 $(x^{k(n+1)}-1)$ 是 $(x^{k}-1)(x^{n+1}-1)$ ，因此根据最小公倍式的性质显然可知。",
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        "page_idx": 11
    },
    {
        "type": "page_number",
        "text": "7",
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    },
    {
        "type": "header",
        "text": "1.2 特殊数域上的一元多项式理论",
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    },
    {
        "type": "text",
        "text": "1.2 特殊数域上的一元多项式理论",
        "text_level": 2,
        "bbox": [
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    },
    {
        "type": "text",
        "text": "1.2.1 复数域上的一元多项式",
        "text_level": 2,
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    },
    {
        "type": "text",
        "text": "定理 1.13 (代数学基本定理)",
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    },
    {
        "type": "text",
        "text": "任意次数大于等于1的多项式在 $\\mathbb{C}$ 上至少有一个根。",
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        "page_idx": 12
    },
    {
        "type": "text",
        "text": "定理 1.14 (复数域因式分解)",
        "text_level": 2,
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        ],
        "page_idx": 12
    },
    {
        "type": "text",
        "text": "$\\mathbb{C}$ 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 $n$ 次多项式在 $\\mathbb{C}$ 上恰有 $n$ 个复根。",
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        "page_idx": 12
    },
    {
        "type": "text",
        "text": "1.2.2 实数域上的一元多项式",
        "text_level": 2,
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        ],
        "page_idx": 12
    },
    {
        "type": "text",
        "text": "定理 1.15 (实数域因式分解)",
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        ],
        "page_idx": 12
    },
    {
        "type": "text",
        "text": "$\\mathbb{R}$ 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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        "page_idx": 12
    },
    {
        "type": "text",
        "text": "1.2.3 有理数域上一元多项式因式分解",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 12
    },
    {
        "type": "text",
        "text": "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。",
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    },
    {
        "type": "text",
        "text": "定义 1.5 (本原多项式)",
        "text_level": 2,
        "bbox": [
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        "page_idx": 12
    },
    {
        "type": "text",
        "text": "若整系数多项式 $g(x)=b_{n}x^{n}+\\cdots+b_{1}x+b_{0}$ 的系数 $b_{0}\\sim b_{n}$ 是互素的 (整体互素而非两两互素)，即",
        "bbox": [
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        "page_idx": 12
    },
    {
        "type": "equation",
        "text": "$$\n(b _ {1}, \\dots , b _ {n}) = 1\n$$",
        "text_format": "latex",
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    },
    {
        "type": "text",
        "text": "则称 $g(x)$ 是一个本原多项式",
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        "page_idx": 12
    },
    {
        "type": "text",
        "text": "定理 1.16 (Gauss 引理)",
        "text_level": 2,
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    },
    {
        "type": "text",
        "text": "两个本原多项式的乘积仍为本原多项式",
        "text_level": 2,
        "bbox": [
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    },
    {
        "type": "text",
        "text": "证明 设 $f(x) = \\sum_{k=0}^{n} a_k x^k, g(x) = \\sum_{k=0}^{m} b_k x^k$ ，设 $h(x) = f(x)g(x) = \\sum_{k=0}^{n+m} d_k x^k$ 。假设 $h(x)$ 不是本原多项式，则存在素数 $p$ 整除所有 $d_k$ ，由于 $f(x), g(x)$ 本原，因此从常数项开始搜索，存在 $i, j$ ，使得",
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    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} p \\mid a _ {0}, \\dots , p \\mid a _ {i - 1}, p \\nmid a _ {i} \\\\ p \\mid b _ {0}, \\dots , p \\mid b _ {j - 1}, p \\nmid b _ {j} \\end{array} \\right.\n$$",
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    },
    {
        "type": "text",
        "text": "而 $d_{i + j} = a_i b_j + \\sum_{k=0, k \\neq i}^{i+j} a_k b_{i+j-k}$ ，除了第一项外均至少有一项被 $p$ 整除，第一项不被 $p$ 整除，因此 $p \\nmid d_{i+j}$ ，这与假设矛盾。",
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    },
    {
        "type": "text",
        "text": "定理 1.17 (有理多项式变本原多项式)",
        "text_level": 2,
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    },
    {
        "type": "text",
        "text": "设 $f(x)$ 是有理多项式， $g(x)$ 为本原多项式，则 $f(x)$ 可表示为有理数 $r$ 与本原多项式 $g(x)$ 的乘积",
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        "page_idx": 12
    },
    {
        "type": "text",
        "text": "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": "header",
        "text": "1.2 特殊数域上的一元多项式理论",
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        "page_idx": 12
    },
    {
        "type": "page_number",
        "text": "8",
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    },
    {
        "type": "text",
        "text": "练习1.14 本原多项式 (1) $f(x), g(x)$ 是本原多项式， $g(x) \\mid f(x)$ ，证明： $\\frac{f(x)}{g(x)}$ 也是本原多项式 (2) $f(x)$ 是本原多项式。证明： $f(x+1)$ 也是本原多项式",
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    },
    {
        "type": "text",
        "text": "(2) $f(x)$ 是本原多项式，证明： $f(x+1)$ 也是本原多项式。",
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        "page_idx": 13
    },
    {
        "type": "text",
        "text": "证明 (1) 设 $f(x) = g(x)q(x)$ ，由于 $f(x), g(x)$ 为本原多项式， $q(x)$ 必然是有理系数的，设 $q(x) = rq_1(x)$ ，这里 $q_1(x)$ 为本原多项式，因此",
        "bbox": [
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        "page_idx": 13
    },
    {
        "type": "equation",
        "text": "$$\nf (x) = r g (x) q _ {1} (x)\n$$",
        "text_format": "latex",
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        "page_idx": 13
    },
    {
        "type": "text",
        "text": "根据 Gauss 引理可知 $g(x)q_{1}(x)$ 仍为本原多项式，因此 $r = \\pm 1$ ，因此 $q(x) = \\pm q_{1}(x)$ 是本原多项式。",
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    },
    {
        "type": "text",
        "text": "(2) 直接展开得到 $f(x+1)=b_{n}x^{n}+\\cdots+b_{1}x+b_{0}$ ，其中 $b_{i}=\\sum_{k=i}^{n}\\binom{k}{i}a_{k}$ 。下面证明互素，假设 $(b_{0},b_{1},\\cdots,b_{n})=d$ ，则",
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    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} d \\mid b _ {n} \\Rightarrow d \\mid a _ {n} \\\\ d \\mid b _ {n - 1} \\Rightarrow d \\mid \\binom {n} {1} a _ {n} + a _ {n - 1} \\Rightarrow d \\mid a _ {n - 1} \\end{array} \\right.\n$$",
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    },
    {
        "type": "text",
        "text": "同理可以得到 $d \\mid a_{n-2}, \\cdots, d \\mid a_{1}, d \\mid a_{0}$ ，这与 $f(x)$ 是本原多项式矛盾。",
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    },
    {
        "type": "text",
        "text": "定理 1.18 (整系数多项式因式分解)",
        "text_level": 2,
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    },
    {
        "type": "text",
        "text": "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",
        "bbox": [
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        "page_idx": 13
    },
    {
        "type": "text",
        "text": "证明 设 $f(x) = g(x)h(x)$ ，这里 $g(x), h(x) \\in \\mathbb{Q}[x]$ ，则存在本原多项式 $f_{1}(x), h_{1}(x), g_{1}(x)$ 使得",
        "bbox": [
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        "page_idx": 13
    },
    {
        "type": "equation",
        "text": "$$\nf (x) = a f _ {1} (x), g (x) = r g _ {1} (x), h (x) = s h _ {1} (x), a \\in \\mathbb {Z}, r, s \\in \\mathbb {Q}\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 13
    },
    {
        "type": "text",
        "text": "从而 $af_{1}(x) = rsg_{1}(x)h_{1}(x)$ ，根据Gauss引理可知 $g_{1}(x)h_{1}(x)$ 是本原多项式，不妨设最高项系数均正，则 $f_{1}(x) = g_{1}(x)h_{1}(x), a = rs$ ，得到 $rs \\in \\mathbb{Z}$ ，于是",
        "bbox": [
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        ],
        "page_idx": 13
    },
    {
        "type": "equation",
        "text": "$$\nf (x) = (r s g _ {1} (x)) h _ {1} (x)\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 13
    },
    {
        "type": "text",
        "text": "其中 $rsg_{1}(x), h_{1}(x)$ 都是整系数多项式。",
        "bbox": [
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        ],
        "page_idx": 13
    },
    {
        "type": "text",
        "text": "练习 1.15 整系数多项式整数根 (1) 若 $\\alpha \\neq \\pm1$ 为整系数多项式 $f(x)$ 的整数根，证明： $\\frac{f(1)}{\\alpha-1}, \\frac{f(-1)}{\\alpha+1}$ 都是整数 (2) ZJU2017.1: $f(x)$ 为整系数多项式， $f(0), f(1)$ 为奇数，证明： $f(x)$ 无整数根",
        "bbox": [
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        "page_idx": 13
    },
    {
        "type": "text",
        "text": "(3)ZJU2019.4: $a_{1}, \\cdots, a_{n}$ 为互不相同的整数, $a_{1} \\cdots a_{n} + 1$ 不是某个整数的平方, 证明 $f(x) = (x + a_{1}) \\cdots (x + a_{n}) + 1$ 不能表示为 Q 上两个次数大于等于 1 的多项式的乘积。",
        "bbox": [
            87,
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        ],
        "page_idx": 13
    },
    {
        "type": "text",
        "text": "证明 (1) 由于 $\\alpha$ 是 $f(x)$ 的根，因此 $f(x) = (x - \\alpha)(b_{n-1}x^{n-1} + \\cdots + b_1x + b_0)$ ，这里 $b_0, \\cdots, b_{n-1}$ 也都是整数，因此",
        "bbox": [
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        ],
        "page_idx": 13
    },
    {
        "type": "equation",
        "text": "$$\nf (1) = (1 - \\alpha) (b _ {n - 1} + \\dots + b _ {1} + b _ {0}) \\Rightarrow \\frac {f (1)}{\\alpha - 1} = - b _ {0} - \\dots - b _ {n - 1} \\in \\mathbb {Z}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 13
    },
    {
        "type": "text",
        "text": "(2) 若 $f$ 有整数根，则 $f(x) = (x - a)f_0(x)$ ，这里 $f_0(x)$ 为整系数多项式，而 $f(0) = (-a)f_0(0), f(1) = (1 - a)f_0(1)$ ，而根据 $f(0), f(1)$ 为奇数得到 $-a, 1 - a$ 同时为奇数，矛盾。",
        "bbox": [
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        ],
        "page_idx": 13
    },
    {
        "type": "text",
        "text": "(3) 若 $f(x)$ 在 $Q[x]$ 上可约，则 $f(x)$ 在 $\\mathbb{Z}[x]$ 也可约，不妨设 $f(x) = g(x)h(x), g(x), h(x) \\in \\mathbb{Z}[x]$ ，则 $f(-a_i) = g(-a_i)h(-a_i) = 1$ ，这说明 $g(-a_i) = h(-a_i) = \\pm 1$ ，因此 $g(x) - h(x)$ 有 $n$ 个不同的根，而 $\\partial(g(x) - h(x)) < n$ ，因此 $g(x) \\equiv h(x)$ ，根据条件 $f(x) = g^2(x)$ ，常数项为整数的平方，这与条件不是平方矛盾。",
        "bbox": [
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        ],
        "page_idx": 13
    },
    {
        "type": "page_number",
        "text": "9",
        "bbox": [
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        ],
        "page_idx": 13
    },
    {
        "type": "header",
        "text": "1.2 特殊数域上的一元多项式理论",
        "bbox": [
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        ],
        "page_idx": 13
    },
    {
        "type": "text",
        "text": "1.2.4 一元有理多项式的根",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "定理 1.19 (有理多项式的根)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "$f(x)=a_{n}x^{n}+\\cdots+a_{1}x+a_{0}$ 是整系数多项式，若既约分数 $\\alpha=\\frac{q}{p}$ 是 $f(x)$ 的有理根，则必有 $q\\mid a_{0},p\\mid a_{n}$",
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "证明 假设 $\\frac{q}{p}$ 是解，则代入多项式得到：",
        "bbox": [
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            208
        ],
        "page_idx": 14
    },
    {
        "type": "equation",
        "text": "$$\na _ {n} \\frac {q ^ {n}}{p ^ {n}} + \\dots + a _ {1} \\frac {q}{p} + a _ {0} = 0\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "两边乘上 $p^n$ 并移项得到：",
        "bbox": [
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            280,
            258
        ],
        "page_idx": 14
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} a _ {n} q ^ {n} = p (- a _ {n - 1} q ^ {n - 1} - a _ {n - 2} p q ^ {n - 2} - \\dots - a _ {0} p ^ {n - 1}) \\\\ a _ {0} p ^ {n} = q (- a _ {n} q ^ {n - 1} - a _ {n - 1} p q ^ {n - 2} - \\dots - a _ {n - 1} p ^ {n - 1}) \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "因为 $(p,q) = 1$ ，因此 $(p,q^n) = 1,(q,p^n) = 1$ ，因此根据上面两个式子得到",
        "bbox": [
            90,
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        ],
        "page_idx": 14
    },
    {
        "type": "equation",
        "text": "$$\np \\mid a _ {0}, q \\mid a _ {n}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "练习 1.16 计算有理根 (1) 证明: $f(x) = x^{3} + 2x^{2} + x + 1$ 在有理数域上不可约",
        "bbox": [
            60,
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "(2) 求 $f(x) = 3x^{4} + 5x^{3} + x^{2} + 5x - 2$ 的有理根",
        "bbox": [
            122,
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            478,
            409
        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "证明 (1) 有理数域不可约即无有理根。而有理根只可能是 $\\pm 1, f(\\pm 1) \\neq 0$ ，因此不可约。",
        "bbox": [
            90,
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            426
        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "(2) 全部列出来验证，结果为 $\\frac{1}{3}, -2$",
        "bbox": [
            122,
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "定理 1.20 (Eisenstein 判别法)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "$f(x)=a_{n}x^{n}+\\cdots+a_{1}x+a_{0}$ 是整系数多项式，若存在素数 p 使得：(1) $p\\nmid a_{n}(2)p\\mid a_{n-1},\\cdots,a_{0}(3)p^{2}\\nmid a_{0}$ ，则 $f(x)$ 在有理数域上是不可约多项式。",
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "练习 1.17 使用 Eisenstein 判别法 证明在 Q 不可约",
        "bbox": [
            60,
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "(1) $f(x)=2x^{4}+3x^{3}-9x^{2}-3x+6$",
        "bbox": [
            122,
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            389,
            601
        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "(2) $f(x)=x^{n}+2x+2$",
        "bbox": [
            122,
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "证明 (1) 取 $p = 3$ 即可",
        "bbox": [
            90,
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "(2) 取 $p = 2$",
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "1.3 多元多项式",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "1.3.1 基本概念",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "定义 1.6 (字典排序)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "对于多元多项式 $p(x_{1}, x_{2}, \\cdots, x_{n})$ 的乘积项，先按 $x_{1}$ 次数由高到低排序，再根据 $x_{2}, x_{3}, \\cdots$ 次数排序，将字典排序得到的第一项称为首项。",
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "定理 1.21 (乘积保首项)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "$f(x_{1},\\cdots,x_{n}),g(x_{1},\\cdots,x_{n})$ 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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        ],
        "page_idx": 14
    },
    {
        "type": "header",
        "text": "1.3 多元多项式",
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "page_number",
        "text": "10",
        "bbox": [
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        ],
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "1.3.2 对称多项式",
        "text_level": 2,
        "bbox": [
            95,
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        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "定义 1.7 (对称多项式)",
        "text_level": 2,
        "bbox": [
            122,
            118,
            285,
            133
        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "若多元多项式 $f(x_{1},\\cdots,x_{n})$ 互换任意 $x_{i},x_{j}$ 后多项式保持不变，则称其为对称多项式",
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "定理 1.22 (对称多项式的性质)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "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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        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "定义 1.8 (初等对称多项式)",
        "text_level": 2,
        "bbox": [
            122,
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            319,
            253
        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "将形如 $\\sigma_{i}=\\sum_{k_{1}<k_{2}<\\cdots<k_{i}}x_{k_{1}}x_{k_{2}}\\cdots x_{k_{i}}$ 的对称多项式称为初等对称多项式：",
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} \\sigma_ {1} = x _ {1} + x _ {2} + \\dots + x _ {n} \\\\ \\sigma_ {2} = x _ {1} x _ {2} + x _ {1} x _ {3} + \\dots + x _ {n - 1} x _ {n} \\\\ \\quad \\vdots \\\\ \\sigma_ {n} = x _ {1} x _ {2} \\dots x _ {n} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "定理 1.23 (Vieta 定理)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "对于 $f(x)=x^{n}+a_{n-1}x^{n-1}+\\cdots+a_{0}$ ，假设其解为 $x_{1},\\cdots,x_{n}$ ，则",
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} a _ {n - 1} = - \\sigma_ {1} \\\\ a _ {n - 2} = \\sigma_ {2} \\\\ \\vdots \\\\ a _ {n - i} = (- 1) ^ {i} \\sigma_ {i} \\\\ \\vdots \\\\ a _ {0} = (- 1) ^ {n} \\sigma_ {n} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "定理 1.24 (复合定理)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "任意对称多项式 $f(x_{1},\\cdots,x_{n})$ 可唯一地写成初等多项式的多项式 $\\varphi(\\sigma_{1},\\cdots,\\sigma_{n})$",
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "证明 设 $f(x_{1},\\cdots,x_{n})$ 的首项为 $ax_{1}^{l_{1}}x_{2}^{l_{2}}\\cdots x_{n}^{l_{n}}$ ，且 $l_{1}\\geq l_{2}\\geq\\cdots l_{n}$ （根据对称多项式可交换性可知该性质），做多项式 $\\varphi_{1}=a\\sigma_{1}^{l_{1}-l_{2}}\\sigma_{2}^{l_{2}-l_{3}}\\cdots\\sigma_{n}^{l_{n}}$ ，得到 $\\varphi_{1}$ 的首项为 $ax_{1}^{l_{1}-l_{2}}(x_{1}x_{2})^{l_{2}-l_{3}}\\cdots(x_{1}\\cdots x_{n})^{l_{n}}=ax_{1}^{l_{1}}\\cdots x_{n}^{l_{n}}$ ，取 $f_{1}(x)=f(x)-\\varphi_{1}$ ，以此类推可以得到结果。",
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "练习 1.18 (1) 将 $f(x_{1}, x_{2}, x_{3}) = x_{1}^{3} + x_{2}^{3} + x_{3}^{3}$ 表达为 $\\sigma_{1}, \\sigma_{2}, \\sigma_{3}$ 的多项式",
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "解(1)根据首项可知 $l_{1} = 3, l_{2} = l_{3} = 0$ ， $\\varphi_{1} = \\sigma_{1}^{l_{1}}\\sigma_{2}^{l_{2}}\\sigma_{3}^{l_{3}} = (x_{1} + x_{2} + x_{3})^{3}$ ，得到",
        "bbox": [
            87,
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        ],
        "page_idx": 15
    },
    {
        "type": "equation",
        "text": "$$\nf _ {1} (x) = f (x) - \\varphi_ {1} = - 3 (x _ {1} ^ {2} x _ {2} + x _ {1} ^ {2} x _ {3} + \\dots) - 6 x _ {1} x _ {2} x _ {3}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "$f_{1}(x)$ 首项为 $-3x_{1}^{2}x_{2}$ ，因此 $\\varphi_{2}=-3\\sigma_{1}^{2-1}\\sigma_{2}=-3(x_{1}^{2}x_{2}+x_{2}^{2}x_{1}+\\cdots)-9x_{1}x_{2}x_{3}$ ，因此 $f_{2}(x_{1},x_{2},x_{3})=3\\sigma_{3}$ 。综上 $f(x_{1},x_{2},x_{3})=3\\sigma_{3}-3\\sigma_{1}\\sigma_{2}+\\sigma_{1}^{3}$",
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "header",
        "text": "1.3 多元多项式",
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "page_number",
        "text": "11",
        "bbox": [
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        ],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "1.3.3 判别式",
        "text_level": 2,
        "bbox": [
            95,
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "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。",
        "bbox": [
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "定义 1.9 (判别式)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "给定多项式 $f(x)=x^{n}+a_{n-1}x^{n-1}+\\cdots+a_{0}\\in\\mathbb{C}[x]$ ，设 $x_{1},\\cdots,x_{n}$ 是 $f(x)$ 的根，则定义判别式为",
        "bbox": [
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        ],
        "page_idx": 16
    },
    {
        "type": "equation",
        "text": "$$\nD (f) := \\prod_ {1 \\leq i <   j \\leq n} (x _ {i} - x _ {j}) ^ {n}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "显然 $f(x)$ 有重根当且仅当 $D(f) = 0$",
        "bbox": [
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "定理 1.25 (判别式的计算)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "由于 $D(f)$ 是 $x_{1},\\cdots,x_{n}$ 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 $a_{0},\\cdots,a_{n-1}$ 的多项式",
        "bbox": [
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "练习 1.19 判别式的计算 (1) 计算 $f(x)=x^{2}+px+q$ 的判别式",
        "bbox": [
            58,
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "解(1)设 $g(x_{1},x_{2}) = (x_{1} - x_{2})^{2}$ ，可以将 $g(x_{1},x_{2})$ 表达为初等对称多项式的多项式：",
        "bbox": [
            87,
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            382
        ],
        "page_idx": 16
    },
    {
        "type": "equation",
        "text": "$$\ng (x _ {1}, x _ {2}) = \\sigma_ {1} ^ {2} - 4 \\sigma_ {2}\n$$",
        "text_format": "latex",
        "bbox": [
            418,
            388,
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "而根据 Vieta 定理， $p = -\\sigma_{1}, q = \\sigma_{2}$ ，因此 $D(f) = p^{2} - 4q$",
        "bbox": [
            90,
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "1.3.4 结式",
        "text_level": 2,
        "bbox": [
            94,
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            194,
            464
        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "给定多项式 $f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\\cdots+a_{0}, g(x)=b_{m}x^{m}+\\cdots+b_{0}$ ，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",
        "bbox": [
            90,
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "定义 1.10 (结式)",
        "text_level": 2,
        "bbox": [
            122,
            520,
            245,
            535
        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "考虑两个多项式 $f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\\cdots+a_{0}, g(x)=b_{m}x^{m}+b_{m-1}x^{m-1}+\\cdots+b_{0}$ ，则定义两个矩阵：",
        "bbox": [
            114,
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            573
        ],
        "page_idx": 16
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c c c c c c c} a _ {n} & a _ {n - 1} & a _ {n - 2} & \\dots & a _ {0} & 0 & \\dots & 0 \\\\ 0 & a _ {n} & a _ {n - 1} & a _ {n - 2} & \\dots & a _ {0} & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\ddots & \\ddots & & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & a _ {n} & a _ {n - 1} & a _ {n - 2} & \\dots & a _ {0} \\end{array} \\right] _ {m \\times (n + m)}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 16
    },
    {
        "type": "equation",
        "text": "$$\nB = \\left[ \\begin{array}{c c c c c c c c} b _ {m} & b _ {m - 1} & b _ {m - 2} & \\dots & b _ {0} & 0 & \\dots & 0 \\\\ 0 & b _ {m} & b _ {m - 1} & b _ {m - 2} & \\dots & b _ {0} & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\ddots & \\ddots & & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & b _ {m} & b _ {m - 1} & b _ {m - 2} & \\dots & b _ {0} \\end{array} \\right] _ {n \\times (n + m)}\n$$",
        "text_format": "latex",
        "bbox": [
            248,
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "则定义它们的结式 $R(f,g) = \\left| \\begin{array}{l}A\\\\ B \\end{array} \\right|$",
        "bbox": [
            114,
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "定理 1.26 (结式判定定理)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "给定 $f(x)=a_{n}x^{n}+\\cdots+a_{0}, g(x)=b_{m}x^{m}+\\cdots+b_{0}, a_{n}, b_{m}\\neq0$ ，则 f, g 不互素的充要条件为 $R(f,g)=0$",
        "bbox": [
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "证明 设 $\\partial (f) = n, \\partial (g) = m$ ，首先 $f(x), g(x)$ 不互素当且仅当 $\\exists u(x), v(x)$ 满足 $\\partial (u) < \\partial (g), \\partial (v) < \\partial (f)$ ，使得",
        "bbox": [
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        ],
        "page_idx": 16
    },
    {
        "type": "page_number",
        "text": "12",
        "bbox": [
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        ],
        "page_idx": 16
    },
    {
        "type": "header",
        "text": "1.3 多元多项式",
        "bbox": [
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        ],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "#",
        "bbox": [
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        ],
        "page_idx": 17
    },
    {
        "type": "text",
        "text": "$u(x)f(x) = v(x)g(x)$ ，不妨设",
        "bbox": [
            90,
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        ],
        "page_idx": 17
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} u (x) = u _ {m - 1} x ^ {m - 1} + \\dots + u _ {0} \\\\ v (x) = v _ {n - 1} x ^ {n - 1} + \\dots + v _ {0} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 17
    },
    {
        "type": "text",
        "text": "将 $u(x)f(x) = v(x)g(x)$ 展开对比系数得到",
        "bbox": [
            90,
            153,
            403,
            168
        ],
        "page_idx": 17
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} a _ {n} u _ {m - 1} = b _ {m} v _ {n - 1} \\\\ a _ {n - 1} u _ {m - 1} + a _ {n} u _ {m - 2} = b _ {m - 1} v _ {n - 1} + b _ {m} v _ {n - 2} \\\\ \\vdots \\\\ a _ {0} u _ {0} = b _ {0} v _ {0} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 17
    },
    {
        "type": "text",
        "text": "将上述线性方程组看成 $u_{0}, \\cdots, u_{m-1}, v_{0}, \\cdots, v_{n-1}$ 的线性方程组，则系数矩阵的转置 $C^{T}$ 为结式定义中的矩阵：",
        "bbox": [
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        ],
        "page_idx": 17
    },
    {
        "type": "equation",
        "text": "$$\nC ^ {T} = \\binom{A}{- B}\n$$",
        "text_format": "latex",
        "bbox": [
            436,
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        ],
        "page_idx": 17
    },
    {
        "type": "text",
        "text": "因此有解当且仅当 $|C^T| = 0$ ，也等价于 $R(f,g) = 0$",
        "bbox": [
            90,
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            468,
            346
        ],
        "page_idx": 17
    },
    {
        "type": "text",
        "text": "定理 1.27 (结式计算公式)",
        "text_level": 2,
        "bbox": [
            122,
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            312,
            372
        ],
        "page_idx": 17
    },
    {
        "type": "text",
        "text": "给定前面的 $f(x), g(x)$ ， $\\alpha_{1}, \\cdots, \\alpha_{n}$ 和 $\\beta_{1}, \\cdots, \\beta_{m}$ 分别是 $f(x), g(x)$ 的复根，则",
        "bbox": [
            114,
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            695,
            392
        ],
        "page_idx": 17
    },
    {
        "type": "equation",
        "text": "$$\nR (f, g) = a _ {n} ^ {m} b _ {m} ^ {n} \\prod_ {i = 1} ^ {n} \\prod_ {j = 1} ^ {m} (\\alpha_ {i} - \\beta_ {i})\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 17
    },
    {
        "type": "text",
        "text": "定理 1.28 (判别式计算公式)",
        "text_level": 2,
        "bbox": [
            124,
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            474
        ],
        "page_idx": 17
    },
    {
        "type": "text",
        "text": "$\\mathbb{C}[x]$ 中多项式 $f(x) = a_{n}x^{n} + a_{n - 1}x^{n - 1} + \\dots +a_{0}, a_{n}\\neq 0$ ，则 $f(x)$ 的判别式",
        "bbox": [
            114,
            480,
            694,
            495
        ],
        "page_idx": 17
    },
    {
        "type": "equation",
        "text": "$$\nD (f) = (- 1) ^ {\\frac {n (n - 1)}{2}} a _ {0} ^ {- (2 n - 1)} R (f, f ^ {\\prime})\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 17
    },
    {
        "type": "header",
        "text": "1.3 多元多项式",
        "bbox": [
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        ],
        "page_idx": 17
    },
    {
        "type": "page_number",
        "text": "13",
        "bbox": [
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        ],
        "page_idx": 17
    },
    {
        "type": "image",
        "img_path": "images/a3107efcc333af95e29aadb82eccf9ae18205fa0aa42d2c811430f586e035cee.jpg",
        "image_caption": [],
        "image_footnote": [],
        "content": "",
        "bbox": [
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        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "第2章 矩阵",
        "text_level": 2,
        "bbox": [
            425,
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            111
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "2.1 矩阵基础",
        "text_level": 2,
        "bbox": [
            94,
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            240,
            165
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "2.1.1 矩阵概念与运算",
        "text_level": 2,
        "bbox": [
            94,
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            292,
            200
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "略",
        "bbox": [
            124,
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            227
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "2.1.2 矩阵的逆",
        "text_level": 2,
        "bbox": [
            94,
            249,
            233,
            266
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "定义 2.1 (矩阵的逆)",
        "text_level": 2,
        "bbox": [
            121,
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            270,
            297
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "若 $AB = BA = E$ ，则称 $B$ 为 $A$ 的逆，记 $A^{-1}$",
        "bbox": [
            115,
            302,
            460,
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        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "定理 2.1 (矩阵逆的基本性质)",
        "text_level": 2,
        "bbox": [
            122,
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            337,
            356
        ],
        "page_idx": 18
    },
    {
        "type": "equation",
        "text": "$$\n(1) (k A) ^ {- 1} = k ^ {- 1} A ^ {- 1} (2) (A B) ^ {- 1} = B ^ {- 1} A ^ {- 1} (3) \\left(A ^ {T}\\right) ^ {- 1} = \\left(A ^ {- 1}\\right) ^ {T} (4) | A ^ {- 1} | = | A | ^ {- 1}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "证明 (2) $(AB)^{-1}AB = E$ ，因此 $(AB)^{-1} = B^{-1}A^{-1}$",
        "bbox": [
            90,
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            412
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "(3) $(A^{-1})^T A^T = (AA^{-1})^T = E$",
        "bbox": [
            122,
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            431
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "(4) $|A^{-1}A|=|A^{-1}|\\cdot|A|=1$ ，因此 $|A^{-1}|=|A|^{-1}$",
        "bbox": [
            122,
            434,
            482,
            450
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "定理 2.2 (可逆充要条件)",
        "text_level": 2,
        "bbox": [
            122,
            461,
            304,
            475
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "矩阵 $A$ 可以的充要条件如下：",
        "bbox": [
            115,
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            495
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "\\- $|A| \\neq 0$",
        "bbox": [
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            216,
            514
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "\\- $A$ 可表示为初等矩阵的积",
        "bbox": [
            139,
            518,
            347,
            532
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "定理 2.3 (利用初等行变换求逆)",
        "text_level": 2,
        "bbox": [
            122,
            556,
            354,
            571
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "若 $A$ 可逆，则利用初等行变换可以得到 $(A,E)\\rightarrow (E,A^{-1})$ 。特别地， $(A,B)\\rightarrow (E,A^{-1}B)$",
        "bbox": [
            114,
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            784,
            592
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "笔记 同样可以通过初等列变换得到： $\\begin{pmatrix} A \\\\ B \\end{pmatrix} \\rightarrow \\begin{pmatrix} E \\\\ BA^{-1} \\end{pmatrix}$",
        "bbox": [
            87,
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        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "练习 2.1 利用初等变换求逆 (1) 计算下面矩阵的逆:",
        "bbox": [
            58,
            653,
            463,
            668
        ],
        "page_idx": 18
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c c} 1 & 1 & - 1 \\\\ 2 & 1 & 0 \\\\ 1 & - 1 & 0 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            436,
            671,
            559,
            730
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "解结果为 $\\frac{1}{3}\\left[ \\begin{array}{ccc}0 & 1 & 1\\\\ 0 & 1 & -2\\\\ -3 & 2 & -1 \\end{array} \\right]$",
        "bbox": [
            87,
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            793
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "练习 2.2 逆矩阵理论练习 (1) A 为方阵，若 $A^{k}=O$ ，证明 E-A 可逆，且 $(E-A)^{-1}=E+A+A^{2}+\\cdots+A^{k-1}$",
        "bbox": [
            58,
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            814
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "(2) $A, B$ 为 $n$ 阶方阵，若 $A + B = AB$ ，证明： $A - E$ 可逆，并求逆矩阵",
        "bbox": [
            122,
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        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "(3) 若 $A + B$ 可逆，证明 $A^{-1} + B^{-1}$ 也可逆，并求其逆",
        "bbox": [
            122,
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            852
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "解(1)直接做乘法验证:",
        "bbox": [
            89,
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            265,
            870
        ],
        "page_idx": 18
    },
    {
        "type": "equation",
        "text": "$$\n(E - A) \\left(E + A + A ^ {2} + \\dots + A ^ {k - 1}\\right) = E + A + A ^ {2} + \\dots + A ^ {k - 1} - A - A ^ {2} - \\dots - A ^ {k} = E - A ^ {k} = E\n$$",
        "text_format": "latex",
        "bbox": [
            127,
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            892
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "(2) 由于 $A + B = AB$ ，则 $(A - E)(B - E) = AB - (A + B) + E = E$ ，因此 $(A - E)^{-1} = B - E$",
        "bbox": [
            122,
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            917
        ],
        "page_idx": 18
    },
    {
        "type": "text",
        "text": "(3) 根据 $B(A^{-1} + B^{-1})A = (A + B)$ 得到",
        "bbox": [
            122,
            84,
            433,
            99
        ],
        "page_idx": 19
    },
    {
        "type": "equation",
        "text": "$$\n(A ^ {- 1} + B ^ {- 1}) B = A ^ {- 1} (A + B) \\Rightarrow (A ^ {- 1} + B ^ {- 1}) B (A + B) ^ {- 1} = A ^ {- 1}\n$$",
        "text_format": "latex",
        "bbox": [
            252,
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        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "因此 $(A^{-1} + B^{-1})^{-1} = B(A + B)^{-1}A$",
        "bbox": [
            92,
            129,
            374,
            146
        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "定义 2.2 ((代数) 余子式)",
        "text_level": 2,
        "bbox": [
            122,
            173,
            302,
            187
        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "$A = (a_{ij})$ 为 $n\\times n$ 矩阵，划去 $i$ 行 $j$ 列的行列式称为余子式 $M_{ij}$ ，代数余子式为 $(-1)^{i + j}M_{ij}$",
        "bbox": [
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        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "定义 2.3 (伴随矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            234,
            268,
            248
        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "$A = (a_{ij})_n$ ， $A_{ij}$ 是 $a_{ij}$ 的代数余子式，则其伴随为 (注意行列颠倒!!):",
        "bbox": [
            114,
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        ],
        "page_idx": 19
    },
    {
        "type": "equation",
        "text": "$$\nA ^ {*} = \\left[ \\begin{array}{c c c c} A _ {1 1} & A _ {2 1} & \\dots & A _ {n 1} \\\\ A _ {1 2} & A _ {2 2} & \\dots & A _ {n 2} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ A _ {1 n} & A _ {2 n} & \\dots & A _ {n n} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            376,
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        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "定理 2.4 (利用伴随求逆)",
        "text_level": 2,
        "bbox": [
            122,
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            302,
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        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "$A$ 及其伴随矩阵 $A^{*}$ 满足 $AA^{*} = |A|E$ ，若 $A$ 可逆，则 $A^{-1} = \\frac{1}{|A|} A^{*}$",
        "bbox": [
            114,
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        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "证明 考虑 $B = AA^{*}$ ，则 $B_{ij} = \\sum_{k=1}^{n} a_{ik} A_{kj}$ ，根据行列式的展开式可知",
        "bbox": [
            90,
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            479
        ],
        "page_idx": 19
    },
    {
        "type": "equation",
        "text": "$$\nB _ {i j} = \\left\\{ \\begin{array}{l l} | A |, & i = j \\\\ 0, & i \\neq j \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "因此 $AA^{*} = |A|E$",
        "bbox": [
            92,
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        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "2.1.3 伴随矩阵的性质",
        "text_level": 2,
        "bbox": [
            95,
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            292,
            586
        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "定理 2.5 (伴随运算的性质)",
        "text_level": 2,
        "bbox": [
            122,
            602,
            319,
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        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "$A$ 为 $n$ 阶方阵，则",
        "bbox": [
            115,
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            636
        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "1. $(A^{*})^{T} = (A^{T})^{*}$",
        "bbox": [
            137,
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            655
        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "2. $(A^{*})^{-1} = (A^{-1})^{*}$",
        "bbox": [
            136,
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            285,
            674
        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "3. $(AB)^{*} = B^{*}A^{*}$",
        "bbox": [
            136,
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            272,
            692
        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "证明 (1) 设 $A = (a_{ij})$ ，则",
        "bbox": [
            92,
            712,
            284,
            728
        ],
        "page_idx": 19
    },
    {
        "type": "equation",
        "text": "$$\n(A ^ {*}) ^ {T} = \\left[ \\begin{array}{c c c c} A _ {1 1} & A _ {1 2} & \\dots & A _ {1 n} \\\\ A _ {2 1} & A _ {2 2} & \\dots & A _ {2 n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ A _ {n 1} & A _ {n 2} & \\dots & A _ {n n} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            364,
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            807
        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "同理设 $(A^T)^* = (B_{ij})$ ，则 $B_{ij} = A_{ji}^T$ ，因此行列式大小相同。",
        "bbox": [
            90,
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            537,
            833
        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "(2) $A^{*} = |A|A^{-1}$ ，则 $(A^{*})^{-1} = \\frac{1}{|A|} A$ 因此",
        "bbox": [
            122,
            835,
            431,
            860
        ],
        "page_idx": 19
    },
    {
        "type": "equation",
        "text": "$$\n(A ^ {- 1}) ^ {*} = | A ^ {- 1} | A = \\frac {1}{| A |} A\n$$",
        "text_format": "latex",
        "bbox": [
            401,
            863,
            594,
            895
        ],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "因此结论成立",
        "bbox": [
            90,
            898,
            194,
            912
        ],
        "page_idx": 19
    },
    {
        "type": "page_number",
        "text": "15",
        "bbox": [
            490,
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            965
        ],
        "page_idx": 19
    },
    {
        "type": "header",
        "text": "2.1 矩阵基础",
        "bbox": [
            805,
            55,
            905,
            68
        ],
        "page_idx": 19
    },
    {
        "type": "equation",
        "text": "$$\n(3) (A B) ^ {*} = | A B | (A B) ^ {- 1} = | B | (B) ^ {- 1} | A | A ^ {- 1} = B ^ {*} A ^ {*}\n$$",
        "text_format": "latex",
        "bbox": [
            122,
            83,
            517,
            99
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "定理 2.6 (伴随矩阵的行列式)",
        "bbox": [
            122,
            111,
            336,
            126
        ],
        "page_idx": 20
    },
    {
        "type": "equation",
        "text": "$$\n| A ^ {*} | = | A | ^ {n - 1}\n$$",
        "text_format": "latex",
        "bbox": [
            115,
            131,
            218,
            147
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "证明 $|A^{*}| = \\left| |A| A^{-1} \\right| = |A|^{n} \\cdot |A^{-1}| = |A|^{n - 1}$",
        "bbox": [
            90,
            166,
            428,
            183
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "定理 2.7 (伴随矩阵的秩)",
        "text_level": 2,
        "bbox": [
            122,
            193,
            304,
            209
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "$A^{*}$ 为 $A$ 的伴随矩阵，则",
        "bbox": [
            115,
            214,
            300,
            229
        ],
        "page_idx": 20
    },
    {
        "type": "equation",
        "text": "$$\nr (A ^ {*}) = \\left\\{ \\begin{array}{l l} n, & r (A) = n \\\\ 1, & r (A) = n - 1 \\\\ 0, & r (A) <   n - 1 \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
            391,
            228,
            603,
            294
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "证明 $r(A) = n$ 时， $|A^{*}| = |A|^{n - 1} \\neq 0$ ，因此满秩。 $r(A) = n - 1$ 时，由于 $AA^{*} = |A|E = O$ ，根据秩不等式：",
        "bbox": [
            90,
            315,
            892,
            332
        ],
        "page_idx": 20
    },
    {
        "type": "equation",
        "text": "$$\nr (A A ^ {*}) \\geq r (A) + r (A ^ {*}) - n = r (A ^ {*}) - 1 \\Rightarrow r (A ^ {*}) \\leq 1\n$$",
        "text_format": "latex",
        "bbox": [
            297,
            338,
            699,
            355
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "而由于 $r(A) = n - 1$ ，其至少有一个 $n - 1$ 阶代数余子式不为0，因此 $r(A^{*})\\geq 1$ ，综上 $r(A^{*}) = 1$ 。其余显然",
        "bbox": [
            90,
            362,
            892,
            378
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "定理 2.8 (伴随矩阵的特征值)",
        "text_level": 2,
        "bbox": [
            121,
            388,
            337,
            403
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "矩阵 A 的特征值为 $\\lambda_{1},\\cdots,\\lambda_{n}$ ，则 $A^{*}$ 的特征值 $\\lambda_{j}^{*}=\\prod_{i\\neq j}\\lambda_{i}$",
        "bbox": [
            114,
            410,
            552,
            441
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "证明 考虑 $A = P^{-1}JP$ ，由于 $J$ 是上三角阵，根据初等变换法， $J^{-1}$ 是上三角阵， $J^{*} = |J|J^{-1}$ 也是上三角阵，而 $J$ 对角元素的代数余子式为 $\\prod_{i\\neq j}\\lambda_i$ ，因此 $J^{*}$ 的对角元为 $\\prod_{i\\neq j}\\lambda_i$ ，代数余子式在初等变换下不变 (位置可能变化，",
        "bbox": [
            90,
            460,
            907,
            511
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "但值整体相同)，因此结论成立。",
        "bbox": [
            92,
            508,
            327,
            521
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "2.1.4 广义逆矩阵",
        "text_level": 2,
        "bbox": [
            94,
            545,
            253,
            562
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "定义 2.4 (广义逆)",
        "text_level": 2,
        "bbox": [
            122,
            577,
            253,
            592
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "$A_{m\\times n}$ 是数域 $\\mathbb{P}$ 上的矩阵，若存在 $G_{n\\times m}$ 使得",
        "bbox": [
            115,
            598,
            458,
            613
        ],
        "page_idx": 20
    },
    {
        "type": "equation",
        "text": "$$\nA G A = A\n$$",
        "text_format": "latex",
        "bbox": [
            458,
            621,
            537,
            634
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "则称 $G$ 是 $A$ 的广义逆矩阵。",
        "bbox": [
            115,
            643,
            321,
            657
        ],
        "page_idx": 20
    },
    {
        "type": "image",
        "img_path": "images/ae1bf8e0a1c16b0d253249710c607fe306d87bcf8e814a8530bbe1480fa8f84e.jpg",
        "image_caption": [],
        "image_footnote": [],
        "content": "",
        "bbox": [
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        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "笔记 广义逆矩阵一般不唯一。",
        "text_level": 2,
        "bbox": [
            87,
            677,
            307,
            690
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "定理 2.9 (广义逆的表达式)",
        "text_level": 2,
        "bbox": [
            122,
            702,
            321,
            717
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "设 $A_{m\\times n}$ 的秩为 $r$ ，设 $A = P\\left( \\begin{array}{cc}E_r & O\\\\ O & O \\end{array} \\right)Q$ ，其中 $P_{m\\times m},Q_{n\\times n}$ 为可逆矩阵，则 $A$ 全部的广义逆矩阵为",
        "bbox": [
            114,
            724,
            880,
            759
        ],
        "page_idx": 20
    },
    {
        "type": "equation",
        "text": "$$\nG = Q ^ {- 1} \\left( \\begin{array}{c c} E _ {r} & C \\\\ D & F \\end{array} \\right) P ^ {- 1}\n$$",
        "text_format": "latex",
        "bbox": [
            396,
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            802
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "这里 C, D, F 为任意 $r \\times (m - r)$ , $(n - r) \\times r$ , $(m - r) \\times (n - r)$ 阶矩阵。",
        "bbox": [
            114,
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            640,
            822
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "证明 (1) 验证 $G$ 是广义逆: 直接乘起来就行, 得到 $AGA = A$",
        "bbox": [
            90,
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            858
        ],
        "page_idx": 20
    },
    {
        "type": "header",
        "text": "2.1 矩阵基础",
        "bbox": [
            805,
            55,
            905,
            68
        ],
        "page_idx": 20
    },
    {
        "type": "page_number",
        "text": "16",
        "bbox": [
            490,
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            509,
            965
        ],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "(2) 验证广义逆都是 $G$ ：设 $G = Q^{-1}\\left( \\begin{array}{cc}B & C\\\\ D & F \\end{array} \\right)P^{-1}$ ，乘法得到",
        "bbox": [
            122,
            82,
            601,
            120
        ],
        "page_idx": 21
    },
    {
        "type": "equation",
        "text": "$$\nA G A = P \\left( \\begin{array}{c c} B & O \\\\ O & O \\end{array} \\right) Q\n$$",
        "text_format": "latex",
        "bbox": [
            405,
            126,
            591,
            164
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "因此得到 $B = E_{r}$",
        "bbox": [
            92,
            168,
            221,
            181
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "定理 2.10 (非方阵线性方程组解的表达式)",
        "text_level": 2,
        "bbox": [
            122,
            193,
            428,
            208
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "$A_{m\\times n}$ ， $G$ 是 $A$ 的广义逆，则 $AX = b$ 的通解可表示为",
        "bbox": [
            115,
            212,
            519,
            227
        ],
        "page_idx": 21
    },
    {
        "type": "equation",
        "text": "$$\nX = G b + (E _ {n} - G A) Y\n$$",
        "text_format": "latex",
        "bbox": [
            408,
            236,
            586,
            250
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "这里 $Y$ 是任一 $n$ 维向量",
        "bbox": [
            114,
            259,
            295,
            273
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "证明 (1) 证明是解：设 $Z$ 是 $AZ = b$ 的解，则此时有 $b = AZ = AGAZ = AGb$ 。将 $X$ 表达式代入 $AX$ 验证：",
        "bbox": [
            90,
            293,
            885,
            307
        ],
        "page_idx": 21
    },
    {
        "type": "equation",
        "text": "$$\nA X = A G b + (A - A G A) Y = b + O = b\n$$",
        "text_format": "latex",
        "bbox": [
            347,
            316,
            650,
            330
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "(2) 证明解都有该形式: 设 $X_{0}$ 是 $AX = b$ 的解, 则",
        "bbox": [
            122,
            338,
            500,
            353
        ],
        "page_idx": 21
    },
    {
        "type": "equation",
        "text": "$$\nX _ {0} = G b + X _ {0} - G b = G b + X _ {0} - G A X _ {0} = G b + (E _ {n} - G A) X _ {0}\n$$",
        "text_format": "latex",
        "bbox": [
            260,
            362,
            734,
            376
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "2.1.5 Moore-Penrose 广义逆",
        "text_level": 2,
        "bbox": [
            94,
            404,
            347,
            420
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "定义 2.5 (共轭转置)",
        "text_level": 2,
        "bbox": [
            122,
            436,
            270,
            450
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "$A$ 是 $\\mathbb{C}$ 上 $n\\times m$ 矩阵，则定义 $A^{H} = (\\overline{A})^{T}$",
        "bbox": [
            114,
            455,
            433,
            472
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "定义 2.6 (Moore-Penrose 广义逆)",
        "text_level": 2,
        "bbox": [
            122,
            498,
            366,
            512
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "$A$ 是 $\\mathbb{C}$ 上 $m \\times n$ 阶矩阵，若存在 $\\mathbb{C}$ 上 $n \\times m$ 阶矩阵 $G$ 满足：",
        "bbox": [
            115,
            518,
            566,
            532
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "1. AGA = A",
        "bbox": [
            136,
            536,
            236,
            550
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "2. $GAG = G$",
        "bbox": [
            136,
            555,
            238,
            568
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "3. $(AG)^{H} = AG$",
        "bbox": [
            136,
            571,
            263,
            588
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "4. $(GA)^{H} = GA$",
        "bbox": [
            136,
            590,
            263,
            606
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "则称 $G$ 为 $A$ 的M-P广义逆",
        "bbox": [
            115,
            609,
            317,
            624
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "定理 2.11 (M-P 广义逆的存在唯一性)",
        "text_level": 2,
        "bbox": [
            122,
            649,
            398,
            663
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "$A_{m\\infty n}$ 的 M-P 广义逆总是存在且唯一的，且若 $\\beta_{1},\\cdots,\\beta_{r}$ 是 A 列向量的极大线性无关组，记 $B=\\left[\\beta_{1},\\cdots,\\beta_{r}\\right]$ ，分解 A=BC，则 M-P 广义逆的具体表达式为：",
        "bbox": [
            114,
            668,
            882,
            702
        ],
        "page_idx": 21
    },
    {
        "type": "equation",
        "text": "$$\nA ^ {+} = C ^ {H} (C C ^ {H}) ^ {- 1} (B ^ {H} B) ^ {- 1} B ^ {H}\n$$",
        "text_format": "latex",
        "bbox": [
            376,
            708,
            621,
            725
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "证明 存在性：设 A 的秩为 r，取 $\\beta_{1},\\cdots,\\beta_{r}$ 为 A 列向量的极大线性无关组，此时",
        "bbox": [
            90,
            746,
            694,
            760
        ],
        "page_idx": 21
    },
    {
        "type": "equation",
        "text": "$$\nA = (\\alpha_ {1}, \\dots , \\alpha_ {n}) = (\\beta_ {1}, \\dots , \\beta_ {r}) \\left( \\begin{array}{c c c} c _ {1 1} & \\dots & c _ {1 n} \\\\ \\vdots & & \\vdots \\\\ c _ {r 1} & \\dots & c _ {r n} \\end{array} \\right) = B C\n$$",
        "text_format": "latex",
        "bbox": [
            278,
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            824
        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "此时 $r(BC)=r\\leq r(C)\\leq\\min\\{r,n\\}\\leq r$ ，得到 $r(C)=r$ 行满秩，B 显然列满秩，因此 $B^{H}B,CC^{H}$ 是 r 阶可逆矩阵 (根据 $r(A^{H}A),r(AA^{H})$ 的性质)。令",
        "bbox": [
            90,
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            864
        ],
        "page_idx": 21
    },
    {
        "type": "equation",
        "text": "$$\nG = C ^ {H} (C C ^ {H}) ^ {- 1} (B ^ {H} B) ^ {- 1} B ^ {H}\n$$",
        "text_format": "latex",
        "bbox": [
            379,
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        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "可验证 $G$ 满足上述所有条件",
        "bbox": [
            92,
            895,
            302,
            910
        ],
        "page_idx": 21
    },
    {
        "type": "header",
        "text": "2.1 矩阵基础",
        "bbox": [
            805,
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            68
        ],
        "page_idx": 21
    },
    {
        "type": "page_number",
        "text": "17",
        "bbox": [
            490,
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            507,
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        ],
        "page_idx": 21
    },
    {
        "type": "text",
        "text": "唯一性：设 $X$ 是另一个广义逆，则进行下面步骤，本质是用 $AXA = A$ 消去 $X$",
        "bbox": [
            122,
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            707,
            99
        ],
        "page_idx": 22
    },
    {
        "type": "equation",
        "text": "$$\n\\begin{array}{l} \\boldsymbol {X} = \\boldsymbol {X A X} = \\boldsymbol {X} (\\boldsymbol {A X}) ^ {\\mathrm{H}} = \\boldsymbol {X X} ^ {\\mathrm{H}} \\boldsymbol {A} ^ {\\mathrm{H}} = \\boldsymbol {X X} ^ {\\mathrm{H}} (\\boldsymbol {A G A}) ^ {\\mathrm{H}} \\\\ \\quad = \\boldsymbol {X X} ^ {\\mathrm{H}} \\boldsymbol {A} ^ {\\mathrm{H}} (\\boldsymbol {A G}) ^ {\\mathrm{H}} = \\boldsymbol {X} (\\boldsymbol {A X}) ^ {\\mathrm{H}} (\\boldsymbol {A G}) ^ {\\mathrm{H}} = \\boldsymbol {X A X A G} \\\\ \\quad = \\boldsymbol {X A G} = \\boldsymbol {X} (\\boldsymbol {A G A}) \\boldsymbol {G} = (\\boldsymbol {X A}) ^ {\\mathrm{H}} (\\boldsymbol {G A}) ^ {\\mathrm{H}} \\boldsymbol {G} \\\\ \\quad = (\\boldsymbol {G A X A}) ^ {\\mathrm{H}} \\boldsymbol {G} = (\\boldsymbol {G A}) ^ {\\mathrm{H}} \\boldsymbol {G} = (\\boldsymbol {G A}) \\boldsymbol {G} \\\\ \\quad = \\boldsymbol {G}. \\end{array}\n$$",
        "text_format": "latex",
        "bbox": [
            289,
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            210
        ],
        "page_idx": 22
    },
    {
        "type": "text",
        "text": "练习2.3计算M-P逆 $(1)A = \\left( \\begin{array}{ccc}1 & 0 & -1\\\\ 1 & 2 & 0\\\\ 0 & 2 & 1 \\end{array} \\right)$ 的M-P逆",
        "bbox": [
            58,
            229,
            505,
            288
        ],
        "page_idx": 22
    },
    {
        "type": "text",
        "text": "解(1) 先进行分解，得到",
        "bbox": [
            90,
            288,
            275,
            303
        ],
        "page_idx": 22
    },
    {
        "type": "equation",
        "text": "$$\nB = \\left( \\begin{array}{c c} 1 & 0 \\\\ 1 & 1 \\\\ 0 & 1 \\end{array} \\right), C = \\left( \\begin{array}{c c c} 1 & 0 & - 1 \\\\ 0 & 2 & 1 \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            357,
            300,
            636,
            359
        ],
        "page_idx": 22
    },
    {
        "type": "text",
        "text": "最终结果得到",
        "bbox": [
            92,
            368,
            196,
            382
        ],
        "page_idx": 22
    },
    {
        "type": "equation",
        "text": "$$\nA ^ {+} = \\frac {1}{9} \\left( \\begin{array}{c c c} 3 & 2 & - 1 \\\\ 0 & 2 & 2 \\\\ - 3 & - 1 & 2 \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            391,
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            439
        ],
        "page_idx": 22
    },
    {
        "type": "text",
        "text": "定理 2.12 (最小二乘解的表达式)",
        "bbox": [
            122,
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            361,
            464
        ],
        "page_idx": 22
    },
    {
        "type": "text",
        "text": "$AX = b$ 最小二乘解为 $X = A^{+}b + (E_{n} - A^{+}A)Y$ ，其中 $A^{+}b$ 是唯一的极小最小二乘解。",
        "bbox": [
            115,
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            486
        ],
        "page_idx": 22
    },
    {
        "type": "text",
        "text": "2.2 分块矩阵",
        "text_level": 2,
        "bbox": [
            94,
            530,
            240,
            549
        ],
        "page_idx": 22
    },
    {
        "type": "text",
        "text": "2.2.1 常见矩阵拆分技巧",
        "text_level": 2,
        "bbox": [
            95,
            569,
            312,
            586
        ],
        "page_idx": 22
    },
    {
        "type": "text",
        "text": "定理 2.13 (点态分解)",
        "text_level": 2,
        "bbox": [
            122,
            602,
            278,
            615
        ],
        "page_idx": 22
    },
    {
        "type": "text",
        "text": "$A$ 是 $n \\times n$ 矩阵，则",
        "bbox": [
            117,
            621,
            270,
            634
        ],
        "page_idx": 22
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c c c} a _ {1} b _ {1} & a _ {1} b _ {2} & \\dots & a _ {1} b _ {n} \\\\ a _ {2} b _ {1} & a _ {2} b _ {2} & \\dots & a _ {2} b _ {n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a _ {n} b _ {1} & a _ {n} b _ {2} & \\dots & a _ {n} b _ {n} \\end{array} \\right] \\Rightarrow A = \\left( \\begin{array}{c} a _ {1} \\\\ a _ {2} \\\\ \\vdots \\\\ a _ {n} \\end{array} \\right) \\left( \\begin{array}{c c c c} b _ {1} & b _ {2} & \\dots & b _ {n} \\end{array} \\right)\\tag{❤️}\n$$",
        "text_format": "latex",
        "bbox": [
            233,
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        ],
        "page_idx": 22
    },
    {
        "type": "text",
        "text": "定理 2.14 ( $A^{T}A$ 的和式表达)",
        "text_level": 2,
        "bbox": [
            122,
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            334,
            760
        ],
        "page_idx": 22
    },
    {
        "type": "text",
        "text": "$A$ 为 $n\\times m$ 矩阵，将 $A$ 做下面的行划分，则 $A^T A = \\sum_{i = 1}^{n}\\alpha_i^T\\alpha_i$",
        "bbox": [
            115,
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            802
        ],
        "page_idx": 22
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left( \\begin{array}{c} \\alpha_ {1} \\\\ \\alpha_ {2} \\\\ \\vdots \\\\ \\alpha_ {s} \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            445,
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        ],
        "page_idx": 22
    },
    {
        "type": "header",
        "text": "2.2 分块矩阵",
        "bbox": [
            805,
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        ],
        "page_idx": 22
    },
    {
        "type": "page_number",
        "text": "18",
        "bbox": [
            490,
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        ],
        "page_idx": 22
    },
    {
        "type": "text",
        "text": "#",
        "bbox": [
            882,
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            900,
            726
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "2.2.2 分块初等变换",
        "text_level": 2,
        "bbox": [
            94,
            85,
            273,
            102
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "定义 2.7 (分块矩阵初等变换)",
        "text_level": 2,
        "bbox": [
            122,
            117,
            337,
            133
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "分块矩阵初等变换分为行变换和列变换：",
        "bbox": [
            115,
            137,
            413,
            152
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "\\- 行变换: (1) 左乘矩阵 $P$ , 加到另一行 (2) 互换两行位置 (3) 将可逆矩阵左乘至一行",
        "bbox": [
            139,
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            172
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "\\- 列变换：(1) 右乘矩阵 $P$ ，加到另一列 (2) 互换两列位置 (3) 将可逆矩阵右乘至一列",
        "bbox": [
            139,
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        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "定义 2.8 (分块初等矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            214,
            304,
            229
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "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：",
        "bbox": [
            115,
            235,
            752,
            250
        ],
        "page_idx": 23
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c} E & 0 \\\\ P & E \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            436,
            253,
            554,
            291
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "定理 2.15 (初等变换的分块初等矩阵表达)",
        "text_level": 2,
        "bbox": [
            122,
            317,
            428,
            332
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "分块初等矩阵左乘一个矩阵即做行变换，右乘即做列变换：",
        "bbox": [
            115,
            337,
            547,
            351
        ],
        "page_idx": 23
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} E & 0 \\\\ P & E \\end{array} \\right] \\left[ \\begin{array}{c c} A _ {1} & A _ {2} \\\\ A _ {3} & A _ {4} \\end{array} \\right] = \\left[ \\begin{array}{c c} A _ {1} & A _ {2} \\\\ P A _ {1} + A _ {3} & P A _ {2} + A _ {4} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            299,
            354,
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            394
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "2.2.3 分块打洞技巧总结",
        "text_level": 2,
        "bbox": [
            94,
            432,
            312,
            450
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "定理 2.16 (分块打洞)",
        "text_level": 2,
        "bbox": [
            122,
            464,
            278,
            480
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "对于分块矩阵 $\\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix}$ 可通过分块初等变换转换为：",
        "bbox": [
            114,
            485,
            519,
            525
        ],
        "page_idx": 23
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} A & B \\\\ O & D - C A ^ {- 1} B \\end{array} \\right] \\quad \\left[ \\begin{array}{c c} A - B D ^ {- 1} C & O \\\\ O & D \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            327,
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            567
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "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。",
        "bbox": [
            115,
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            581,
            586
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "推论 2.1 (两种特殊形式)",
        "text_level": 2,
        "bbox": [
            122,
            609,
            302,
            625
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "对称形式和归纳形式：",
        "bbox": [
            115,
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            278,
            643
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "\\- 对称形式:",
        "bbox": [
            139,
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            669
        ],
        "page_idx": 23
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} A & B \\\\ B ^ {T} & D \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A & O \\\\ O & D - B ^ {T} A ^ {- 1} B \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            245,
            644,
            529,
            680
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "归纳形式：",
        "bbox": [
            139,
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            236,
            707
        ],
        "page_idx": 23
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} A & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A & \\alpha \\\\ 0 & a _ {n n} - \\beta^ {T} A ^ {- 1} \\alpha \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            245,
            682,
            542,
            720
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "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",
        "bbox": [
            58,
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        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "(2) $A$ 是 $n$ 阶实方阵, $A$ 所有顺序主子式大于 0 , 且非主对角元素为负, 证明: $A^{-1}$ 的元素均为正数",
        "bbox": [
            122,
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            810
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "证明 $(1)n = 1$ 显然成立。设 $n - 1$ 时成立，考虑如下分块，由于 $A$ 顺序主子式非零，因此 $A_{1}$ 顺序主子式非零，存在 $n - 1$ 阶下三角矩阵 $B_{1}$ 满足 $B_{1}A_{1}$ 为上三角阵，因此做对应分块初等变换",
        "bbox": [
            90,
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        ],
        "page_idx": 23
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] \\Rightarrow \\left[ \\begin{array}{c c} B _ {1} & \\\\ & 1 \\end{array} \\right] \\cdot \\left[ \\begin{array}{c c} E _ {n - 1} & O \\\\ - \\beta^ {T} A _ {1} ^ {- 1} & 1 \\end{array} \\right] \\cdot \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] = \\left[ \\begin{array}{c c} B _ {1} A _ {1} & B _ {1} \\alpha \\\\ O & a _ {n n} - \\beta^ {T} A _ {1} ^ {- 1} \\alpha \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            132,
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            889
        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "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。",
        "bbox": [
            90,
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            544,
            907
        ],
        "page_idx": 23
    },
    {
        "type": "header",
        "text": "2.2 分块矩阵",
        "bbox": [
            805,
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        ],
        "page_idx": 23
    },
    {
        "type": "page_number",
        "text": "19",
        "bbox": [
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        ],
        "page_idx": 23
    },
    {
        "type": "text",
        "text": "#",
        "bbox": [
            884,
            639,
            900,
            651
        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "#",
        "bbox": [
            884,
            807,
            900,
            820
        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "❤️",
        "bbox": [
            882,
            450,
            900,
            463
        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "(2) $n = 1$ 显然。设 $n - 1$ 时成立，对 $A$ 做下述分块，则 $A_{1}^{-1}$ 元素都是正的",
        "bbox": [
            122,
            85,
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            99
        ],
        "page_idx": 24
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] \\Rightarrow \\left[ \\begin{array}{c c} E _ {n - 1} & O \\\\ - \\beta^ {T} A _ {1} ^ {- 1} & 1 \\end{array} \\right] \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {n - 1} & - A _ {1} ^ {- 1} \\alpha \\\\ O & 1 \\end{array} \\right] = \\left[ \\begin{array}{c c} A _ {1} & O \\\\ O & a _ {n n} - \\beta^ {T} A _ {1} ^ {- 1} \\alpha \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            122,
            103,
            873,
            143
        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "根据 $|A| > 0, |A_1| > 0$ 得到 $a_{nn} - \\beta^T A_1^{-1}\\alpha > 0$ 。记上述等式为 $BAC = D$ ，则 $A^{-1} = CD^{-1}B$ ，根据矩阵乘法乘开即可获得结论。",
        "bbox": [
            90,
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            180
        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "2.2.4 分块矩阵行列式",
        "text_level": 2,
        "bbox": [
            94,
            202,
            292,
            219
        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "定理 2.17 (分块对角矩阵的行列式)",
        "text_level": 2,
        "bbox": [
            122,
            235,
            379,
            250
        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "根据 Laplace 多行展开定理可以得到:",
        "bbox": [
            115,
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            269
        ],
        "page_idx": 24
    },
    {
        "type": "equation",
        "text": "$$\n\\left| \\begin{array}{c c} A & 0 \\\\ B & C \\end{array} \\right| = | A | | B |,\n$$",
        "text_format": "latex",
        "bbox": [
            243,
            294,
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            332
        ],
        "page_idx": 24
    },
    {
        "type": "equation",
        "text": "$$\n\\left| \\begin{array}{c c c c} A _ {1 1} & 0 & \\dots & 0 \\\\ A _ {2 1} & A _ {2 2} & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ A _ {n 1} & A _ {n 2} & \\dots & A _ {n n} \\end{array} \\right| = | A _ {1 1} | | A _ {2 2} | \\dots | A _ {n n} |\\tag{❤️}\n$$",
        "text_format": "latex",
        "bbox": [
            416,
            274,
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            354
        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "定理 2.18 (一般分块矩阵的行列式)",
        "text_level": 2,
        "bbox": [
            122,
            380,
            379,
            394
        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "设 $A, D$ 分别为 $r, s$ 阶矩阵， $A, D$ 可逆，则",
        "bbox": [
            115,
            400,
            433,
            414
        ],
        "page_idx": 24
    },
    {
        "type": "equation",
        "text": "$$\n\\left| \\begin{array}{c c} A & B \\\\ C & D \\end{array} \\right| = | D | | A - B D ^ {- 1} C | = | A | | D - C A ^ {- 1} B |\n$$",
        "text_format": "latex",
        "bbox": [
            317,
            418,
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            456
        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "证明 (1) 如果 $D$ 可逆，则把 $B$ 消掉 (左乘 $BD^{-1}$ )，取行列式即可",
        "bbox": [
            90,
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            492
        ],
        "page_idx": 24
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} A & B \\\\ C & D \\end{array} \\right] \\Rightarrow \\left[ \\begin{array}{c c} A - B D ^ {- 1} C & 0 \\\\ C & D \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            364,
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        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "定理 2.19 (行列式计算技巧 1: 配凑分块矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            550,
            458,
            565
        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "矩阵 A 可分解为 $A = J - u^{T}u$ ，要计算 $|A|$ ，则可以构造分块矩阵：",
        "bbox": [
            115,
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            610,
            585
        ],
        "page_idx": 24
    },
    {
        "type": "equation",
        "text": "$$\n\\left| \\begin{array}{c c} J & u \\\\ u ^ {T} & 1 \\end{array} \\right| = | 1 | \\cdot | J - u ^ {T} u | = | J | \\cdot | 1 - u J ^ {- 1} u ^ {T} | \\Rightarrow | A | = | J | \\cdot | 1 - u J ^ {- 1} u ^ {T} |\n$$",
        "text_format": "latex",
        "bbox": [
            223,
            590,
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            627
        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "注意这里 $u$ 是列向量",
        "bbox": [
            115,
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            272,
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        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "2.2.5 分块矩阵求逆",
        "text_level": 2,
        "bbox": [
            95,
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            273,
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        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "一般分块矩阵求逆都先转换为分块对角矩阵，具体如下。",
        "bbox": [
            124,
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        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "定理 2.20 (上三角分块矩阵求逆)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "当下面的 $A_{1}, A_{2}$ 都是方阵，则 $A$ 可逆当且仅当 $A_{1}, A_{2}$ 都可逆，且逆矩阵如下：",
        "bbox": [
            115,
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            695,
            771
        ],
        "page_idx": 24
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c} A _ {1} & A _ {3} \\\\ O & A _ {2} \\end{array} \\right] \\Rightarrow A ^ {- 1} = \\left[ \\begin{array}{c c} A _ {1} ^ {- 1} & - A _ {1} ^ {- 1} A _ {3} A _ {2} ^ {- 1} \\\\ 0 & A _ {2} ^ {- 1} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            302,
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        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "证明 根据分块矩阵初等变换可知",
        "bbox": [
            90,
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            848
        ],
        "page_idx": 24
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} E & - A _ {2} ^ {- 1} A _ {3} \\\\ O & E \\end{array} \\right] \\left[ \\begin{array}{c c} A _ {1} & A _ {3} \\\\ O & A _ {2} \\end{array} \\right] = \\left[ \\begin{array}{c c} A _ {1} & O \\\\ O & A _ {2} \\end{array} \\right] \\Rightarrow \\left[ \\begin{array}{c c} E & - A _ {2} ^ {- 1} A _ {3} \\\\ O & E \\end{array} \\right] ^ {- 1} \\left[ \\begin{array}{c c} A _ {1} & A _ {3} \\\\ O & A _ {2} \\end{array} \\right] ^ {- 1} = \\left[ \\begin{array}{c c} A _ {1} ^ {- 1} & O \\\\ O & A _ {2} ^ {- 1} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            114,
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        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "因此可以得到结论。",
        "bbox": [
            92,
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        ],
        "page_idx": 24
    },
    {
        "type": "page_number",
        "text": "20",
        "bbox": [
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        ],
        "page_idx": 24
    },
    {
        "type": "header",
        "text": "2.2 分块矩阵",
        "bbox": [
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        ],
        "page_idx": 24
    },
    {
        "type": "text",
        "text": "定理 2.21 (一般分块矩阵可逆条件)",
        "text_level": 2,
        "bbox": [
            122,
            86,
            379,
            102
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "分块矩阵 $H = \\left( \\begin{array}{ll}A & B\\\\ C & D \\end{array} \\right)$ 可逆当且仅当 $A - BD^{-1}C$ 和 $D - CA^{-1}B$ 都可逆",
        "bbox": [
            114,
            107,
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            146
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "练习2.5分块矩阵求逆 $A, B$ 时 $n, m$ 阶可逆方阵，求 $\\left[ \\begin{array}{cc}O & A\\\\ B & O \\end{array} \\right]$ 的逆矩阵",
        "bbox": [
            58,
            177,
            636,
            216
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "解 先做交换行的变换: $\\left[ \\begin{array}{ll} O & A \\\\ B & O \\end{array} \\right]\\left[ \\begin{array}{ll} O & E \\\\ E & O \\end{array} \\right] = \\left[ \\begin{array}{l l} A & \\\\ & B \\end{array} \\right]$ , 两侧求逆即可得到:",
        "bbox": [
            87,
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            256
        ],
        "page_idx": 25
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} O & A \\\\ B & O \\end{array} \\right] ^ {- 1} = \\left[ \\begin{array}{c c} A ^ {- 1} & O \\\\ O & B ^ {- 1} \\end{array} \\right] \\left[ \\begin{array}{c c} O & E \\\\ E & O \\end{array} \\right] = \\left[ \\begin{array}{c c} O & B ^ {- 1} \\\\ A ^ {- 1} & O \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            268,
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        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "2.3 矩阵的秩",
        "text_level": 2,
        "bbox": [
            94,
            323,
            240,
            342
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "2.3.1 秩的概念",
        "text_level": 2,
        "bbox": [
            94,
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            231,
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        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "定义 2.9 (矩阵的秩)",
        "text_level": 2,
        "bbox": [
            122,
            394,
            270,
            409
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "矩阵的秩等于:",
        "bbox": [
            115,
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            228,
            429
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "\\- 矩阵行列向量的秩",
        "bbox": [
            139,
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            297,
            447
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "\\- 最大非零子式的阶数",
        "bbox": [
            139,
            451,
            312,
            466
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "2.3.2 初等变换标准型与矩阵分解",
        "text_level": 2,
        "bbox": [
            94,
            504,
            394,
            521
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "定理 2.22 (初等变换标准型)",
        "text_level": 2,
        "bbox": [
            122,
            536,
            327,
            551
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "设 $A$ 是秩为 $r$ 的 $s \\times n$ 矩阵，则存在 $s$ 阶可逆矩阵 $P$ 和 $n$ 阶可逆矩阵 $Q$ 使得",
        "bbox": [
            115,
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            684,
            571
        ],
        "page_idx": 25
    },
    {
        "type": "equation",
        "text": "$$\nA = P \\left[ \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right] Q\n$$",
        "text_format": "latex",
        "bbox": [
            418,
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            613
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "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": [
            89,
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        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "定理 2.23 (行列满秩的标准型)",
        "text_level": 2,
        "bbox": [
            122,
            659,
            346,
            675
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "$A$ 是 $m\\times n$ 矩阵， $A$ 列满秩当且仅当存在 $m$ 阶可逆矩阵 $P$ 使得 $A = P\\left[ \\begin{array}{c}E_{n}\\\\ O \\end{array} \\right]$ 。 $A$ 行满秩当且仅当存在 $n$ 阶可逆矩阵 $Q$ ，使得 $A = \\left[ \\begin{array}{ll}E_m & O \\end{array} \\right]Q$",
        "bbox": [
            112,
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            885,
            747
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "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": [
            89,
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            905,
            796
        ],
        "page_idx": 25
    },
    {
        "type": "header",
        "text": "2.3 矩阵的秩",
        "bbox": [
            805,
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            68
        ],
        "page_idx": 25
    },
    {
        "type": "page_number",
        "text": "21",
        "bbox": [
            489,
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            965
        ],
        "page_idx": 25
    },
    {
        "type": "text",
        "text": "2.3.3 利用初等变换标准型进行矩阵分解",
        "text_level": 2,
        "bbox": [
            92,
            85,
            453,
            102
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "定理 2.24 (矩阵分解)",
        "text_level": 2,
        "bbox": [
            122,
            118,
            278,
            133
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "$s\\times n$ 矩阵 $\\left[ \\begin{array}{cc}E_r & O\\\\ O & O \\end{array} \\right]$ 可分解为",
        "bbox": [
            114,
            140,
            357,
            175
        ],
        "page_idx": 26
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right] = \\left[ \\begin{array}{c} E _ {r} \\\\ O \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {r} & O \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            371,
            180,
            625,
            221
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "若为方阵时，可以分解为：",
        "bbox": [
            115,
            224,
            312,
            237
        ],
        "page_idx": 26
    },
    {
        "type": "equation",
        "text": "$$\n{\\left[ \\begin{array}{l l} E _ {r} & O \\\\ O & O \\end{array} \\right]} = {\\left[ \\begin{array}{l l} E _ {r} & O \\\\ O & O \\end{array} \\right]} {\\left[ \\begin{array}{l l} E _ {r} & O \\\\ O & O \\end{array} \\right]}\n$$",
        "text_format": "latex",
        "bbox": [
            356,
            235,
            640,
            277
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "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": [
            58,
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            311
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "(2) 设 $B_{1}, B_{2}$ 为数域 $P$ 上 $s \\times n$ 列满秩矩阵，证明：存在 $P$ 上 $s$ 阶可逆矩阵 $C$ ，使得 $B_{2} = CB_{1}$ (即 $B_{1}, B_{2}$ 只需要行变换就可以相互转换)",
        "bbox": [
            90,
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            905,
            348
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "(3) $C_{1}, C_{2}$ 为P上 $s \\times n$ 行满秩矩阵，证明： $C_{1}, C_{2}$ 可以仅通过列变换相互转换。",
        "bbox": [
            122,
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            367
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "证明 (1) 使用等价标准型:",
        "bbox": [
            90,
            370,
            287,
            385
        ],
        "page_idx": 26
    },
    {
        "type": "equation",
        "text": "$$\nA = P \\left[ \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right] Q = P \\left[ \\begin{array}{c} E _ {r} \\\\ O \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {r} & O \\end{array} \\right] Q\n$$",
        "text_format": "latex",
        "bbox": [
            319,
            382,
            675,
            423
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "(2) 根据列满秩初等标准型， $\\exists P$ 使得 $PB_{1} = \\left[ \\begin{array}{c} E_{n} \\\\ O \\end{array} \\right]$ ，同理存在 $Q$ 使得 $QB_{2} = \\left[ \\begin{array}{c} E_{n} \\\\ O \\end{array} \\right]$ ，结论成立。",
        "bbox": [
            122,
            430,
            863,
            470
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "(3) 和 (2) 一样",
        "bbox": [
            122,
            468,
            231,
            483
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "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,
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            677,
            520
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "证明 (1) 利用等价标准型得到:",
        "bbox": [
            90,
            524,
            319,
            538
        ],
        "page_idx": 26
    },
    {
        "type": "equation",
        "text": "$$\nA = P \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) Q = P Q Q ^ {- 1} \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) Q\n$$",
        "text_format": "latex",
        "bbox": [
            317,
            542,
            678,
            580
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "PQ 为可逆矩阵，后面为幂等",
        "bbox": [
            90,
            583,
            310,
            599
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "2.3.4 秩不等式",
        "text_level": 2,
        "bbox": [
            94,
            621,
            233,
            638
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "引理 2.1 (乘可逆矩阵不改变秩)",
        "text_level": 2,
        "bbox": [
            122,
            653,
            354,
            668
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "$P, Q$ 为可逆矩阵，则 $r(A) = r(PAQ)$",
        "bbox": [
            115,
            673,
            393,
            689
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "证明 相当于做了一系列初等变换，因此不改变秩",
        "bbox": [
            90,
            708,
            457,
            722
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "练习 2.8 非方阵情况 A, B 分别为 $m \\times n, n \\times s$ 矩阵，证明：",
        "bbox": [
            58,
            746,
            527,
            760
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "(1) 若 $r(A) = n$ ，则 $r(AB) = r(B)$",
        "bbox": [
            122,
            765,
            381,
            780
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "(2) 若 $r(B) = n$ ，则 $r(AB) = r(A)$",
        "bbox": [
            122,
            783,
            381,
            797
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "证明 (1) 存在可逆矩阵 $P$ 使得 $PA = \\left[ \\begin{array}{c} A_{1} \\\\ O \\end{array} \\right]$ ，这里 $A_{1}$ 为可逆矩阵，因此",
        "bbox": [
            87,
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            638,
            835
        ],
        "page_idx": 26
    },
    {
        "type": "equation",
        "text": "$$\nr (A B) = r (\\left[ \\begin{array}{c} A _ {1} \\\\ O \\end{array} \\right] B) = r \\binom{A _ {1} B}{O} = r (A _ {1} B) = r (B)\n$$",
        "text_format": "latex",
        "bbox": [
            289,
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            879
        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "(2) 同理 (转置即可)",
        "bbox": [
            122,
            882,
            270,
            897
        ],
        "page_idx": 26
    },
    {
        "type": "header",
        "text": "2.3 矩阵的秩",
        "bbox": [
            805,
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            68
        ],
        "page_idx": 26
    },
    {
        "type": "page_number",
        "text": "22",
        "bbox": [
            489,
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        ],
        "page_idx": 26
    },
    {
        "type": "text",
        "text": "引理 2.2 (分块矩阵的秩)",
        "text_level": 2,
        "bbox": [
            124,
            86,
            304,
            102
        ],
        "page_idx": 27
    },
    {
        "type": "equation",
        "text": "$$\n\\begin{array}{l} \\bullet r \\left( \\begin{array}{c c} A & O \\\\ O & B \\end{array} \\right) = r (A) + r (B) \\\\ \\bullet r \\left( \\begin{array}{c c} A & O \\\\ C & B \\end{array} \\right) \\geq r (A) + r (B) \\end{array}\n$$",
        "text_format": "latex",
        "bbox": [
            139,
            107,
            373,
            184
        ],
        "page_idx": 27
    },
    {
        "type": "text",
        "text": "证明 (1) 根据初等变换等价标准型有 $P_{1}AQ_{1} = \\left( \\begin{array}{cc}E_{r_1} & O\\\\ O & O \\end{array} \\right),P_{2}BQ_{2} = \\left( \\begin{array}{cc}E_{r_2} & O\\\\ O & O \\end{array} \\right)$ ，因此",
        "bbox": [
            90,
            197,
            776,
            235
        ],
        "page_idx": 27
    },
    {
        "type": "equation",
        "text": "$$\n\\left( \\begin{array}{c c} P _ {1} & O \\\\ O & P _ {2} \\end{array} \\right) \\left( \\begin{array}{c c} A & O \\\\ O & B \\end{array} \\right) \\left( \\begin{array}{c c} Q _ {1} & O \\\\ O & Q _ {2} \\end{array} \\right) = \\left( \\begin{array}{c c c c} E _ {r _ {1}} & O & O & O \\\\ O & O & O & O \\\\ O & O & E _ {r _ {2}} & O \\\\ O & O & O & O \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 27
    },
    {
        "type": "text",
        "text": "(2) 根据初等变换等价标准型有 $P_{1}AQ_{1} = \\left( \\begin{array}{cc}E_{r_1} & O\\\\ O & O \\end{array} \\right), P_{2}BQ_{2} = \\left( \\begin{array}{cc}E_{r_2} & O\\\\ O & O \\end{array} \\right)$ ，因此",
        "bbox": [
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        ],
        "page_idx": 27
    },
    {
        "type": "equation",
        "text": "$$\n\\left( \\begin{array}{c c} P _ {1} & O \\\\ O & P _ {2} \\end{array} \\right) \\left( \\begin{array}{c c} A & O \\\\ C & B \\end{array} \\right) \\left( \\begin{array}{c c} Q _ {1} & O \\\\ O & Q _ {2} \\end{array} \\right) = \\left( \\begin{array}{c c c c} E _ {r _ {1}} & O & O & O \\\\ O & O & O & O \\\\ C _ {1 1} & C _ {1 2} & E _ {r _ {2}} & O \\\\ C _ {2 1} & C _ {2 2} & O & O \\end{array} \\right) \\to \\left( \\begin{array}{c c c c} E _ {r _ {1}} & O & O & O \\\\ O & O & O & O \\\\ O & O & E _ {r _ {2}} & O \\\\ O & C _ {2 2} & O & O \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "text",
        "text": "定理 2.25 (矩阵秩基本不等式)",
        "text_level": 2,
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "text",
        "text": "• $r(AB) \\leq \\min\\{r(A), r(B)\\}$",
        "bbox": [
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        ],
        "page_idx": 27
    },
    {
        "type": "text",
        "text": "• $r(A+B) \\leq r(A) + r(B)$",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "text",
        "text": "\\- Sylvester 不等式: $A, B$ 是 $n$ 阶方阵, 则 $r(A) + r(B) \\leq r(AB) + n$",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "text",
        "text": "\\- Frobenius 不等式: $A, B, C$ 是 $n$ 阶方阵, 则 $r(ABC) \\geq r(AB) + r(BC) - r(B)$",
        "bbox": [
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        ],
        "page_idx": 27
    },
    {
        "type": "text",
        "text": "\\- $A, B$ 为 $s \\times n, n \\times m$ 阶矩阵， $AB = O$ ，则 $r(A) + r(B) \\neq n$",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "text",
        "text": "\\- $A$ 是 $m \\times n$ 的实矩阵，则 $r(A^T A) = r(A)$",
        "bbox": [
            139,
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        ],
        "page_idx": 27
    },
    {
        "type": "text",
        "text": "证明 (1) 将 $B$ 写为行向量形式: $B = \\left( \\begin{array}{c} \\beta_{1} \\\\ \\beta_{2} \\\\ \\vdots \\\\ \\beta_{n} \\end{array} \\right)$ , 则",
        "bbox": [
            90,
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        ],
        "page_idx": 27
    },
    {
        "type": "equation",
        "text": "$$\nA B = \\left( \\begin{array}{c c c} a _ {1 1} & \\dots & a _ {1 n} \\\\ \\vdots & & \\vdots \\\\ a _ {n 1} & \\dots & a _ {n n} \\end{array} \\right) \\left( \\begin{array}{c} \\beta_ {1} \\\\ \\vdots \\\\ \\beta_ {n} \\end{array} \\right) = \\left( \\begin{array}{c} a _ {1 1} \\beta_ {1} + \\dots + a _ {1 n} \\beta_ {n} \\\\ \\vdots \\\\ a _ {n 1} \\beta_ {1} + \\dots + a _ {n n} \\beta_ {n} \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "text",
        "text": "因此 AB 每个行向量都可被 B 行向量线性表出，从而 $r(AB) \\leq r(B)$ ，同理可知 $r(AB) \\leq r(A)$",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "text",
        "text": "(2) 用列向量角度， $A + B$ 的列向量显然可以被 $A, B$ 的列向量线性表出，从而结论成立。",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "text",
        "text": "(3) 是 (4) 的推论",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "text",
        "text": "(4)Frobenius: 即构造 $r(ABC) + r(B) \\geq r(AB) + r(BC)$",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} A B C & \\\\ & B \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A B C & O \\\\ B C & B \\end{array} \\right] \\to \\left[ \\begin{array}{c c} O & - A B \\\\ B C & B \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "text",
        "text": "根据分块矩阵秩不等式可知",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "text",
        "text": "练习 2.9 秩不等式的推广 (1) A, B 为 n 阶方阵，则 $r(A - ABA) = r(A) + r(E - BA) - n$",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "header",
        "text": "2.3 矩阵的秩",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "page_number",
        "text": "23",
        "bbox": [
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        "page_idx": 27
    },
    {
        "type": "text",
        "text": "(2) $A, B$ 是 $n$ 阶方阵， $AB = BA = O, r(A^2) = r(A)$ ，证明： $r(A + B) = r(A) + r(B)$ 证明 (1) 等号需要构造单位矩阵，即等式中的 $n$ 用 $E_n$ 代替：",
        "bbox": [
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        "page_idx": 28
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} A & \\\\ & E _ {n} - B A \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A & O \\\\ B A & E _ {n} - B A \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A & A \\\\ B A & E _ {n} \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A - A B A & O \\\\ O & E _ {n} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 28
    },
    {
        "type": "text",
        "text": "从右往左推非常难，因此一般从无 $E_{n}$ 的一边推另一边(后面的练习也会体现这一点)",
        "bbox": [
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        "page_idx": 28
    },
    {
        "type": "text",
        "text": "(2) 注意从复杂的一侧 $A + B$ 开始，否则 $A, B$ 不知道怎么搞，",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "equation",
        "text": "$$\n\\left( \\begin{array}{c c} A + B & O \\\\ O & O \\end{array} \\right) \\to \\left( \\begin{array}{c c} A + B & O \\\\ A ^ {2} + A B & O \\end{array} \\right) = \\left( \\begin{array}{c c} A + B & O \\\\ A ^ {2} & O \\end{array} \\right) \\to \\left( \\begin{array}{c c} A + B & A ^ {2} \\\\ A ^ {2} & A ^ {3} \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "text",
        "text": "由于 $r(A) = r(A^2)$ ，因此矩阵方程 $A^2 X = A$ 有解，故 $\\exists P$ 使得 $A^2 P = A$ ，故第二列可右乘 $P$ 消去第一列：",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "equation",
        "text": "$$\n\\left( \\begin{array}{c c} A + B & A ^ {2} \\\\ A ^ {2} & A ^ {3} \\end{array} \\right) \\to \\left( \\begin{array}{c c} B & A ^ {2} \\\\ O & A ^ {3} \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "text",
        "text": "因此根据分块矩阵的秩得到 $r(A + B) \\geq r(A^3) + r(B) = r(A) + r(B)$ ，而不等式另一侧显然.",
        "bbox": [
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        "page_idx": 28
    },
    {
        "type": "text",
        "text": "练习 2.10 用秩不等式研究矩阵性质 (1) 证明零矩阵: $A_{m \\times n}, B_{n \\times s}, r(B) = n$ ，证明：若 AB = O，则 A = O",
        "bbox": [
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        "page_idx": 28
    },
    {
        "type": "text",
        "text": "(2) 判定可逆: $n$ 为奇数, $A, B$ 为 $n$ 阶矩阵, $A^2 = O$ , 证明: $AB - BA$ 不可逆证明 (1) 由于 $r(A) + r(B) \\leq r(AB) + n$ , 而 $r(B) = n$ , 得到 $r(A) = 0$ , 即 $A = O$",
        "bbox": [
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        "page_idx": 28
    },
    {
        "type": "text",
        "text": "(2) $n=2k+1$ ，根据 $A^{2}=O$ 得到 $2r(A)\\leq2k+1+0$ ，从而 $r(A)\\leq k$ ，此时",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "equation",
        "text": "$$\nr (A B - B A) \\leq r (A B) + r (B A) \\leq r (A) + r (A) \\leq 2 k\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 28
    },
    {
        "type": "text",
        "text": "因此不可逆",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "text",
        "text": "2.3.5 空间分解法研究矩阵秩",
        "text_level": 2,
        "bbox": [
            92,
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        ],
        "page_idx": 28
    },
    {
        "type": "text",
        "text": "定理 2.26 (多项式矩阵的秩关系)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "text",
        "text": "$f(x) = f_{1}(x)f_{2}(x)$ 是多项式， $(f_{1}(x),f_{2}(x)) = 1$ ，则 $f(A) = O$ 当且仅当 $r(f_{1}(A)) + r(f_{2}(A)) = n$ 。常见的推论如下：",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "text",
        "text": "\\- $A^2 = A$ 当且仅当 $r(A - E) + r(A) = n$",
        "bbox": [
            139,
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        ],
        "page_idx": 28
    },
    {
        "type": "text",
        "text": "\\- $A^2 = E$ 当且仅当 $r(A - E) + r(A + E) = n$",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "text",
        "text": "证明 根据 $(f_{1}(x), f_{2}(x)) = 1$ ，有 $u(x)f_{1}(x) + v(x)f_{2}(x) = 1$ ，因此 $u(A)f_{1}(A) + v(A)f_{2}(A) = E_{n}$ ，根据分块矩阵初等变换：",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} f _ {1} (A) & \\\\ & f _ {2} (A) \\end{array} \\right] \\to \\left[ \\begin{array}{c c} f _ {1} (A) & f _ {1} (A) u (A) + v (A) f _ {2} (A) \\\\ O & f _ {2} (A) \\end{array} \\right] = \\left[ \\begin{array}{c c} f _ {1} (A) & E _ {n} \\\\ O & f _ {2} (A) \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 28
    },
    {
        "type": "text",
        "text": "再做初等变换：",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} f _ {1} (A) & E _ {n} \\\\ O & f _ {2} (A) \\end{array} \\right] \\to \\left[ \\begin{array}{c c} O & E _ {n} \\\\ - f _ {2} (A) f _ {1} (A) & f _ {2} (A) \\end{array} \\right] \\to \\left[ \\begin{array}{c c} O & E _ {n} \\\\ - f (A) & O \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 28
    },
    {
        "type": "text",
        "text": "因此得到 $r(f_{1}(A))+r(f_{2}(A))=n+r(f(A))$",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "text",
        "text": "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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        ],
        "page_idx": 28
    },
    {
        "type": "text",
        "text": "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： $r(A) + r(B) \\geq r(AB) + r(A + B)$ 证明 设 AX = 0, BX = 0 的解空间分别为 $V_{1}, V_{2}$ ，设 ABX = BAX = 0 的解空间为 $W_{1}$ ， $(A + B)X = 0$ 的解空间为 $W_{2}$ 。显然 $V_{1} \\subseteq W_{1}, V_{2} \\subseteq W_{1}$ ，从而 $V_{1} + V_{2} \\subseteq W_{1}$ ，显然 $V_{1} \\cap V_{2} \\subseteq W_{2}$ ，根据维数公式：",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "equation",
        "text": "$$\nd i m V _ {1} + d i m V _ {2} = d i m (V _ {1} + V _ {2}) + d i m (V _ {1} \\cap V _ {2}) \\leq d i m W _ {1} + d i m W _ {2}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "page_number",
        "text": "24",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "header",
        "text": "2.3 矩阵的秩",
        "bbox": [
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        ],
        "page_idx": 28
    },
    {
        "type": "text",
        "text": "因此推出：",
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "equation",
        "text": "$$\n(n - r (A)) + (n - r (B)) \\leq (n - r (A B)) + (n - r (A + B))\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "化简得到结论。",
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "image",
        "img_path": "images/ce3a524e4cfc4924fcc83503ac164d9bab0392e84942ca5de982e270059db9a0.jpg",
        "image_caption": [],
        "image_footnote": [],
        "content": "",
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "笔记 这种 $r(A), r(B), r(A + B), r(AB)$ 相关问题绝大多数从线性空间角度都能迎刃而解",
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "2.4 矩阵的迹",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "定理 2.27 (矩阵迹的性质)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "$A, B$ 为 $n$ 阶方阵，则",
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "equation",
        "text": "$$\n\\begin{array}{l} \\bullet \\operatorname{tr} (A) = \\operatorname{tr} (k A) \\\\ \\bullet \\operatorname{tr} (A + B) = \\operatorname{tr} (A) + \\operatorname{tr} (B) \\\\ \\bullet \\operatorname{tr} (A B) = \\operatorname{tr} (B A) \\end{array}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "若 $A$ 为实方阵，则 $A = O$ 当且仅当 $\\operatorname{tr}(A^T A) = 0$",
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "证明 显然",
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "2.5 几种技巧矩阵",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "2.5.1 初等矩阵",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "定义 2.10 (初等矩阵)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "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 $P(i,j), P(i(c)), P(i,j(k))$ 表示 $i,j$ 行互换， $i$ 行乘 $c$ ， $P_{ij} = k$ 的矩阵",
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "定理 2.28 (初等矩阵乘法)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "$AP$ 相当于做一次列初等变换， $PA$ 相当于做一次行初等变换，",
        "bbox": [
            115,
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            590
        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "2.5.2 基本矩阵：仅一个元素为1的矩阵",
        "text_level": 2,
        "bbox": [
            95,
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            453,
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "定义 2.11 (基本矩阵)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "基本矩阵 $E_{ij}$ 表示 $n \\times n$ 矩阵，但是仅有 $(i,j)$ 元素为1，其余元素为0的矩阵。",
        "bbox": [
            115,
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            696
        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "定理 2.29 (基本矩阵乘法)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "基本矩阵乘法有以下性质：",
        "bbox": [
            115,
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            755
        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "\\- $AE_{ij}$ 将 $A$ 的第 $i$ 列移动到第 $j$ 列，",
        "bbox": [
            139,
            760,
            411,
            775
        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "\\- $E_{ij}A$ 将 $A$ 的第 $j$ 行移动到第 $i$ 行。",
        "bbox": [
            139,
            778,
            413,
            793
        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "\\- 基础矩阵分解： $E_{ij} = e_i e_j^T$",
        "bbox": [
            141,
            797,
            357,
            813
        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "\\- 基本矩阵相乘： $E_{ik}E_{kj}=E_{ij}$ ， $k\\neq l$ 时 $E_{ik}E_{lj}=0$",
        "bbox": [
            139,
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        ],
        "page_idx": 29
    },
    {
        "type": "header",
        "text": "2.4 矩阵的迹",
        "bbox": [
            805,
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        ],
        "page_idx": 29
    },
    {
        "type": "page_number",
        "text": "25",
        "bbox": [
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        ],
        "page_idx": 29
    },
    {
        "type": "text",
        "text": "练习2.12用基础矩阵研究幂次(1)设 $A = \\left( \\begin{array}{cccc}0 & 1 & 0 & \\dots & 0\\\\ 0 & 0 & 1 & \\dots & 0\\\\ \\vdots & \\vdots & \\vdots & & \\vdots \\\\ 0 & 0 & 0 & \\dots & 1\\\\ 1 & 0 & 0 & \\dots & 0 \\end{array} \\right)$ ，证明： $A^k = \\left( \\begin{array}{cc}O & E_{n - k}\\\\ E_k & O \\end{array} \\right)$ 。",
        "bbox": [
            58,
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        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "(2) $A$ 是 $n$ 阶上三角阵且主对角上元素均为0，证明 $A^n = O$",
        "bbox": [
            122,
            181,
            561,
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        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "证明 (1) 写为 $A = (e_{n}, e_{1}, \\cdots, e_{n-1})$ ，则 $A^{2} = (A e_{n}, A e_{1}, \\cdots, A e_{n-1})$ 。显然 $Ae_{i}$ 为 A 的第 i 列，因此",
        "bbox": [
            89,
            198,
            842,
            215
        ],
        "page_idx": 30
    },
    {
        "type": "equation",
        "text": "$$\nA ^ {2} = (e _ {n - 1}, e _ {n}, e _ {1} \\dots , e _ {n - 2})\n$$",
        "text_format": "latex",
        "bbox": [
            393,
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        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "以此类推容易得到结论",
        "bbox": [
            90,
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            263,
            259
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "(2) $A=\\sum_{i<j}a_{ij}E_{ij}$ ，而 $j\\neq k$ 时 $E_{ij}E_{kl}=O$ ，因此 $A^{n}$ 展开式中非零项只能是 $E_{ij_{1}}E_{j_{1}j_{2}}\\cdots E_{j_{n-1}j_{n}}$ 这样的项，但满足 $1\\leq i<j_{1}<\\cdots<j_{n}\\leq n$ 的项不存在，因此 $A^{n}=O$",
        "bbox": [
            89,
            262,
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        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "练习2.13用二次型研究矩阵性质(1)证明： $n$ 阶对称矩阵 $A$ 为零矩阵当且仅当 $\\forall \\alpha \\in \\mathbb{R}^n$ 有 $\\alpha^T A\\alpha = 0$",
        "bbox": [
            60,
            328,
            836,
            343
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "(2) 证明： $n$ 阶方阵 $A$ 是反对称矩阵当且仅当 $\\forall \\alpha \\in \\mathbb{R}^n$ 有 $\\alpha^T A\\alpha = 0$",
        "bbox": [
            121,
            347,
            625,
            361
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "证明 (1) 左推右显然。先设 $\\alpha = e_i$ ，则 $e_i^T Ae_i = a_{ii} = 0$ ，再取 $\\alpha = e_i + e_j$ ，则",
        "bbox": [
            90,
            365,
            672,
            380
        ],
        "page_idx": 30
    },
    {
        "type": "equation",
        "text": "$$\n0 = (e _ {i} + e _ {j}) ^ {T} A (e _ {i} + e _ {j}) = a _ {i i} + a _ {j j} + a _ {i j} + a _ {j i}\n$$",
        "text_format": "latex",
        "bbox": [
            322,
            388,
            673,
            405
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "而 $a_{ij} = a_{ji}$ ，因此得到 $a_{ij} = 0, \\forall i, j$ (2) 左推右用 $\\alpha^T A\\alpha = (\\alpha^T A\\alpha)^T = -\\alpha^T A\\alpha$ 即可。右推左：根据条件得到 $\\alpha^T A^T\\alpha = 0$ ，因此 $\\alpha^T (A + A^T)\\alpha = 0$ ，由于 $A + A^T$ 为对称矩阵，根据上题结论得到 $A + A^T = O$ ，因此 $A^T = -A$",
        "bbox": [
            90,
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            464
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "2.5.3 置换矩阵",
        "text_level": 2,
        "bbox": [
            94,
            485,
            231,
            502
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "定义 2.12 (置换矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            517,
            278,
            533
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "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 $P = [\\epsilon_{i_{1}}, \\cdots, \\epsilon_{i_{n}}] = [\\epsilon_{j_{1}}^{T}, \\cdots, \\epsilon_{j_{n}}^{T}]^{T}$ ，其中 $\\epsilon_{i}$ 表示仅第 i 元素为 1 的列向量， $i_{k}, j_{k}$ 分别表示列、行的重排顺序。",
        "bbox": [
            114,
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            897,
            574
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "定理 2.30 (置换矩阵是正交矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            598,
            361,
            613
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "置换矩阵是正交矩阵。",
        "bbox": [
            117,
            618,
            278,
            632
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "证明 证明 $P^T P = I$",
        "text_level": 2,
        "bbox": [
            90,
            651,
            243,
            665
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "定理 2.31 (置换矩阵的作用)",
        "text_level": 2,
        "bbox": [
            122,
            676,
            329,
            690
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "左乘置换矩阵相当于把行原本的 $j_{1}, j_{2}, \\cdots, j_{n}$ 行换到 $1, 2, \\cdots, n$ 行，右乘相当于把列原本的 $i_{1}, \\cdots, i_{n}$ 列换到 $1, 2, \\cdots, n$ 列。",
        "bbox": [
            114,
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            887,
            731
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "定理 2.32 (置换矩阵正交相似)",
        "text_level": 2,
        "bbox": [
            122,
            755,
            344,
            770
        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "$i_{1}\\cdots i_{n}$ 是 $1\\sim n$ 的排列， $A=(a_{ij})$ ，取 $P=[\\epsilon_{i_{1}},\\cdots,\\epsilon_{i_{n}}]$ ，则：",
        "bbox": [
            114,
            775,
            581,
            791
        ],
        "page_idx": 30
    },
    {
        "type": "equation",
        "text": "$$\nP ^ {T} A P = P ^ {- 1} A P = \\left[ \\begin{array}{c c c c} a _ {i _ {1} i _ {1}} & a _ {i _ {1} i _ {2}} & \\dots & a _ {i _ {1} i _ {n}} \\\\ a _ {i _ {2} i _ {1}} & a _ {i _ {2} i _ {2}} & \\dots & a _ {i _ {2} i _ {n}} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a _ {i _ {n} i _ {1}} & a _ {i _ {n} i _ {2}} & \\dots & a _ {i _ {n} i _ {n}} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            312,
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        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "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": [
            115,
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            900,
            895
        ],
        "page_idx": 30
    },
    {
        "type": "header",
        "text": "2.5 几种技巧矩阵",
        "bbox": [
            771,
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        ],
        "page_idx": 30
    },
    {
        "type": "page_number",
        "text": "26",
        "bbox": [
            489,
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        ],
        "page_idx": 30
    },
    {
        "type": "text",
        "text": "练习 2.14 置换矩阵正交相似例子 $i_{1}\\cdots i_{n}$ 是 $1,2,\\cdots,n$ 的排列， $A=\\mathrm{diag}\\{\\lambda_{1},\\cdots,\\lambda_{n}\\},B=\\mathrm{diag}\\{\\lambda_{i_{1}},\\cdots,\\lambda_{i_{n}}\\}$ ，证明 A,B 是正交相似的",
        "bbox": [
            60,
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            118
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "证明 取 $P = [\\epsilon_{i_{1}}, \\cdots, \\epsilon_{i_{n}}]$ 即可。",
        "bbox": [
            90,
            121,
            329,
            137
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "定义 2.13 (特殊的置换矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            147,
            329,
            162
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "特殊置换矩阵 $K = \\left[ \\begin{array}{cccc}  &  &  & 1 \\\\  &  & 1 &  \\\\  & \\ddots &  &  \\\\ 1 &  &  &  \\\\ \\end{array} \\right]$ ， $K^T AK$ 的功能是绕中心转180度(行列全部倒排)",
        "bbox": [
            112,
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            250
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "2.6 特殊矩阵性质总结",
        "text_level": 2,
        "bbox": [
            94,
            292,
            337,
            312
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "2.6.1 对角矩阵",
        "text_level": 2,
        "bbox": [
            94,
            331,
            231,
            348
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "定理 2.33 (对角矩阵乘法)",
        "text_level": 2,
        "bbox": [
            121,
            363,
            312,
            378
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "矩阵 $A$ 右乘一个对角阵，则每列乘上对角相应元素。 $A$ 左乘对角阵，则每行乘上对角相应元素。",
        "bbox": [
            114,
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            818,
            399
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "定理 2.34 (对角矩阵求逆)",
        "bbox": [
            122,
            422,
            312,
            437
        ],
        "page_idx": 31
    },
    {
        "type": "equation",
        "text": "$$\n\\operatorname{diag} \\left\\{d _ {1}, \\dots , d _ {n} \\right\\} ^ {- 1} = \\operatorname{diag} \\left\\{d _ {1} ^ {- 1}, \\dots , d _ {n} ^ {- 1} \\right\\}\n$$",
        "text_format": "latex",
        "bbox": [
            114,
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            418,
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        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "2.6.2 上、下三角矩阵",
        "text_level": 2,
        "bbox": [
            94,
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            292,
            513
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "定理 2.35 (上下三角乘法、逆、伴随)",
        "text_level": 2,
        "bbox": [
            121,
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            396,
            544
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "$A, B$ 是上三角矩阵，则",
        "bbox": [
            114,
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            292,
            563
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "\\- $AB$ 仍然是上三角矩阵，且 $AB$ 对角元为 $a_{11}b_{11},\\dots ,a_{nn}b_{nn}$",
        "bbox": [
            139,
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            583
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "\\- $A^{-1}$ 也是上三角阵，且 $A^{-1}$ 对角元为 $a_{11}^{-1}, \\cdots, a_{nn}^{-1}$",
        "bbox": [
            139,
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            602
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "\\- $A^{*}$ 也是上三角阵，且 $A^{*}$ 的对角元为 $a_{22}a_{33}\\cdots a_{nn}, a_{11}a_{33}\\cdots a_{nn}, \\cdots, a_{11}a_{22}\\cdots a_{n-1,n-1}$ (也是特征值)",
        "bbox": [
            139,
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            639
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "证明 (1) 令 $C = AB$ ，若 $c_{ij} = \\sum_{k=1}^{n} a_{ik} b_{kj} = \\sum_{k=1}^{i-1} a_{ik} b_{kj} + \\sum_{k=i}^{n} a_{ik} b_{kj}$ ，当 $i > j$ 时，显然 $c_{ij} = 0$ 。而 $c_{ii} = \\sum_{k=1}^{i-1} a_{ik} b_{ki} + a_{ii} b_{ii} + \\sum_{k=i+1}^{n} a_{ik} b_{ki} = a_{ii} b_{ii}$",
        "bbox": [
            87,
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        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "(2) 根据行变换可以将 $(A, I) \\to (I, A^{-1})$ ，当 $A$ 为上三角时，先用最后一行消去 $A$ 最后一列其余行的元素，以此类推可以发现 $A^{-1}$ 为上三角，根据矩阵乘法显然有 $A^{-1}$ 对角为 $A$ 对角的逆。",
        "bbox": [
            90,
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            907,
            760
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "(3) 根据 $A^{*} = |A|A^{-1}$ 即可",
        "bbox": [
            122,
            764,
            322,
            778
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "定理 2.36 (n 个上三角矩阵的乘积)",
        "text_level": 2,
        "bbox": [
            122,
            790,
            376,
            804
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "$A_{1},\\cdots,A_{n}$ 时 n 个对角为 0 的 n 阶上三角矩阵，证明 $A_{1}A_{2}\\cdots A_{n}=O$",
        "bbox": [
            114,
            810,
            638,
            825
        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "证明 用数学归纳法，设 $n - 1$ 时成立，取 $A_{n} = \\left[ \\begin{array}{cc}B_{n - 1} & *\\\\ 0 & 0 \\end{array} \\right]$ ，此时",
        "bbox": [
            89,
            837,
            601,
            876
        ],
        "page_idx": 31
    },
    {
        "type": "equation",
        "text": "$$\nA _ {1} \\dots A _ {n} = \\left[ \\begin{array}{c c} B _ {1} \\dots B _ {n - 1} & * \\\\ 0 & 0 \\end{array} \\right] \\cdot \\left[ \\begin{array}{c c} B _ {n} & * \\\\ 0 & 0 \\end{array} \\right] = \\left[ \\begin{array}{c c} 0 & * \\\\ 0 & 0 \\end{array} \\right] \\left[ \\begin{array}{c c} B _ {n} & * \\\\ 0 & 0 \\end{array} \\right] = O\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 31
    },
    {
        "type": "header",
        "text": "2.6 特殊矩阵性质总结",
        "bbox": [
            737,
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        ],
        "page_idx": 31
    },
    {
        "type": "page_number",
        "text": "27",
        "bbox": [
            489,
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        ],
        "page_idx": 31
    },
    {
        "type": "text",
        "text": "2.6.3 实对称矩阵",
        "text_level": 2,
        "bbox": [
            94,
            85,
            253,
            102
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "定义 2.14 (实对称矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            118,
            295,
            133
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "每个元素都是实数，且满足 $A^T = A$ 的矩阵 $A$ 称为实对称矩阵",
        "bbox": [
            115,
            139,
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            154
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "定理 2.37 (实对称矩阵的矩阵性质)",
        "text_level": 2,
        "bbox": [
            122,
            179,
            379,
            193
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "实对称矩阵 A 具有以下性质：",
        "bbox": [
            115,
            199,
            332,
            212
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "\\- $A^{-1}$ 也是实对称矩阵",
        "bbox": [
            141,
            217,
            315,
            231
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "证明 (1) 由于 $(AA^{-1})^T = E_n$ ，得到 $(A^{-1})^T A^T = E_n$ ，而 $A$ 实对称，因此 $(A^{-1})^T A = E_n$ ，这说明 $(A^{-1})^T = A^{-1}$",
        "bbox": [
            90,
            250,
            899,
            266
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "定理 2.38 (实对称矩阵的特征值性质)",
        "text_level": 2,
        "bbox": [
            122,
            277,
            396,
            292
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "实对称矩阵有如下性质：",
        "bbox": [
            115,
            297,
            295,
            311
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "\\- 每个特征值都是实数",
        "bbox": [
            139,
            316,
            312,
            329
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "\\- 不同特征值的特征向量正交",
        "bbox": [
            139,
            334,
            363,
            348
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "\\- 一定正交相似 (相似且合同) 于对角矩阵",
        "bbox": [
            139,
            353,
            448,
            367
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "证明 (1) 共轭转置法：设 $\\lambda \\in \\mathbb{C}$ 是特征值， $\\xi \\in \\mathbb{C}^n$ 是特征向量，一方面根据 $A\\xi = \\lambda\\xi$ 两侧同时左乘 $\\overline{\\xi}^T$ 得到",
        "bbox": [
            89,
            384,
            882,
            403
        ],
        "page_idx": 32
    },
    {
        "type": "equation",
        "text": "$$\n\\overline {{\\xi}} ^ {T} A \\xi = \\lambda \\overline {{\\xi}} ^ {T} \\xi\n$$",
        "text_format": "latex",
        "bbox": [
            445,
            407,
            549,
            426
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "另一方面直接对 $A\\xi = \\lambda \\xi$ 取共轭转置，再同时右乘 $\\xi$ 得到",
        "bbox": [
            90,
            432,
            517,
            448
        ],
        "page_idx": 32
    },
    {
        "type": "equation",
        "text": "$$\n\\overline {{\\xi}} ^ {T} A = \\overline {{\\xi}} ^ {T} \\overline {{\\lambda}} \\Rightarrow \\overline {{\\xi}} ^ {T} A \\xi = \\overline {{\\lambda}} \\overline {{\\xi}} ^ {T} \\xi\n$$",
        "text_format": "latex",
        "bbox": [
            389,
            451,
            605,
            473
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "因此比较得到 $\\lambda = \\overline{\\lambda}$",
        "bbox": [
            90,
            480,
            243,
            494
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "(2) 不同特征值的特征向量正交：设 $\\lambda_{1}, \\lambda_{2}$ 是两个互异特征值， $\\xi_{1}, \\xi_{2}$ 分别属于 $\\lambda_{1}, \\lambda_{2}$ 的实特征向量，则",
        "bbox": [
            122,
            498,
            880,
            513
        ],
        "page_idx": 32
    },
    {
        "type": "equation",
        "text": "$$\n\\lambda_ {1} (\\xi_ {1}, \\xi_ {2}) = (\\lambda_ {1} \\xi_ {1}) ^ {T} \\xi_ {2} = (A \\xi_ {1}) ^ {T} \\xi_ {2} = \\xi_ {1} ^ {T} (A \\xi_ {2}) = \\lambda_ {2} (\\xi_ {1}, \\xi_ {2})\n$$",
        "text_format": "latex",
        "bbox": [
            285,
            521,
            710,
            537
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "从而得到 $(\\lambda_{2} - \\lambda_{1})(\\xi_{1},\\xi_{2}) = 0$ ，而 $\\lambda_1\\neq \\lambda_2$ ，从而 $(\\xi_1,\\xi_2) = 0$",
        "bbox": [
            90,
            545,
            547,
            558
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "(3) 做数学归纳法，n=1 显然，假设 n-1 成立，任取特征值 $\\lambda$ ，对应单位特征向量 $\\alpha_{1}$ ，将 $\\alpha_{1}$ 扩充为 $R^{n}$ 的一组标准正交基 $\\alpha_{1},\\cdots,\\alpha_{n}$ ，令 $T=(\\alpha_{1},\\cdots,\\alpha_{n})$ ，则",
        "bbox": [
            90,
            563,
            905,
            596
        ],
        "page_idx": 32
    },
    {
        "type": "equation",
        "text": "$$\nA T = T \\left( \\begin{array}{c c} \\lambda & \\alpha^ {T} \\\\ 0 & A _ {n - 1} \\end{array} \\right) \\Rightarrow T ^ {T} A T = \\left( \\begin{array}{c c} \\lambda & \\alpha^ {T} \\\\ 0 & A _ {n - 1} \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            309,
            600,
            685,
            638
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "由于 $T^T A T$ 对称，因此 $\\alpha^T = 0$ ，再根据归纳假设，存在正交矩阵 $P_{1}$ 使得 $P_{1}^{T}A_{n - 1}P_{1}$ 为对角阵，因此令 $P = \\mathrm{diag}(1,P_1)$ ，此时",
        "bbox": [
            90,
            642,
            905,
            677
        ],
        "page_idx": 32
    },
    {
        "type": "equation",
        "text": "$$\nP ^ {T} T ^ {T} A T P = \\left( \\begin{array}{c c} \\lambda & 0 \\\\ 0 & P ^ {T} A _ {n - 1} P \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            368,
            677,
            623,
            714
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "显然 $TP = Q$ ，则 $Q$ 也是正交阵。",
        "bbox": [
            92,
            722,
            341,
            738
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "2.6.4 反对称矩阵",
        "text_level": 2,
        "bbox": [
            94,
            760,
            253,
            777
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "定理 2.39 (反对称矩阵的行列式)",
        "text_level": 2,
        "bbox": [
            122,
            793,
            363,
            807
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "奇数阶反对称矩阵的行列式必为 0。",
        "bbox": [
            117,
            813,
            374,
            827
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "证明 根据 $A^T = -A$ 得到 $|A| = (-1)^n |A|$ ，当 $n$ 为奇数时 $|A| = -|A|$ ，因此 $|A| = 0$",
        "bbox": [
            90,
            846,
            709,
            862
        ],
        "page_idx": 32
    },
    {
        "type": "page_number",
        "text": "28",
        "bbox": [
            489,
            953,
            507,
            965
        ],
        "page_idx": 32
    },
    {
        "type": "header",
        "text": "2.6 特殊矩阵性质总结",
        "bbox": [
            737,
            55,
            905,
            68
        ],
        "page_idx": 32
    },
    {
        "type": "text",
        "text": "定理 2.40 (反对称矩阵的矩阵性质)",
        "text_level": 2,
        "bbox": [
            122,
            87,
            378,
            102
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "$A$ 为反对称矩阵，则",
        "bbox": [
            115,
            107,
            272,
            120
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "\\- $A^{-1}$ 为反对称矩阵",
        "bbox": [
            139,
            124,
            297,
            139
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "证明 (1) 类似实对称矩阵的",
        "text_level": 2,
        "bbox": [
            92,
            159,
            295,
            173
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "定理 2.41 (反对称矩阵的二次型)",
        "text_level": 2,
        "bbox": [
            122,
            184,
            361,
            199
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "$A$ 为反对称矩阵当且仅当任意 $X \\in \\mathbb{R}^n$ ，有 $X^T A X = 0$",
        "bbox": [
            114,
            205,
            522,
            219
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "证明 在基础矩阵技巧一节中给出过证明",
        "text_level": 2,
        "bbox": [
            90,
            239,
            389,
            254
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "定理 2.42 (反对称矩阵的特征值、特征向量)",
        "text_level": 2,
        "bbox": [
            122,
            263,
            445,
            279
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "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": [
            115,
            284,
            850,
            299
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "证明 (1) 设 $A$ 的特征值 $\\lambda = a + bi$ ，对应特征向量 $x = u + vi$ ， $A(u + vi) = (a + bi)(u + vi)$ ，因此 $Au + iAv = (au - bv) + (bu + av)i$ ，实虚部分离得到：",
        "bbox": [
            90,
            318,
            905,
            353
        ],
        "page_idx": 33
    },
    {
        "type": "equation",
        "text": "$$\nA u = a u - b v, A v = b u + a v \\Rightarrow u ^ {T} A u = a u ^ {T} u - b u ^ {T} v, v ^ {T} A v = b v ^ {T} u + a v ^ {T} v\n$$",
        "text_format": "latex",
        "bbox": [
            218,
            359,
            779,
            374
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "因此得到 $u^T Au + v^T Av = a(|u|^2 + |v|^2)$ ，由于 $u^T Au \\in \\mathbb{C}$ 是一个数，因此",
        "bbox": [
            90,
            382,
            638,
            398
        ],
        "page_idx": 33
    },
    {
        "type": "equation",
        "text": "$$\nu ^ {T} A u = (u ^ {T} A u) ^ {T} = u ^ {T} A ^ {T} u = - u ^ {T} A u \\Rightarrow u ^ {T} A u = 0\n$$",
        "text_format": "latex",
        "bbox": [
            302,
            405,
            694,
            420
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "同理有 $v^{T}Av = 0$ ，因此 $a(|u|^2 +|v|^2) = 0$ ，由于 $u + vi\\neq 0$ ，得到 $a = 0$ ，即 $\\lambda = a + bi = bi$",
        "bbox": [
            92,
            428,
            773,
            444
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "(2) 设 $\\alpha + \\beta i$ 是特征值 $ki$ 的特征向量，即 $A(\\alpha + i\\beta) = ki(\\alpha + i\\beta)$ ，对比实部虚部，得到",
        "bbox": [
            122,
            447,
            774,
            463
        ],
        "page_idx": 33
    },
    {
        "type": "equation",
        "text": "$$\nA \\alpha = - k \\beta , A \\beta = k \\alpha\n$$",
        "text_format": "latex",
        "bbox": [
            418,
            470,
            578,
            486
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "由于 $\\alpha^T A\\alpha = 0$ ，得到 $\\alpha^T A\\alpha = \\alpha^T (-k\\beta) = -k\\alpha^T\\beta = 0$ ，这说明 $\\alpha, \\beta$ 正交，再由于",
        "bbox": [
            92,
            492,
            707,
            508
        ],
        "page_idx": 33
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha^ {T} A \\beta + \\beta^ {T} A \\alpha = \\left\\{ \\begin{array}{l} \\alpha^ {T} A \\beta - (\\alpha^ {T} A \\beta) ^ {T} = 0 \\\\ k (\\alpha^ {T} \\alpha - \\beta^ {T} \\beta) \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
            329,
            514,
            648,
            558
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "得到 $\\alpha^T\\alpha -\\beta^T\\beta = 0$ ，即模长相等",
        "bbox": [
            90,
            562,
            347,
            577
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "定理 2.43 (反对称矩阵的标准型)",
        "text_level": 2,
        "bbox": [
            122,
            589,
            361,
            604
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "$A$ 为 $n$ 阶反对称矩阵，则",
        "text_level": 2,
        "bbox": [
            117,
            609,
            305,
            623
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "\\- $A$ 正交相似于如下准对角阵",
        "bbox": [
            139,
            627,
            361,
            642
        ],
        "page_idx": 33
    },
    {
        "type": "equation",
        "text": "$$\n\\operatorname{diag} \\left\\{\\left( \\begin{array}{c c} 0 & b _ {1} \\\\ - b _ {1} & 0 \\end{array} \\right), \\dots , \\left( \\begin{array}{c c} 0 & b _ {s} \\\\ - b _ {s} & 0 \\end{array} \\right), 0, \\dots , 0 \\right\\}\n$$",
        "text_format": "latex",
        "bbox": [
            327,
            644,
            707,
            684
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "\\- $A$ 合同于标准型",
        "bbox": [
            141,
            687,
            280,
            701
        ],
        "page_idx": 33
    },
    {
        "type": "equation",
        "text": "$$\n\\operatorname{diag} \\left\\{\\left( \\begin{array}{c c} 0 & 1 \\\\ - 1 & 0 \\end{array} \\right), \\dots , \\left( \\begin{array}{c c} 0 & 1 \\\\ - 1 & 0 \\end{array} \\right), 0, \\dots , 0 \\right\\}\n$$",
        "text_format": "latex",
        "bbox": [
            339,
            699,
            697,
            739
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "定理 2.44 (将矩阵拆为对称 + 反对称)",
        "text_level": 2,
        "bbox": [
            122,
            763,
            396,
            778
        ],
        "page_idx": 33
    },
    {
        "type": "equation",
        "text": "$$\nA = \\frac {1}{2} (A + A ^ {T}) + \\frac {1}{2} (A - A ^ {T})\n$$",
        "text_format": "latex",
        "bbox": [
            115,
            784,
            339,
            812
        ],
        "page_idx": 33
    },
    {
        "type": "header",
        "text": "2.6 特殊矩阵性质总结",
        "bbox": [
            737,
            55,
            905,
            68
        ],
        "page_idx": 33
    },
    {
        "type": "page_number",
        "text": "29",
        "bbox": [
            489,
            953,
            507,
            964
        ],
        "page_idx": 33
    },
    {
        "type": "text",
        "text": "2.6.5 Hermite 矩阵",
        "text_level": 2,
        "bbox": [
            95,
            86,
            268,
            102
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "定义 2.15 (Hermite 矩阵)",
        "bbox": [
            122,
            118,
            309,
            133
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "$\\overline{A}^T = A$ 的矩阵称为Hermite矩阵",
        "bbox": [
            115,
            140,
            363,
            156
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "定理 2.45 (Hermite 矩阵的性质)",
        "bbox": [
            122,
            181,
            359,
            195
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "Hermite 矩阵的特征值都是实数",
        "bbox": [
            115,
            200,
            347,
            215
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "2.6.6 正交矩阵",
        "text_level": 2,
        "bbox": [
            95,
            253,
            231,
            269
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "定义 2.16 (正交矩阵)",
        "bbox": [
            122,
            285,
            278,
            300
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "若矩阵 $T$ 满足 $TT^{T} = I$ ，则称 $T$ 为正交矩阵。",
        "bbox": [
            115,
            306,
            457,
            321
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "定理 2.46 (正交矩阵的性质)",
        "text_level": 2,
        "bbox": [
            122,
            346,
            327,
            361
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "$A, B$ 为 $n$ 阶正交矩阵，则",
        "bbox": [
            115,
            366,
            310,
            380
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "\\- 求逆: $A^{-1} = A^{T}$",
        "bbox": [
            141,
            382,
            287,
            398
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "\\- $A^{-1}, A^T, AB$ 都是正交矩阵",
        "bbox": [
            141,
            401,
            361,
            417
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "\\- $S = \\left[ \\begin{array}{ll}A & 0\\\\ 0 & B \\end{array} \\right]$ 也是正交矩阵",
        "bbox": [
            141,
            418,
            379,
            453
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "\\- $A$ 正交当且仅当 $A$ 的列向量是 $\\mathbb{R}^n$ 的一组标准正交基。",
        "bbox": [
            139,
            454,
            554,
            468
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "证明 (1) 显然",
        "bbox": [
            92,
            487,
            194,
            502
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "(2) $(AB)^T AB = I$",
        "bbox": [
            124,
            506,
            257,
            521
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "(3) 直接用 $S^T S$ 验证即可",
        "bbox": [
            122,
            524,
            312,
            539
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "(4) $A=\\left[\\alpha_{1},\\cdots,\\alpha_{n}\\right]$ ，计算 $A^{T}A=I$ 可得出结论。",
        "bbox": [
            122,
            543,
            487,
            558
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "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": [
            58,
            580,
            793,
            613
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "定理 2.47 (分块上三角正交矩阵)",
        "text_level": 2,
        "bbox": [
            124,
            640,
            361,
            655
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "若 $A = \\left[ \\begin{array}{ll}A_{1} & A_{2}\\\\ 0 & A_{4} \\end{array} \\right]$ 是正交矩阵，则 $A_{2} = 0$ ， $A_{1},A_{4}$ 是正交矩阵",
        "bbox": [
            115,
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            596,
            700
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "证明 根据分块矩阵转置以及乘法可以得到结论",
        "bbox": [
            90,
            720,
            438,
            734
        ],
        "page_idx": 34
    },
    {
        "type": "equation",
        "text": "$$\nA ^ {T} A = \\left[ \\begin{array}{c c} A _ {1} ^ {T} & 0 \\\\ A _ {2} ^ {T} & A _ {4} ^ {T} \\end{array} \\right] \\left[ \\begin{array}{c c} A _ {1} & A _ {2} \\\\ 0 & A _ {4} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            368,
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        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "定理 2.48 (正交矩阵的行列式与特征值)",
        "text_level": 2,
        "bbox": [
            124,
            793,
            413,
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        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "$A$ 是 $n$ 阶正交矩阵，则 $A$ 的特征值模长为1，复特征值成对出现， $|A| = \\pm 1$ ，进一步地",
        "bbox": [
            114,
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        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "若 $|A| = -1: -1$ 一定是 $A$ 的特征值",
        "bbox": [
            139,
            831,
            430,
            846
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "若 $A$ 是奇数阶的，且 $|A| = 1$ ，则1一定是 $A$ 的特征值",
        "bbox": [
            139,
            850,
            563,
            864
        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "\\- 复特征值的特征向量实部和虚部模长相等且正交",
        "bbox": [
            139,
            868,
            514,
            883
        ],
        "page_idx": 34
    },
    {
        "type": "header",
        "text": "2.6 特殊矩阵性质总结",
        "bbox": [
            737,
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        ],
        "page_idx": 34
    },
    {
        "type": "page_number",
        "text": "30",
        "bbox": [
            490,
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        ],
        "page_idx": 34
    },
    {
        "type": "text",
        "text": "证明 (1) 主命题: $A\\alpha = \\lambda \\alpha$ ，两侧取共轭得到 $A\\overline{\\alpha} = \\overline{\\lambda}\\overline{\\alpha}$ ，两侧取转置得到 $\\overline{\\alpha}^T A^T = \\overline{\\lambda}\\overline{\\alpha}^T$ ，两侧右乘 $\\alpha$ 得到 $\\overline{\\alpha}^T A^{-1}\\alpha = \\overline{\\lambda}\\overline{\\alpha}^T\\alpha$ ，因此 $\\frac{1}{\\lambda}\\overline{\\alpha}^T\\alpha = \\overline{\\lambda}\\overline{\\alpha}^T\\alpha$ ，因此 $\\lambda \\overline{\\lambda} = 1$ ，即模长为 1。根据 $|A^T A| = 1$ 可知行列式为 $\\pm 1$",
        "bbox": [
            87,
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        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "(2) $|A|=-1$ 情况：根据 $|A|$ 是特征值乘积可知至少有 1 个 -1",
        "bbox": [
            122,
            121,
            576,
            137
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "(3) $|A|=1$ 情况：同理",
        "bbox": [
            122,
            140,
            290,
            155
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "笔记在计算 $A$ 的特征值时，如果 $A$ 的特殊性需要 $A^T$ 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)。",
        "bbox": [
            89,
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        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "定理 2.49 (n 阶正交矩阵的标准型)",
        "text_level": 2,
        "bbox": [
            122,
            202,
            376,
            217
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "$A$ 是 $n$ 阶正交矩阵，则 $A$ 总是正交相似于",
        "bbox": [
            114,
            222,
            428,
            236
        ],
        "page_idx": 35
    },
    {
        "type": "equation",
        "text": "$$\n\\operatorname{diag} \\left\\{\\lambda_ {1}, \\dots , \\lambda_ {r}, \\left( \\begin{array}{c c} \\cos \\theta_ {1} & - \\sin \\theta_ {1} \\\\ \\sin \\theta_ {1} & \\cos \\theta_ {1} \\end{array} , \\dots , \\left( \\begin{array}{c c} \\cos \\theta_ {m} & - \\sin \\theta_ {m} \\\\ \\sin \\theta_ {m} & \\cos \\theta_ {m} \\end{array} \\right)\\right) \\right\\}\n$$",
        "text_format": "latex",
        "bbox": [
            243,
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        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "2.6.7 Unitary 矩阵",
        "text_level": 2,
        "bbox": [
            95,
            318,
            263,
            335
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "定理 2.50 (Unitary 矩阵的性质)",
        "bbox": [
            122,
            350,
            354,
            365
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "Unitary 矩阵的特征值模长均为 1",
        "bbox": [
            115,
            370,
            356,
            385
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "2.6.8 幂零矩阵",
        "text_level": 2,
        "bbox": [
            95,
            424,
            231,
            441
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "定义 2.17 (幂零矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            456,
            278,
            470
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "若存在整数 l，使得 $A^{l}=O$ ，则称 A 为幂零矩阵。",
        "bbox": [
            115,
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            482,
            493
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "定理 2.51 (幂零矩阵充要条件)",
        "text_level": 2,
        "bbox": [
            122,
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            344,
            532
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "$A$ 是幂零矩阵当且仅当",
        "text_level": 2,
        "bbox": [
            115,
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            285,
            551
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "1. 特征值: $A$ 的特征值只有 0",
        "bbox": [
            136,
            556,
            361,
            569
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "2. 特征多项式: $A$ 的特征多项式为 $x^{n}$",
        "bbox": [
            134,
            574,
            418,
            587
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "3. 最小多项式： $A$ 的最小多项式为 $x^{k}, k \\leq n$",
        "bbox": [
            134,
            592,
            468,
            606
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "4. 迹 (重点): 对 $\\forall k > 0$ , $\\operatorname{tr}(A^{k}) = 0$",
        "bbox": [
            134,
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            626
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "2.6.9 幂等矩阵",
        "text_level": 2,
        "bbox": [
            95,
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            231,
            681
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "定义 2.18 (幂等矩阵)",
        "text_level": 2,
        "bbox": [
            122,
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            277,
            712
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "若 $n$ 阶矩阵 $A$ 满足 $A^2 = A$ ，则称 $A$ 为幂等矩阵",
        "bbox": [
            115,
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        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "定理 2.52 (幂等矩阵充要条件)",
        "text_level": 2,
        "bbox": [
            122,
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            344,
            772
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "$n$ 阶方阵 $A$ 是幂等矩阵当且仅当 $r(A) + r(A - E) = n$",
        "bbox": [
            114,
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            515,
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        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "证明 由于 $A^2 = A$ ，即 $A(A - E) = O$ ，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": [
            90,
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        ],
        "page_idx": 35
    },
    {
        "type": "equation",
        "text": "$$\nn = n - r (A) + n - r (A - E)\n$$",
        "text_format": "latex",
        "bbox": [
            386,
            837,
            608,
            852
        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "因此 $r(A) + r(A - E) = n$",
        "bbox": [
            92,
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            287,
            873
        ],
        "page_idx": 35
    },
    {
        "type": "header",
        "text": "2.6 特殊矩阵性质总结",
        "bbox": [
            737,
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        ],
        "page_idx": 35
    },
    {
        "type": "page_number",
        "text": "31",
        "bbox": [
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        ],
        "page_idx": 35
    },
    {
        "type": "text",
        "text": "定理 2.53 (幂等矩阵的性质)",
        "text_level": 2,
        "bbox": [
            122,
            87,
            327,
            101
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "$A$ 为 $n$ 阶幂等矩阵，则",
        "bbox": [
            115,
            107,
            290,
            120
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "\\- $A$ 的特征值只有1和0",
        "bbox": [
            139,
            124,
            327,
            139
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "\\- $A$ 一定可以对角化，且标准型为 $\\mathrm{diag}\\{E_r, O\\}$",
        "bbox": [
            139,
            143,
            487,
            159
        ],
        "page_idx": 36
    },
    {
        "type": "equation",
        "text": "$$\nr (A) = \\operatorname{tr} (A)\n$$",
        "text_format": "latex",
        "bbox": [
            139,
            162,
            257,
            177
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "证明 (1) 设 $\\lambda$ 是 $A$ 的特征值，故 $A\\alpha = \\lambda \\alpha$ ， $A^2\\alpha = \\lambda^2\\alpha$ ，由于 $A = A^2$ ，得到 $\\lambda^2 = \\lambda$ ，因此 $\\lambda = 0,1$",
        "bbox": [
            90,
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        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "(2) 根据最小多项式理论， $f(A)=A(A-E)=O$ ，因此最小多项式只有一次项乘积，故可对角化。",
        "bbox": [
            122,
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        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "(3) 根据秩、迹在相似下的不变性可知",
        "bbox": [
            122,
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            405,
            250
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "定理 2.54 (加减乘法的幂等性)",
        "text_level": 2,
        "bbox": [
            122,
            275,
            344,
            291
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "$A, B$ 为两个 $n$ 阶幂等矩阵，则",
        "bbox": [
            115,
            297,
            344,
            310
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "1. $A + B$ 为幂等矩阵当且仅当 $AB = BA = O$",
        "bbox": [
            134,
            315,
            478,
            329
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "2. $A - B$ 为幂等矩阵当且仅当 $AB = BA = B$",
        "bbox": [
            134,
            334,
            480,
            347
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "3. 若 $AB = BA$ ，则 $AB$ 为幂等矩阵。反之不一定",
        "bbox": [
            134,
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            509,
            366
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "4. 如 E-A-B 可逆，则 $r(A)=r(B)$",
        "bbox": [
            134,
            370,
            430,
            385
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "证明 (1) 右推左：由于 $AB = BA = O$ ，因此 $(A + B)^2 = A^2 + AB + BA + B^2 = A + B$ 。左推右： $(A + B)^2 = A + AB + BA + B = A + B$ ，因此 $AB + BA = O$ ，得到",
        "bbox": [
            87,
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        ],
        "page_idx": 36
    },
    {
        "type": "equation",
        "text": "$$\nA B = A B ^ {2} = (- B A) B = - B A B = B (B A) = B A\n$$",
        "text_format": "latex",
        "bbox": [
            310,
            447,
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            462
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "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",
        "bbox": [
            58,
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        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "证明 (1) 利用等价标准型得到:",
        "bbox": [
            90,
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            521
        ],
        "page_idx": 36
    },
    {
        "type": "equation",
        "text": "$$\nA = P \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) Q = P Q Q ^ {- 1} \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) Q\n$$",
        "text_format": "latex",
        "bbox": [
            317,
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            564
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "PQ 为可逆矩阵，后面为幂等",
        "bbox": [
            90,
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            309,
            583
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "定理 2.55 (幂等矩阵列的性质)",
        "text_level": 2,
        "bbox": [
            124,
            609,
            344,
            624
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "$A_{1},\\cdots,A_{m}$ 为 n 阶方阵，且 $A_{1}+A_{2}+\\cdots+A_{m}=E_{n}$ ，则下面三个命题等价：",
        "bbox": [
            115,
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            645
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "\\- $A_{1}, \\cdots, A_{m}$ 为幂等矩阵",
        "bbox": [
            139,
            648,
            336,
            663
        ],
        "page_idx": 36
    },
    {
        "type": "equation",
        "text": "$$\n\\bullet \\mathrm{r} (A _ {1}) + \\dots + \\mathrm{r} (A _ {m}) = n\n$$",
        "text_format": "latex",
        "bbox": [
            139,
            667,
            339,
            682
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "\\- $\\forall i \\neq j$ 有 $A_{i}A_{j} = 0$",
        "bbox": [
            139,
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            302,
            701
        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "证明 $(1) \\Rightarrow (2)$ : 由于 $r(A_{i}) = \\operatorname{tr}(A_{i})$ , 因此",
        "bbox": [
            90,
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        ],
        "page_idx": 36
    },
    {
        "type": "equation",
        "text": "$$\nr \\left(A _ {1}\\right) + \\dots + r \\left(A _ {m}\\right) = \\operatorname{tr} \\left(A _ {1}\\right) + \\dots + \\operatorname{tr} \\left(A _ {m}\\right) = \\operatorname{tr} \\left(A _ {1} + \\dots + A _ {m}\\right) = \\operatorname{tr} \\left(E _ {n}\\right) = n\n$$",
        "text_format": "latex",
        "bbox": [
            201,
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        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "(2) $\\Leftarrow$ (3): 比较经典，先通过分块矩阵的初等变换凑出 E:",
        "bbox": [
            122,
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        ],
        "page_idx": 36
    },
    {
        "type": "equation",
        "text": "$$\n\\left( \\begin{array}{c c c c c} A _ {1} & & & & \\\\ & A _ {2} & & & \\\\ & & A _ {3} & & \\\\ & & & \\ddots & \\\\ & & & & A _ {m} \\end{array} \\right) \\left( \\begin{array}{c c c c c} A _ {1} & A _ {2} & A _ {3} & \\dots & A _ {m} \\\\ A _ {2} & & & & \\\\ & & A _ {3} & & \\\\ & & & \\ddots & \\\\ & & & & A _ {m} \\end{array} \\right) \\to \\left( \\begin{array}{c c c c c} E & A _ {2} & A _ {3} & \\dots & A _ {m} \\\\ A _ {2} & A _ {2} & & \\\\ A _ {3} & & A _ {3} & \\\\ \\vdots & & & \\ddots \\\\ A _ {m} & & & A _ {m} \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            151,
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        ],
        "page_idx": 36
    },
    {
        "type": "header",
        "text": "2.6 特殊矩阵性质总结",
        "bbox": [
            737,
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        ],
        "page_idx": 36
    },
    {
        "type": "page_number",
        "text": "32",
        "bbox": [
            489,
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        ],
        "page_idx": 36
    },
    {
        "type": "text",
        "text": "再通过 E 消去其他部分:",
        "bbox": [
            90,
            85,
            275,
            99
        ],
        "page_idx": 37
    },
    {
        "type": "equation",
        "text": "$$\n\\rightarrow \\left(\\begin{array}{c c c c c}E&O&O&\\dots&O\\\\A _ {2}&A _ {2} - A _ {2} ^ {2}&- A _ {2} A _ {3}&\\dots&- A _ {2} A _ {m}\\\\A _ {3}&- A _ {3} A _ {2}&A _ {3} - A _ {3} ^ {2}&\\dots&- A _ {3} A _ {m}\\\\\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\A _ {m}&- A _ {m} A _ {2}&- A _ {m} A _ {3}&\\dots&A _ {m} - A _ {m} ^ {2}\\end{array}\\right)\\rightarrow \\left(\\begin{array}{c c c c c}E&O&O&\\dots&O\\\\O&A _ {2} - A _ {2} ^ {2}&- A _ {2} A _ {3}&\\dots&- A _ {2} A _ {m}\\\\O&- A _ {3} A _ {2}&A _ {3} - A _ {3} ^ {2}&\\dots&- A _ {3} A _ {m}\\\\\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\O&- A _ {m} A _ {2}&- A _ {m} A _ {3}&\\dots&A _ {m} - A _ {m} ^ {2}\\end{array}\\right)\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 37
    },
    {
        "type": "text",
        "text": "根据 $r(A_{1}) + r(A_{2}) + \\cdots + r(A_{m}) = n$ ，得到",
        "bbox": [
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        "page_idx": 37
    },
    {
        "type": "equation",
        "text": "$$\n\\left( \\begin{array}{c c c c} A _ {2} - A _ {2} ^ {2} & - A _ {2} A _ {3} & \\dots & - A _ {2} A _ {m} \\\\ - A _ {3} A _ {2} & A _ {3} - A _ {3} ^ {2} & \\dots & - A _ {3} A _ {m} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ - A _ {m} A _ {2} & - A _ {m} A _ {3} & \\dots & A _ {m} - A _ {m} ^ {2} \\end{array} \\right) = O\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 37
    },
    {
        "type": "text",
        "text": "这说明 $\\forall i \\neq j$ 有 $A_{i}^{2} = A_{i}$ ，且 $A_{i}A_{j} = O(i, j = 2, 3, \\cdots, m)$ ，再对 $A_{1} + \\cdots + A_{m} = E_{n}$ 两边同乘 $A_{i}$ 得到 $A_{1}A_{i} + A_{i} = A_{i}$",
        "bbox": [
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        ],
        "page_idx": 37
    },
    {
        "type": "text",
        "text": "因此 $A_{1}A_{i} = O$ ，结论成立",
        "bbox": [
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        ],
        "page_idx": 37
    },
    {
        "type": "text",
        "text": "(3) $\\Rightarrow(1)$ ：直接对 $A_{1}+\\cdots+A_{m}=E_{n}$ 两边右乘 $A_{i}$ 得到",
        "bbox": [
            122,
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        ],
        "page_idx": 37
    },
    {
        "type": "equation",
        "text": "$$\nA _ {1} A _ {i} + A _ {2} A _ {i} + \\dots + A _ {m} A _ {i} = A _ {i} ^ {2} = A _ {i}\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 37
    },
    {
        "type": "text",
        "text": "因此都是幂等矩阵",
        "bbox": [
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        ],
        "page_idx": 37
    },
    {
        "type": "image",
        "img_path": "images/116dae9c4751e133d34c8783fe26e5db95777c31f641faaaf1b2315ed7ef9957.jpg",
        "image_caption": [],
        "image_footnote": [],
        "content": "",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "笔记 此定理在 ZJU2020 中考过",
        "bbox": [
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        ],
        "page_idx": 37
    },
    {
        "type": "text",
        "text": "2.6.10 对角占优矩阵",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 37
    },
    {
        "type": "header",
        "text": "2.6 特殊矩阵性质总结",
        "bbox": [
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        ],
        "page_idx": 37
    },
    {
        "type": "page_number",
        "text": "33",
        "bbox": [
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        "page_idx": 37
    },
    {
        "type": "text",
        "text": "第3章 行列式",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "3.1 行列式基本概念与性质",
        "text_level": 2,
        "bbox": [
            94,
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            384,
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "定理 3.1 (行列式初等变换)",
        "text_level": 2,
        "bbox": [
            122,
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            319,
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "矩阵初等变换相对行列式的变换为:",
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "\\- 互换两行/列：取反号",
        "bbox": [
            139,
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            239
        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "\\- 同行/列可同时提出公因式",
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "\\- 倍加：不改变行列式大小",
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "定理 3.2 (行列式分拆)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "行列式可以关于一行或者一列进行分拆，即",
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "equation",
        "text": "$$\n\\left| \\begin{array}{c c c c} a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ b _ {s 1} + c _ {s 1} & b _ {s 2} + c _ {s 2} & \\dots & b _ {s n} + c _ {s n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right| = \\left| \\begin{array}{c c c c} a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ b _ {s 1} & b _ {s 2} & \\dots & b _ {s n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right| + \\left| \\begin{array}{c c c c} a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ c _ {s 1} & c _ {s 2} & \\dots & c _ {s n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right|.\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "3.2 行列式展开",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "定义 3.1 (余子式与代数余子式)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "将矩阵 A 去除第 i 行，第 j 列后矩阵的行列式称为 $a_{ij}$ 的余子式 $M_{ij}$ ，代数余子式 $A_{ij} := (-1)^{i+j} M_{ij}$",
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "定理 3.3 (行列式展开)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "固定列 $j$ ，则按列展开为 $|A| = \\sum_{i=1}^{n} a_{ij} A_{ij}$ 。固定行 $i$ ，则按行展开为 $|A| = \\sum_{j=1}^{n} a_{ij} A_{ij}$",
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "定理 3.4 (不同行相乘为 0)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "$A$ 的第 $i$ 行元素与 $k$ 行的代数余子式乘积和为0: $\\sum_{j=1}^{n} a_{ij} A_{kj} = 0$",
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "3.3 特殊的行列式",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "3.3.1 爪型行列式",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "定理 3.5 (正爪型行列式)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "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",
        "bbox": [
            117,
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        ],
        "page_idx": 38
    },
    {
        "type": "text",
        "text": "定理 3.6 (异爪型行列式)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "异爪型行列式(一横两斜)分以下几种情况:",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "\\- 行和为0型：全部加到第一行，按第一列展开",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "\\- 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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            158
        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "练习3.1异爪型行列式(1)计算 $|A| = \\left| \\begin{array}{cccccc}0 & -1 & -1 & \\dots & -1 & -1\\\\ 1 & 1 & 0 & \\dots & 0 & 0\\\\ 0 & 1 & 1 & \\dots & 0 & 0\\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 0 & 0 & 0 & \\dots & 1 & 1 \\end{array} \\right|$",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "(2) 设 $s(x) = \\begin{cases} \\frac{x}{|x|}, & x \\neq 0 \\\\ 0, & x = 0 \\end{cases}$ ，已知 $A = (a_{ij})$ ， $a_{ij} = s(i - j)$ ，求 $|A|$",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "解(1)按最后一列展开，得到 $D_{n} = (-1)^{n + 2} + D_{n - 1}$ ，因此 $n$ 为奇数时 $D_{n} = 0$ ， $n$ 为偶数时 $D_{n} = 1$",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "(2) 上下相减后变为 (1)",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "3.3.2 大对角型 (三斜线)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "3.3.3 行和相同型",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "定理 3.7 (行和相同型)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "行和相同型行列式分以下两种情况：",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "\\- 内部大量相同：加到第一列，提出公因式，消去内部大量元素",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "\\- 循环型：上下相减 (例题见循环型一节)",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "练习 3.2 内部大量相同型 计算行列式",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "equation",
        "text": "$$\n\\left| \\begin{array}{c c c c c} a & b & b & \\dots & b \\\\ b & a & b & \\dots & b \\\\ b & b & a & \\dots & b \\\\ \\vdots & \\vdots & & & \\vdots \\\\ b & b & b & \\dots & a \\end{array} \\right|\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "解加到第一列，提出 $[a + (n - 1)b]$ ，用第一列消去其他元素，最终得到答案 $(a - b)^{n - 1}[a + (n - 1)b]$",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "3.3.4 循环型",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "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。",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "text",
        "text": "练习3.3循环型 计算 $|A|, |B|$",
        "bbox": [
            60,
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        ],
        "page_idx": 39
    },
    {
        "type": "equation",
        "text": "$$\n| A | = \\left| \\begin{array}{c c c c c c} 0 & 1 & 2 & \\dots & n - 2 & n - 1 \\\\ 1 & 0 & 1 & \\dots & n - 3 & n - 2 \\\\ 2 & 1 & 0 & \\dots & n - 4 & n - 3 \\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ n - 2 & n - 3 & n - 4 & \\dots & 0 & 1 \\\\ n - 1 & n - 2 & n - 3 & \\dots & 1 & 0 \\end{array} \\right| \\quad | B | = \\left| \\begin{array}{c c c c c c} 1 & 2 & 3 & \\dots & n - 1 & n \\\\ n & 1 & 2 & \\dots & n - 2 & n - 1 \\\\ n - 1 & n & 1 & \\dots & n - 3 & n - 2 \\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 3 & 4 & 5 & \\dots & 1 & 2 \\\\ 2 & 3 & 4 & \\dots & n & 1 \\end{array} \\right|\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 39
    },
    {
        "type": "header",
        "text": "3.3 特殊的行列式",
        "bbox": [
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        "page_idx": 39
    },
    {
        "type": "page_number",
        "text": "35",
        "bbox": [
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        "page_idx": 39
    },
    {
        "type": "text",
        "text": "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",
        "bbox": [
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        ],
        "page_idx": 40
    },
    {
        "type": "equation",
        "text": "$$\n\\left| \\begin{array}{c c c c c c} - 1 & 1 & 1 & \\dots & 1 & 1 \\\\ - 1 & - 1 & 1 & \\dots & 1 & 1 \\\\ - 1 & - 1 & - 1 & \\dots & 1 & 1 \\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ - 1 & - 1 & - 1 & \\dots & - 1 & 1 \\\\ n - 1 & n - 2 & n - 3 & \\dots & 1 & 0 \\end{array} \\right|\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 40
    },
    {
        "type": "text",
        "text": "(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。",
        "bbox": [
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        ],
        "page_idx": 40
    },
    {
        "type": "text",
        "text": "3.3.5 行列式加边法",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 40
    },
    {
        "type": "text",
        "text": "定理 3.8 (行列式加边法)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 40
    },
    {
        "type": "text",
        "text": "计算矩阵 $A$ 的行列式 $|A|$ 时，可加上一边：",
        "bbox": [
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        ],
        "page_idx": 40
    },
    {
        "type": "equation",
        "text": "$$\n| A | = \\left| \\begin{array}{c c c c} a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ a _ {2 1} & a _ {2 2} & \\dots & a _ {2 n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right| = \\left| \\begin{array}{c c c c c} 1 & 1 & 1 & 1 & 1 \\\\ 0 & a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ 0 & a _ {2 1} & a _ {2 2} & \\dots & a _ {2 n} \\\\ \\vdots & \\vdots & \\vdots & & \\vdots \\\\ 0 & a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right|\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 40
    },
    {
        "type": "image",
        "img_path": "images/5237bc175ab7cc857bef0b9fce8ebb278fd06b25162610870ed179f93aa53519.jpg",
        "image_caption": [],
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        "bbox": [
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        "page_idx": 40
    },
    {
        "type": "text",
        "text": "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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        "page_idx": 40
    },
    {
        "type": "text",
        "text": "3.4 Cramer 法则",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 40
    },
    {
        "type": "text",
        "text": "定理 3.9 (Cramer 法则)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 40
    },
    {
        "type": "text",
        "text": "$n$ 个方程的 $n$ 元线性方程组 $AX = B$ ，系数矩阵行列式 $|A|\\neq 0$ ，则其唯一解为",
        "bbox": [
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        "page_idx": 40
    },
    {
        "type": "equation",
        "text": "$$\n\\left(\\frac {| B _ {1} |}{| A |}, \\frac {| B _ {2} |}{| A |}, \\dots , \\frac {| B _ {n} |}{| A |}\\right)\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 40
    },
    {
        "type": "text",
        "text": "其中 $B_{i}$ 为 $A$ 将第 $i$ 列换为 $B$ 组成的矩阵。",
        "bbox": [
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        ],
        "page_idx": 40
    },
    {
        "type": "header",
        "text": "3.4 Cramer 法则",
        "bbox": [
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        "page_idx": 40
    },
    {
        "type": "page_number",
        "text": "36",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "第 4 章 线性方程组",
        "text_level": 2,
        "bbox": [
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        "page_idx": 41
    },
    {
        "type": "text",
        "text": "4.1 线性方程组解的判定定理",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 41
    },
    {
        "type": "text",
        "text": "4.1.1 解存在判定定理",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 41
    },
    {
        "type": "text",
        "text": "定理 4.1 (解存在充要条件)",
        "bbox": [
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        ],
        "page_idx": 41
    },
    {
        "type": "text",
        "text": "K 上线性方程组 Ax=b 有解的充要条件为 $r(A)=r(\\overline{A})$ ，这里 $\\overline{A} := [A, b]$",
        "bbox": [
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        "page_idx": 41
    },
    {
        "type": "text",
        "text": "4.1.2 线性方程组同解",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 41
    },
    {
        "type": "text",
        "text": "定理 4.2 (同解定理)",
        "bbox": [
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        ],
        "page_idx": 41
    },
    {
        "type": "text",
        "text": "$A_{s\\times n},B_{m\\times n}$ 为两个矩阵，则 $AX = 0,BX = 0$ 同解当且仅当 $A,B$ 行向量等价",
        "bbox": [
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        "page_idx": 41
    },
    {
        "type": "text",
        "text": "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",
        "bbox": [
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    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} x _ {1} + 2 x _ {2} + 3 x _ {3} = 0 \\\\ 2 x _ {1} + 3 x _ {2} + 5 x _ {3} = 0 \\\\ x _ {1} + x _ {2} + a x _ {3} = 0 \\end{array} \\right. \\quad \\left\\{ \\begin{array}{l} x _ {1} + b x _ {2} + c x _ {3} = 0 \\\\ 2 x _ {1} + b ^ {2} x _ {2} + (c + 1) x _ {3} = 0 \\end{array} \\right.\n$$",
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    },
    {
        "type": "text",
        "text": "(2) $A_{s \\times n}, B_{t \\times n}$ 为两个矩阵， $AX = 0, BX = 0$ 的解空间分别为 $V_{1}, V_{2}$ ，证明： $V_{1} \\subseteq V_{2}$ 当且仅当存在矩阵 $C_{t \\times s}$ 使得 $B = CA$",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "证明 (1) 设方程组为 $AX = 0, BX = 0$ ，即 $AX = 0$ 与 $\\left( \\begin{array}{l} A \\\\ B \\end{array} \\right) X = 0$ 同解，做初等行变换即可。",
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    },
    {
        "type": "text",
        "text": "(2) $V_{1}\\subseteq V_{2}$ 当且仅当AX=0与 $\\left(\\begin{array}{l}A\\\\ B\\end{array}\\right)X=0$ 同解。",
        "bbox": [
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        "page_idx": 41
    },
    {
        "type": "text",
        "text": "4.1.3 解个数判定定理",
        "text_level": 2,
        "bbox": [
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        "page_idx": 41
    },
    {
        "type": "text",
        "text": "定理 4.3 (解个数判定定理)",
        "text_level": 2,
        "bbox": [
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        "page_idx": 41
    },
    {
        "type": "text",
        "text": "若 K 上 Ax = b 满足 $r(A) = r(\\overline{A})$ :",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "若 $r(A) = n$ ：则 $Ax = b$ 有唯一解",
        "bbox": [
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        "page_idx": 41
    },
    {
        "type": "text",
        "text": "若 $r(A) < n$ ：则 $Ax = b$ 有无穷解",
        "bbox": [
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        ],
        "page_idx": 41
    },
    {
        "type": "text",
        "text": "4.2 线性方程组的求解",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 41
    },
    {
        "type": "text",
        "text": "4.2.1 $Ax = 0$ 求解",
        "bbox": [
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        "page_idx": 41
    },
    {
        "type": "text",
        "text": "定义 4.1 (自由未知量)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 41
    },
    {
        "type": "text",
        "text": "消元后多余的 0 = 0 位置对应的变量 $x_{r+1}, \\cdots, x_{n}$ 称为方程组自由未知量",
        "bbox": [
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    },
    {
        "type": "image",
        "img_path": "images/569bb7571fe1b96279850b2e831c71dd14f2853e8248c29c1c1dd0d70ae8dd6c.jpg",
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        "content": "",
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    },
    {
        "type": "text",
        "text": "定理 4.4 (自由变量的个数)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 42
    },
    {
        "type": "text",
        "text": "自由变量的个数为 $n - r(A)$",
        "bbox": [
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        "page_idx": 42
    },
    {
        "type": "text",
        "text": "定义 4.2 (基础解系)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 42
    },
    {
        "type": "text",
        "text": "让自由未知量 $x_{r - 1 + i}$ 取1，其余取0，构造解向量 $\\eta_{i}$ 称为基础解系：",
        "bbox": [
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        ],
        "page_idx": 42
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} \\eta_ {1} = (*, \\dots , *, 1, 0, \\dots , 0) \\\\ \\eta_ {2} = (*, \\dots , *, 0, 1, \\dots , 0) \\\\ \\vdots \\\\ \\eta_ {n - r} = (*, \\dots , *, 0, 0, \\dots , 1) \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 42
    },
    {
        "type": "text",
        "text": "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。",
        "bbox": [
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        "page_idx": 42
    },
    {
        "type": "text",
        "text": "定理 4.5 (齐次线性方程组的解)",
        "bbox": [
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        ],
        "page_idx": 42
    },
    {
        "type": "text",
        "text": "齐次线性方程组 $Ax = 0$ 的任意解是其基础解系的线性组合",
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        ],
        "page_idx": 42
    },
    {
        "type": "text",
        "text": "4.2.2 $Ax = b$ 求解",
        "text_level": 2,
        "bbox": [
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        "page_idx": 42
    },
    {
        "type": "text",
        "text": "定理 4.6 (非齐次线性方程组的解)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 42
    },
    {
        "type": "text",
        "text": "非齐次线性方程组 Ax = b 的任意解是其特解 $\\gamma_{0}$ 加上齐次方程组 Ax = 0 基础解系线性组合，即 $x = \\gamma_{0} + L\\{\\eta_{1}, \\cdots, \\eta_{n-r}\\}$",
        "bbox": [
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        "page_idx": 42
    },
    {
        "type": "text",
        "text": "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",
        "bbox": [
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        "page_idx": 42
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} x _ {1} + x _ {2} - x _ {3} = 1 \\\\ 2 x _ {1} + 3 x _ {2} + k x _ {3} = 3 \\\\ x _ {1} + k x _ {2} + 3 x _ {3} = 2 \\end{array} \\right.\n$$",
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        "page_idx": 42
    },
    {
        "type": "text",
        "text": "(2) 已知 $A = \\left( \\begin{array}{ccc}1 & -1 & -1\\\\ 4 & t & 1\\\\ -2 & 2 & t \\end{array} \\right),b = \\left( \\begin{array}{c}1\\\\ 2\\\\ -t \\end{array} \\right)$ ，当 $t$ 为何值时， $AX = b$ 无解，有唯一解，有无穷多解，并给出有解时对应的解。",
        "bbox": [
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        "page_idx": 42
    },
    {
        "type": "text",
        "text": "解(1)对增广矩阵做初等变换得到",
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        ],
        "page_idx": 42
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c c c} 1 & 1 & - 1 & 1 \\\\ 0 & 1 & k + 2 & 1 \\\\ 0 & 0 & (k + 3) (k - 2) & k - 2 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 42
    },
    {
        "type": "text",
        "text": "先讨论解的情况：有解当且仅当 $r(A) = r(\\overline{A})$ ， $k = -3$ 无解， $k = 2$ 有无穷多解。其余情况有唯一解。 $k = 2$ 时特解以及齐次基础解系：",
        "bbox": [
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        ],
        "page_idx": 42
    },
    {
        "type": "equation",
        "text": "$$\n\\gamma = \\left( \\begin{array}{c} 0 \\\\ 1 \\\\ 0 \\end{array} \\right), y = \\left( \\begin{array}{c} 5 \\\\ - 4 \\\\ 1 \\end{array} \\right) \\Rightarrow x = \\gamma + k _ {1} y\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 42
    },
    {
        "type": "text",
        "text": "唯一解情况直接解出结果即可。",
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        ],
        "page_idx": 42
    },
    {
        "type": "header",
        "text": "4.2 线性方程组的求解",
        "bbox": [
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        "page_idx": 42
    },
    {
        "type": "page_number",
        "text": "38",
        "bbox": [
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        "page_idx": 42
    },
    {
        "type": "text",
        "text": "(2) 做增广矩阵初等变换得到",
        "bbox": [
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        ],
        "page_idx": 43
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c c c} 1 & - 1 & - 1 & 1 \\\\ 0 & t + 4 & 5 & - 2 \\\\ 0 & 0 & t - 2 & 2 - t \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 43
    },
    {
        "type": "text",
        "text": "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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        ],
        "page_idx": 43
    },
    {
        "type": "text",
        "text": "4.2.3 线性方程组公共解的计算",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 43
    },
    {
        "type": "text",
        "text": "练习 4.3 公共解的情况 (1) 线性方程组 I 的通解为 $(1,1,0,0)^{T} + k_{1}(1,0,-1,0)^{T} + k_{2}(2,3,0,1)^{T}$ ，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",
        "bbox": [
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        ],
        "page_idx": 43
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} 7 x _ {1} - 6 x _ {2} + 3 x _ {3} = b \\\\ 8 x _ {1} - 9 x _ {2} + a x _ {4} = 7 \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 43
    },
    {
        "type": "text",
        "text": "解 由于有无穷多组解，因此有无穷组 $k_{1}, k_{2}$ 使得 $(1 + k_{1} + 2k_{2}, 1 + 3k_{2}, -k_{1}, k_{2})$ 为方程组 II 的解。代入方程化简得到",
        "bbox": [
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        ],
        "page_idx": 43
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} 4 k _ {1} - 4 k _ {2} = b - 1 \\\\ 8 k _ {1} + (a - 1 1) k _ {2} = 8 \\end{array} \\right. \\Rightarrow \\left\\{ \\begin{array}{l} 4 k _ {1} - 4 k _ {2} = b - 1 \\\\ 0 k _ {1} + (a - 3) k _ {2} = 1 0 - 2 b \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 43
    },
    {
        "type": "text",
        "text": "需要有无穷多解，得有自由位置量，因此 $a = 3, b = 5$ ， $k_{1} = k_{2} + 1$ ，解系为 $(2,1, -1,0)^{T} + k_{2}(3,3, -1,1)^{T}$ 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。",
        "bbox": [
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        "page_idx": 43
    },
    {
        "type": "text",
        "text": "4.3 最小二乘解",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 43
    },
    {
        "type": "text",
        "text": "4.3.1 $A^T A$ 的研究",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 43
    },
    {
        "type": "text",
        "text": "定理 4.7 ( $A^{T}A$ 与 A 的秩)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 43
    },
    {
        "type": "text",
        "text": "设 A 是 $m \\times n$ 的实矩阵，则 $r(A^{T}A) = r(A)$ 。复数域有 $r(A\\overline{A}^{T}) = r(\\overline{A}^{T}A) = r(A)$ ，但不一定有 $r(A^{T}A) = r(A)$ 。",
        "bbox": [
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        "page_idx": 43
    },
    {
        "type": "text",
        "text": "证明 (1) 只需要考虑 AX = 0 和 $A^{T}AX = 0$ 解的情况。首先根据 AX = 0 可以推出 $A^{T}AX = 0$ 。其次 $A^{T}AX = 0$ 可推出 $X^{T}A^{T}AX = (AX)^{T}(AX) = 0$ ，而 AX 是一个实向量， $\\alpha^{T}\\alpha = 0$ 当且仅当 $\\alpha = 0$ ，因此 AX = 0。故它们同解， $n - r(A) = n - r(A^{T}A)$ 得到 $r(A) = r(A^{T}A)$",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "(2) 若 $AX = 0$ ，则 $\\overline{A}^T A X = 0$ 。若 $\\overline{A}^T A X = 0$ ，则 $\\overline{X}^T \\overline{A}^T A X = (\\overline{A X})^T A X = 0$ ，得到 $A X = 0$ 笔记该定理最重要的结论是最小二乘解一定存在。",
        "bbox": [
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        "page_idx": 43
    },
    {
        "type": "text",
        "text": "练习 $4.4A^{T}A$ 与 A 性质 (1) A 是 R 上任意 $s \\times n$ 矩阵，证明： $r(AA^{T}A) = r(A)$",
        "bbox": [
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        "page_idx": 43
    },
    {
        "type": "text",
        "text": "证明 (1) $r(AA^{T}A)\\leq r(A)$ ，而 $r(AA^{T}A)\\geq r(A^{T}AA^{T}A)=r(A^{T}A)=r(A)$",
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        "page_idx": 43
    },
    {
        "type": "text",
        "text": "4.3.2 正规方程组",
        "text_level": 2,
        "bbox": [
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        ],
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    },
    {
        "type": "text",
        "text": "定义 4.3 (最小二乘解)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 43
    },
    {
        "type": "text",
        "text": "对一个无解的线性方程组 (矛盾方程组) AX = b，若向量 $(x_{1}, \\cdots, x_{s}) \\in \\mathbb{R}^{s}$ 使得距离",
        "bbox": [
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        "page_idx": 43
    },
    {
        "type": "equation",
        "text": "$$\nd = \\sum_ {i = 1} ^ {n} (a _ {i 1} x _ {1} + \\dots + a _ {i s} x _ {s} - b _ {i}) ^ {2}\n$$",
        "text_format": "latex",
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        "page_idx": 43
    },
    {
        "type": "text",
        "text": "最小，则称该向量为最小二乘解。",
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        "page_idx": 43
    },
    {
        "type": "header",
        "text": "4.3 最小二乘解",
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    },
    {
        "type": "page_number",
        "text": "39",
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    },
    {
        "type": "image",
        "img_path": "images/fc5927c4943f5711971e1159904e69cf3096da10f51513e01b79f3f32935c532.jpg",
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    },
    {
        "type": "text",
        "text": "定理 4.8 (正规方程组)",
        "text_level": 2,
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            122,
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            287,
            101
        ],
        "page_idx": 44
    },
    {
        "type": "text",
        "text": "对于任一方程组 $AX = b$ ，最小二乘解一定存在，且 $X_0$ 是最小二乘解当且仅当 $A^T A X = A^T b$",
        "bbox": [
            115,
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        ],
        "page_idx": 44
    },
    {
        "type": "text",
        "text": "证明 设 $\\alpha_{1},\\cdots,\\alpha_{s}$ 是 A 的列向量，定义 $W=L(\\alpha_{1},\\cdots,\\alpha_{s})$ ， $X_{0}$ 是最小二乘解当且仅当 $AX_{0}$ 是 b 在上的投影，即 $(b-AX_{0})\\perp\\alpha_{i}$ ，因此",
        "bbox": [
            90,
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            175
        ],
        "page_idx": 44
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha_ {i} ^ {T} (b - A X _ {0}) \\Rightarrow A ^ {T} (b - A X _ {0}) = 0 \\Rightarrow A ^ {T} A X _ {0} = A ^ {T} b\n$$",
        "text_format": "latex",
        "bbox": [
            300,
            183,
            695,
            199
        ],
        "page_idx": 44
    },
    {
        "type": "text",
        "text": "定理 4.9 (正规方程组解的存在性)",
        "text_level": 2,
        "bbox": [
            122,
            214,
            369,
            229
        ],
        "page_idx": 44
    },
    {
        "type": "text",
        "text": "方程组 $A^T A X = A^T b$ 一定有解。",
        "bbox": [
            115,
            235,
            357,
            249
        ],
        "page_idx": 44
    },
    {
        "type": "text",
        "text": "证明 因为 $r(A^T A) = r(A)$ ，先将增广矩阵拆开得到 $r([A^T A, A^T b]) = r(A^T [A, b])$ ，用秩不等式得到 $r(A^T [A, b]) \\leq r(A^T) = r(A) = r(A^T A)$ ，而显然 $r([A^T A, A^T b]) \\geq r(A^T A)$ ，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。",
        "bbox": [
            89,
            267,
            905,
            321
        ],
        "page_idx": 44
    },
    {
        "type": "text",
        "text": "定理 4.10 (凸闭集最佳逼近元的唯一性)",
        "text_level": 2,
        "bbox": [
            122,
            331,
            413,
            346
        ],
        "page_idx": 44
    },
    {
        "type": "text",
        "text": "$\\mathcal{H}$ 是Hilbert空间， $M\\subset \\mathcal{H}$ 是凸闭集，则 $\\forall x\\in \\mathcal{H}$ 在 $M$ 中存在唯一最佳逼近。",
        "bbox": [
            114,
            351,
            685,
            365
        ],
        "page_idx": 44
    },
    {
        "type": "text",
        "text": "笔记 一般而言最小二乘解不唯一，除非是凸闭集。",
        "bbox": [
            87,
            385,
            458,
            400
        ],
        "page_idx": 44
    },
    {
        "type": "header",
        "text": "4.3 最小二乘解",
        "bbox": [
            788,
            55,
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        ],
        "page_idx": 44
    },
    {
        "type": "page_number",
        "text": "40",
        "bbox": [
            489,
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            964
        ],
        "page_idx": 44
    },
    {
        "type": "text",
        "text": "第 5 章 线性空间与线性变换",
        "text_level": 1,
        "bbox": [
            324,
            87,
            690,
            111
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "5.1 线性空间",
        "text_level": 2,
        "bbox": [
            94,
            146,
            240,
            164
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "5.1.1 线性空间的基本概念",
        "text_level": 2,
        "bbox": [
            92,
            184,
            332,
            200
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "定义 5.1 (线性空间)",
        "text_level": 2,
        "bbox": [
            122,
            217,
            270,
            231
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "$\\mathbb{P}$ 是一个数域， $V$ 是非空集合， $V$ 上定义了二元运算 $+$ 和数乘，若这两种运算满足：",
        "bbox": [
            112,
            236,
            734,
            250
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "二元运算：(1) 交换 (2) 结合 (3) 零元 (4) 负元",
        "bbox": [
            139,
            256,
            487,
            269
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "\\- 数乘：(1)一元(2)结合 $(c_{1}c_{2})\\alpha = c_{1}(c_{2}\\alpha)(3)$ 向量分配 $(c_{1} + c_{2})\\alpha = c_{1}\\alpha + c_{2}\\alpha (4)$ 数分配 $c(\\alpha +\\beta) = c\\alpha +c\\beta$",
        "bbox": [
            139,
            273,
            880,
            307
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "则称 $V$ 是 $\\mathbb{P}$ 上的线性空间",
        "bbox": [
            115,
            311,
            312,
            325
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "5.1.2 线性相关、线性无关与向量组的秩",
        "text_level": 2,
        "bbox": [
            94,
            363,
            453,
            380
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "定义 5.2 (线性相关和线性无关)",
        "text_level": 2,
        "bbox": [
            122,
            395,
            354,
            411
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "$\\alpha_{1},\\cdots,\\alpha_{n}$ 为向量组，若存在不全为0的 $k_{1},\\cdots,k_{n}$ 使得",
        "bbox": [
            114,
            416,
            537,
            431
        ],
        "page_idx": 45
    },
    {
        "type": "equation",
        "text": "$$\nk _ {1} \\alpha_ {1} + k _ {2} \\alpha_ {2} + \\dots + k _ {n} \\alpha_ {n} = 0\n$$",
        "text_format": "latex",
        "bbox": [
            384,
            439,
            611,
            455
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "则称它们线性相关。若该等式成立当且仅当 $k_{1}=k_{2}=\\cdots=k_{n}=0$ ，则称线性无关。",
        "bbox": [
            114,
            462,
            734,
            476
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "练习 5.1 证明线性无关 (1) 幂零基: A 为 V 上的线性变换, $\\xi \\in V$ , 若 $A^{k-1}\\xi \\neq 0, A^{k}\\xi = 0$ , 证明 $\\xi, A\\xi, \\cdots, A^{k-1}\\xi$ 线性无关\n\n证明 设 $k_{0}\\xi + k_{1}A\\xi + \\cdots + k_{n-1}A^{k-1}\\xi = 0$ , 两侧同时作用 $A^{k-1}$ 得到 $k_{0}A^{k-1}\\xi = 0$ , 因此 $k_{0} = 0$ , 以此类推得到 $k_{0}, k_{1}, \\cdots, k_{n-1} = 0$",
        "bbox": [
            58,
            514,
            905,
            588
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "定理 5.1 (极大线性无关组的计算)",
        "text_level": 2,
        "bbox": [
            122,
            612,
            371,
            627
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "给定一列向量 $\\alpha_{1},\\cdots,\\alpha_{n}$ ，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": [
            112,
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            882,
            684
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "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",
        "bbox": [
            58,
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            584,
            737
        ],
        "page_idx": 45
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} \\alpha_ {1} = (1, - 1, 2, 4) \\\\ \\alpha_ {2} = (0, 3, 1, 2) \\\\ \\alpha_ {3} = (3, 0, 7, 1 4) \\\\ \\alpha_ {4} = (1, - 1, 2, 0) \\\\ \\alpha_ {5} = (2, 1, 5, 6) \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
            421,
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            851
        ],
        "page_idx": 45
    },
    {
        "type": "text",
        "text": "解 拼成一个矩阵，并做初等行变换：",
        "bbox": [
            90,
            85,
            359,
            99
        ],
        "page_idx": 46
    },
    {
        "type": "equation",
        "text": "$$\n\\left[\\begin{array}{c c c c c}1&0&3&1&2\\\\- 1&3&0&- 1&1\\\\2&1&7&2&5\\\\4&2&1 4&0&6\\end{array}\\right]\\rightarrow \\left[\\begin{array}{c c c c c}1&0&3&1&- 2\\\\0&1&1&0&1\\\\0&0&0&1&1\\\\0&0&0&0&0\\end{array}\\right]\n$$",
        "text_format": "latex",
        "bbox": [
            314,
            102,
            682,
            177
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "因此极大线性无关组为 $\\alpha_{1},\\alpha_{2},\\alpha_{4}$",
        "bbox": [
            92,
            181,
            337,
            196
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "定义 5.3 (向量组的秩)",
        "text_level": 2,
        "bbox": [
            124,
            223,
            285,
            237
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "$\\alpha_{1},\\cdots,\\alpha_{n}$ 为向量组，称其中极大线性无关组的向量个数为它们的秩。",
        "bbox": [
            114,
            243,
            628,
            258
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "定理 5.2 (源泉定理及其推论)",
        "text_level": 2,
        "bbox": [
            122,
            282,
            336,
            297
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "设 $\\alpha_{1},\\cdots,\\alpha_{r}$ 可以被 $\\beta_{1},\\cdots,\\beta_{s}$ 线性表出，r>s，则 $\\alpha_{1},\\cdots,\\alpha_{r}$ 线性相关。",
        "bbox": [
            114,
            303,
            668,
            318
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "5.1.3 线性空间同构",
        "text_level": 2,
        "bbox": [
            95,
            356,
            273,
            373
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "定义 5.4 (同构)",
        "bbox": [
            122,
            388,
            236,
            403
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "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": [
            115,
            407,
            764,
            424
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "定理 5.3 (有限维线性空间的同构映射)",
        "text_level": 2,
        "bbox": [
            122,
            447,
            403,
            463
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "$V$ 为 $\\mathbb{P}$ 上线性空间， $\\alpha_{1}, \\cdots, \\alpha_{n}$ 为 $V$ 的一组基，构造线性映射 $f$ ，将 $\\forall \\alpha \\in V$ 映射到其坐标，则 $f$ 为 $V \\to \\mathbb{P}^{n}$ 的同构映射",
        "bbox": [
            114,
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            882,
            502
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "定理 5.4 (用坐标研究向量组)",
        "text_level": 2,
        "bbox": [
            124,
            544,
            336,
            558
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "$\\alpha_{1},\\cdots,\\alpha_{n}$ 是 V 的一组基， $A_{n\\times s}$ ，向量组 $\\beta_{1},\\cdots,\\beta_{s}$ 满足",
        "bbox": [
            114,
            565,
            546,
            579
        ],
        "page_idx": 46
    },
    {
        "type": "equation",
        "text": "$$\n(\\beta_ {1}, \\dots , \\beta_ {s}) = (\\alpha_ {1}, \\dots , \\alpha_ {n}) A\n$$",
        "text_format": "latex",
        "bbox": [
            389,
            587,
            606,
            604
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "则 $\\dim(\\beta_{1},\\cdots,\\beta_{s})=r(A)$",
        "bbox": [
            115,
            609,
            309,
            625
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "证明 $A = [X_{1}, \\cdots, X_{s}]$ 即 $\\beta_{1}, \\cdots, \\beta_{s}$ 的坐标，因此 A 的列向量与 $\\beta_{1}, \\cdots, \\beta_{s}$ 等价，从而 $r(A) = \\dim(\\beta_{1}, \\cdots, \\beta_{s})$",
        "bbox": [
            90,
            645,
            900,
            662
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "推论 5.1 (线性无关情况)",
        "text_level": 2,
        "bbox": [
            122,
            671,
            302,
            687
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "$\\alpha_{1},\\cdots,\\alpha_{n}$ 线性无关， $(\\beta_{1},\\cdots,\\beta_{n})=(\\alpha_{1},\\cdots,\\alpha_{n})A$ ，则 $\\beta_{i}$ 线性无关当且仅当 A 可逆。",
        "bbox": [
            114,
            690,
            754,
            708
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "练习5.3同构的应用 $(1)\\beta_{1} = \\alpha_{1} + \\alpha_{2},\\beta_{2} = \\alpha_{1} + \\alpha_{3},\\beta_{3} = \\alpha_{1} + \\alpha_{2}$ ，证明： $\\alpha_{1},\\alpha_{2},\\alpha_{3}$ 和 $\\beta_{1},\\beta_{2},\\beta_{3}$ 等价",
        "bbox": [
            60,
            746,
            838,
            760
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "(2) 证明 n 为奇数时， $\\alpha_{1},\\cdots,\\alpha_{n}$ 线性无关当且仅当 $\\alpha_{1}+\\alpha_{2},\\cdots,\\alpha_{n-1}+\\alpha_{n},a_{n}+\\alpha_{1}$ 线性无关",
        "bbox": [
            122,
            766,
            820,
            780
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "(3) 设 $\\beta_{1} = \\alpha_{2} + \\cdots + \\alpha_{n}, \\beta_{2} = \\alpha_{1} + \\alpha_{3} + \\cdots + \\alpha_{n}, \\beta_{n} = \\alpha_{1} + \\cdots + \\alpha_{n-1}$ ，证明 $\\alpha_{1}, \\cdots, \\alpha_{n}$ 和 $\\beta_{1}, \\cdots, \\beta_{n}$ 有相同的秩",
        "bbox": [
            87,
            784,
            905,
            815
        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "证明 (1) 如下同构映射至坐标, 显然 $A$ 可逆, 因此等价",
        "bbox": [
            90,
            820,
            500,
            834
        ],
        "page_idx": 46
    },
    {
        "type": "equation",
        "text": "$$\n(\\beta_ {1}, \\beta_ {2}, \\beta_ {3}) = (\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}) \\left[ \\begin{array}{c c c} 1 & 1 & 1 \\\\ 1 & & 1 \\\\ 0 & 1 \\end{array} \\right] := (\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}) A\n$$",
        "text_format": "latex",
        "bbox": [
            290,
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            897
        ],
        "page_idx": 46
    },
    {
        "type": "header",
        "text": "5.1 线性空间",
        "bbox": [
            806,
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        ],
        "page_idx": 46
    },
    {
        "type": "page_number",
        "text": "42",
        "bbox": [
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        ],
        "page_idx": 46
    },
    {
        "type": "text",
        "text": "(2) 将后者设为 $\\beta_{1}, \\cdots, \\beta_{n}$ ，如下线性表出",
        "bbox": [
            122,
            85,
            433,
            99
        ],
        "page_idx": 47
    },
    {
        "type": "equation",
        "text": "$$\n(\\beta_ {1}, \\dots , \\beta_ {n}) = (\\alpha_ {1}, \\dots , \\alpha_ {n}) \\left[ \\begin{array}{c c c c c} 1 & & & & 1 \\\\ 1 & 1 & & & \\\\ & 1 & \\ddots & & \\\\ & & \\ddots & 1 & \\\\ & & & 1 & 1 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            312,
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            205
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "用 Gauss 消元法可看出 $n$ 为奇数时矩阵满秩，因此等价。",
        "bbox": [
            90,
            210,
            509,
            225
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "(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。",
        "bbox": [
            121,
            229,
            635,
            244
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "5.1.4 基、坐标",
        "text_level": 2,
        "bbox": [
            92,
            266,
            233,
            282
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "定义 5.5 (基与坐标)",
        "text_level": 2,
        "bbox": [
            122,
            298,
            270,
            313
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "V 是线性空间， $\\epsilon_{1},\\cdots,\\epsilon_{n}$ 是 V 中线性无关的向量组，若 $\\forall\\alpha\\in V$ 可被该向量组线性表出，则称该向量组为 V 的一组基。且若",
        "bbox": [
            112,
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        ],
        "page_idx": 47
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha = (\\epsilon_ {1}, \\dots , \\epsilon_ {n}) X\n$$",
        "text_format": "latex",
        "bbox": [
            428,
            356,
            566,
            370
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "则称 X 为 $\\alpha$ 在 $\\epsilon_{1},\\cdots,\\epsilon_{n}$ 基下的坐标",
        "bbox": [
            114,
            382,
            394,
            398
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "笔记 坐标的计算：直接根据 $\\alpha = (\\epsilon_{1}, \\cdots, \\epsilon_{n})X$ 解线性方程组 (或者用初等变换求逆)",
        "bbox": [
            60,
            417,
            707,
            432
        ],
        "page_idx": 47
    },
    {
        "type": "equation",
        "text": "$$\nX = (\\epsilon_ {1}, \\dots , \\epsilon_ {n}) ^ {- 1} \\alpha\n$$",
        "text_format": "latex",
        "bbox": [
            418,
            439,
            578,
            457
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "练习 5.4 证明基 给定 $\\alpha_{1}=(1,1,\\cdots,1,1)$ , $\\alpha_{2}=(1,1,\\cdots,1,0)$ , $\\cdots$ , $\\alpha_{n}=(1,0,\\cdots,0)$ ，证明其是 $P^{n}$ 的一组基证明 当成列向量排成一个矩阵，满秩即可",
        "bbox": [
            58,
            482,
            890,
            514
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "练习 5.5 坐标的计算 (1) 已知 $\\epsilon_{1}, \\epsilon_{2}, \\epsilon_{3}$ 是 $P^{3}$ 的一组基，计算 $\\alpha = (1, 2, 1)^{T}$ 在这组基下的坐标",
        "bbox": [
            58,
            538,
            773,
            552
        ],
        "page_idx": 47
    },
    {
        "type": "equation",
        "text": "$$\n\\epsilon_ {1} = (1, 1, 1) ^ {T}, \\epsilon_ {2} = (1, 1, - 1) ^ {T}, \\epsilon_ {3} = (1, - 1, - 1) ^ {T}\n$$",
        "text_format": "latex",
        "bbox": [
            315,
            558,
            678,
            577
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "解(1)只需要解方程组 $\\alpha = k_{1}\\epsilon_{1} + k_{2}\\epsilon_{2} + k_{3}\\epsilon_{3}$ ，即 $[\\epsilon_1,\\epsilon_2,\\epsilon_3]X = \\alpha$",
        "bbox": [
            87,
            583,
            589,
            599
        ],
        "page_idx": 47
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c c} 1 & 1 & 1 \\\\ 1 & 1 & - 1 \\\\ 1 & - 1 & - 1 \\end{array} \\right] X = \\left[ \\begin{array}{c} 1 \\\\ 2 \\\\ 1 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            393,
            601,
            603,
            663
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "解得 $k_{1} = 1, k_{2} = \\frac{1}{2}, k_{3} = -\\frac{1}{2}$",
        "bbox": [
            90,
            664,
            310,
            692
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "练习5.6难度进阶(1)求非零向量 $\\alpha \\in \\mathbb{P}^4$ ，使得其在下面两组基中有相同的坐标",
        "bbox": [
            58,
            707,
            680,
            722
        ],
        "page_idx": 47
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} \\epsilon_ {1} = (1, 0, 0, 0) ^ {T} \\\\ \\epsilon_ {2} = (0, 1, 0, 0) ^ {T} \\\\ \\epsilon_ {3} = (0, 0, 1, 0) ^ {T} \\\\ \\epsilon_ {4} = (0, 0, 0, 1) ^ {T} \\end{array} \\right., \\quad \\left\\{ \\begin{array}{l} \\eta_ {1} = (2, 1, - 1, 1) ^ {T} \\\\ \\eta_ {2} = (0, 3, 1, 0) ^ {T} \\\\ \\eta_ {3} = (5, 3, 2, 1) ^ {T} \\\\ \\eta_ {4} = (6, 6, 1, 3) ^ {T} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
            327,
            728,
            653,
            815
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "解 (1) 设 $\\alpha$ 在第一组基下的坐标为 $(x_{1}, x_{2}, x_{3}, x_{4})^{T}$ , 则由于同坐标",
        "bbox": [
            87,
            820,
            581,
            835
        ],
        "page_idx": 47
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha = x _ {1} \\eta_ {1} + x _ {2} \\eta_ {2} + x _ {3} \\eta_ {3} + x _ {4} \\eta_ {4}\n$$",
        "text_format": "latex",
        "bbox": [
            379,
            846,
            613,
            860
        ],
        "page_idx": 47
    },
    {
        "type": "page_number",
        "text": "43",
        "bbox": [
            489,
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            507,
            964
        ],
        "page_idx": 47
    },
    {
        "type": "header",
        "text": "5.1 线性空间",
        "bbox": [
            805,
            55,
            905,
            68
        ],
        "page_idx": 47
    },
    {
        "type": "text",
        "text": "解线性方程组 $\\left[\\eta_{1},\\eta_{2},\\eta_{3},\\eta_{4}\\right]X=\\alpha$ 即可 (注意将 $\\alpha$ 减到左边):",
        "bbox": [
            90,
            85,
            539,
            99
        ],
        "page_idx": 48
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{r r r r} 1 & 0 & 5 & 6 \\\\ 1 & 2 & 3 & 6 \\\\ - 1 & 1 & 1 & 1 \\\\ 1 & 0 & 1 & 2 \\end{array} \\right] X = 0\n$$",
        "text_format": "latex",
        "bbox": [
            406,
            104,
            594,
            179
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "解得 $\\alpha = k(1,1,1,-1)$",
        "bbox": [
            90,
            183,
            257,
            197
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "5.1.5 坐标变换",
        "text_level": 2,
        "bbox": [
            94,
            219,
            233,
            236
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "定义 5.6 (过渡矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            252,
            270,
            267
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "给定 $(\\epsilon_{1},\\cdots,\\epsilon_{n}),(\\eta_{1},\\cdots,\\eta_{n})$ 两组基，若存在矩阵T满足：",
        "bbox": [
            112,
            272,
            552,
            287
        ],
        "page_idx": 48
    },
    {
        "type": "equation",
        "text": "$$\n(\\epsilon_ {1}, \\dots , \\epsilon_ {n}) = (\\eta_ {1}, \\dots , \\eta_ {n}) T\n$$",
        "text_format": "latex",
        "bbox": [
            393,
            294,
            605,
            311
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "则称 T 为 $\\eta_{1},\\cdots,\\eta_{n}$ 到 $\\epsilon_{1},\\cdots,\\epsilon_{n}$ 的过渡矩阵 (不要搞反)",
        "bbox": [
            114,
            318,
            534,
            332
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "笔记 过渡矩阵的计算：直接根据 $(\\epsilon_{1},\\cdots,\\epsilon_{n})=(\\eta_{1},\\cdots,\\eta_{n})T$ 得到",
        "bbox": [
            87,
            353,
            581,
            368
        ],
        "page_idx": 48
    },
    {
        "type": "equation",
        "text": "$$\nT = (\\eta_ {1}, \\dots , \\eta_ {n}) ^ {- 1} (\\epsilon_ {1}, \\dots , \\epsilon_ {n})\n$$",
        "text_format": "latex",
        "bbox": [
            381,
            376,
            611,
            392
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "用初等变换求 $A^{-1}B$ 的方式直接求 T 即可。",
        "bbox": [
            90,
            399,
            411,
            414
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "定理 5.5 (坐标变换)",
        "text_level": 2,
        "bbox": [
            122,
            424,
            270,
            439
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "$V$ 是 $n$ 维线性空间， $\\eta_1,\\dots ,\\eta_n$ 和 $\\epsilon_1,\\dots ,\\epsilon_n$ 是空间两组基，且 $(\\eta_{1},\\dots ,\\eta_{n}) = (\\epsilon_{1},\\dots ,\\epsilon_{n})T$ ，向量 $x\\in V$ 在 $\\epsilon_i,\\eta_i$ 下的坐标分别为 $X,Y$ ，则",
        "bbox": [
            112,
            444,
            885,
            478
        ],
        "page_idx": 48
    },
    {
        "type": "equation",
        "text": "$$\nX = T Y\n$$",
        "text_format": "latex",
        "bbox": [
            463,
            481,
            532,
            495
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "证明 $x=(\\epsilon_{1},\\cdots,\\epsilon_{n})X=(\\eta_{1},\\cdots,\\eta_{n})Y=(\\epsilon_{1},\\cdots,\\epsilon_{n})TY$ 即可。",
        "bbox": [
            90,
            514,
            566,
            529
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "笔记 坐标变换用荡秋千记忆，即 $(\\eta_{1},\\cdots,\\eta_{n})=(\\epsilon_{1},\\cdots,\\epsilon_{n})T$ ，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": [
            89,
            532,
            907,
            565
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "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 $\\xi = (1, 0, 0, -1)$ 在 $\\alpha_{1}, \\alpha_{2}, \\alpha_{3}, \\alpha_{4}$ 下的坐标",
        "bbox": [
            58,
            587,
            905,
            602
        ],
        "page_idx": 48
    },
    {
        "type": "equation",
        "text": "$$\n\\begin{array}{l l l l} \\alpha_ {1} = (1, 1, 1, 1) & \\alpha_ {2} = (1, 1, - 1, - 1) & \\alpha_ {3} = (1, - 1, 1, - 1) & \\alpha_ {4} = (1, - 1, - 1, 1) \\\\ \\beta_ {1} = (1, 1, 0, 1) & \\beta_ {2} = (2, 1, 3, 1) & \\beta_ {3} = (1, 1, 0, 0) & \\beta_ {4} = (0, 1, - 1, - 1) \\end{array}\n$$",
        "text_format": "latex",
        "bbox": [
            208,
            609,
            784,
            643
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "解(1)先取标准基 $\\epsilon_{1}, \\epsilon_{2}, \\epsilon_{3}, \\epsilon_{4}$ ，设",
        "bbox": [
            89,
            650,
            342,
            664
        ],
        "page_idx": 48
    },
    {
        "type": "equation",
        "text": "$$\n(\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}, \\alpha_ {4}) = (\\epsilon_ {1}, \\epsilon_ {2}, \\epsilon_ {3}, \\epsilon_ {4}) A, (\\beta_ {1}, \\beta_ {2}, \\beta_ {3}, \\beta_ {4}) = (\\epsilon_ {1}, \\epsilon_ {2}, \\epsilon_ {3}, \\epsilon_ {4}) B\n$$",
        "text_format": "latex",
        "bbox": [
            258,
            673,
            739,
            688
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "则",
        "bbox": [
            92,
            695,
            112,
            709
        ],
        "page_idx": 48
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{r r r r} 1 & 1 & 1 & 1 \\\\ 1 & 1 & - 1 & - 1 \\\\ 1 & - 1 & 1 & - 1 \\\\ 1 & - 1 & - 1 & 1 \\end{array} \\right], B = \\left[ \\begin{array}{r r r r} 1 & 2 & 1 & 0 \\\\ 1 & 1 & 1 & 1 \\\\ 0 & 3 & 0 & - 1 \\\\ 1 & 1 & 0 & - 1 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            305,
            707,
            687,
            782
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "由此得到",
        "bbox": [
            92,
            791,
            163,
            804
        ],
        "page_idx": 48
    },
    {
        "type": "equation",
        "text": "$$\n(\\beta_ {1}, \\beta_ {2}, \\beta_ {3}, \\beta_ {4}) = (\\epsilon_ {1}, \\epsilon_ {2}, \\epsilon_ {3}, \\epsilon_ {4}) B = (\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}, \\alpha_ {4}) A ^ {- 1} B\n$$",
        "text_format": "latex",
        "bbox": [
            292,
            810,
            705,
            826
        ],
        "page_idx": 48
    },
    {
        "type": "header",
        "text": "5.1 线性空间",
        "bbox": [
            806,
            55,
            905,
            68
        ],
        "page_idx": 48
    },
    {
        "type": "page_number",
        "text": "44",
        "bbox": [
            489,
            954,
            509,
            964
        ],
        "page_idx": 48
    },
    {
        "type": "text",
        "text": "综上得到过渡矩阵为 (具体计算时用行初等变换: $(A|B) \\rightarrow (E|A^{-1}B)$ )",
        "bbox": [
            90,
            85,
            608,
            99
        ],
        "page_idx": 49
    },
    {
        "type": "equation",
        "text": "$$\nA ^ {- 1} B = \\frac {1}{4} \\left[ \\begin{array}{r r r r} 3 & 7 & 2 & - 1 \\\\ 1 & - 1 & 2 & 3 \\\\ - 1 & 3 & 0 & - 1 \\\\ 1 & - 1 & 0 & - 1 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            373,
            105,
            618,
            178
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "(2) $\\xi$ 在 $(\\epsilon_1, \\epsilon_2, \\epsilon_3, \\epsilon_4)$ 下的坐标即本身，因此在 $(\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4)$ 下坐标为 $A^{-1}\\xi$ (具体计算时也和 $(A|B)$ 摆在一起)，结果为 $(0, \\frac{1}{2}, \\frac{1}{2}, 0)^T$",
        "bbox": [
            90,
            181,
            905,
            222
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "5.1.6 和与交、维数定理",
        "text_level": 2,
        "bbox": [
            94,
            243,
            312,
            259
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "定义 5.7 (线性空间和与交)",
        "text_level": 2,
        "bbox": [
            122,
            275,
            319,
            290
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "$V_{1}, V_{2}$ 为线性空间，则",
        "bbox": [
            115,
            296,
            284,
            309
        ],
        "page_idx": 49
    },
    {
        "type": "equation",
        "text": "$$\nV _ {1} \\cap V _ {2} := \\{\\alpha : \\alpha \\in V _ {1}, \\alpha \\in V _ {2} \\}, V _ {1} + V _ {2} := \\{\\alpha = \\alpha_ {1} + \\alpha_ {2}: \\alpha_ {1} \\in V _ {1}, \\alpha_ {2} \\in V _ {2} \\}\n$$",
        "text_format": "latex",
        "bbox": [
            216,
            318,
            778,
            334
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "且显然 $V_{1} \\cap V_{2}, V_{1} + V_{2}$ 都是子空间。",
        "bbox": [
            115,
            341,
            386,
            356
        ],
        "page_idx": 49
    },
    {
        "type": "image",
        "img_path": "images/6704a6fa582ccd0f7b0270cfe13b761ae03c428e9e7a8137480049d96a04f59a.jpg",
        "image_caption": [],
        "image_footnote": [],
        "content": "",
        "bbox": [
            58,
            370,
            78,
            390
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "笔记 $V_{1} \\cup V_{2}$ 由于运算不封闭，一般而言不是子空间。注意分别 $V_{1} \\cup V_{2}$ 和 $V_{1} + V_{2}$",
        "bbox": [
            89,
            376,
            695,
            391
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "练习 5.8 和与交基本计算 (1) 已知 $V_{1} = L(\\alpha_{1}, \\alpha_{2}, \\alpha_{3}), V_{2} = L(\\beta_{1}, \\beta_{2})$ ，其中",
        "bbox": [
            58,
            413,
            642,
            428
        ],
        "page_idx": 49
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha_ {1} = \\left( \\begin{array}{c} 1 \\\\ 2 \\\\ 1 \\\\ 0 \\end{array} \\right), \\alpha_ {2} = \\left( \\begin{array}{c} - 1 \\\\ 1 \\\\ 1 \\\\ 1 \\end{array} \\right), \\alpha_ {3} = \\left( \\begin{array}{c} 0 \\\\ 3 \\\\ 2 \\\\ 1 \\end{array} \\right), \\beta_ {1} = \\left( \\begin{array}{c} 2 \\\\ - 1 \\\\ 0 \\\\ 1 \\end{array} \\right), \\beta_ {2} = \\left( \\begin{array}{c} 1 \\\\ - 1 \\\\ 3 \\\\ 7 \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            218,
            432,
            773,
            506
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "求 $V_{1} + V_{2}, V_{1} \\cap V_{2}$ 的基与维数",
        "bbox": [
            90,
            512,
            321,
            526
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "(2) 已知 $W_{1} = L(\\alpha_{1}, \\alpha_{2}, \\alpha_{3}), W_{2} = L(\\beta_{1}, \\beta_{2})$ ，求 $W_{1} \\cap W_{2}, W_{1} + W_{2}$ 的基与维数。",
        "bbox": [
            122,
            530,
            722,
            545
        ],
        "page_idx": 49
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha_ {1} = \\left( \\begin{array}{r} 1 \\\\ 2 \\\\ - 1 \\\\ - 3 \\end{array} \\right), \\alpha_ {2} = \\left( \\begin{array}{r} - 1 \\\\ - 1 \\\\ 2 \\\\ 1 \\end{array} \\right), \\alpha_ {3} = \\left( \\begin{array}{r} - 1 \\\\ - 3 \\\\ 0 \\\\ 5 \\end{array} \\right), \\beta_ {1} = \\left( \\begin{array}{r} - 1 \\\\ 0 \\\\ 4 \\\\ 2 \\end{array} \\right), \\beta_ {2} = \\left( \\begin{array}{r} 0 \\\\ 5 \\\\ 9 \\\\ - 1 4 \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            201,
            549,
            791,
            623
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "解 (1) $V_{1}+V_{2}=L(\\alpha_{1},\\alpha_{2},\\alpha_{3},\\beta_{1},\\beta_{2})$ ，因此只需要排成列，做初等变换：",
        "bbox": [
            87,
            627,
            615,
            643
        ],
        "page_idx": 49
    },
    {
        "type": "equation",
        "text": "$$\n\\left( \\begin{array}{c c c c c} 1 & - 1 & 0 & 2 & 1 \\\\ 2 & 1 & 3 & - 1 & - 1 \\\\ 1 & 1 & 2 & 0 & 3 \\\\ 0 & 1 & 1 & 1 & 7 \\end{array} \\right) \\to \\left( \\begin{array}{c c c c c} 1 & 0 & 1 & 0 & - 1 \\\\ 0 & 1 & 1 & 0 & 4 \\\\ 0 & 0 & 0 & 1 & 3 \\\\ 0 & 0 & 0 & 0 & 0 \\end{array} \\right) := A\n$$",
        "text_format": "latex",
        "bbox": [
            284,
            648,
            715,
            721
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "因此维数为3，基为 $\\alpha_{1},\\alpha_{2},\\beta_{1}$",
        "bbox": [
            92,
            727,
            312,
            740
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "任取 $\\alpha \\in V_1 \\cap V_2$ ，设 $\\alpha = x_1\\alpha_1 + x_2\\alpha_2 + x_3\\alpha_3 = y_1\\beta_1 + y_2\\beta_2$ ，因此系数需要满足方程组：",
        "bbox": [
            122,
            745,
            786,
            759
        ],
        "page_idx": 49
    },
    {
        "type": "equation",
        "text": "$$\nx _ {1} \\alpha_ {1} + x _ {2} \\alpha_ {2} + x _ {3} \\alpha_ {3} - y _ {1} \\beta_ {1} - y _ {2} \\beta_ {2} = 0\n$$",
        "text_format": "latex",
        "bbox": [
            352,
            769,
            645,
            783
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "设 $x_{4} = -y_{1}, x_{5} = y_{2}$ ，即求解 $AX = [\\alpha_{1}, \\alpha_{2}, \\alpha_{3}, \\beta_{1}, \\beta_{2}]X = 0$ ，解得基础解系为",
        "bbox": [
            90,
            791,
            675,
            804
        ],
        "page_idx": 49
    },
    {
        "type": "equation",
        "text": "$$\n\\eta_ {1} = \\left( \\begin{array}{c} - 1 \\\\ - 1 \\\\ 1 \\\\ 0 \\\\ 0 \\end{array} \\right), \\eta_ {2} = \\left( \\begin{array}{c} 1 \\\\ - 4 \\\\ 0 \\\\ - 3 \\\\ 1 \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            384,
            810,
            608,
            904
        ],
        "page_idx": 49
    },
    {
        "type": "page_number",
        "text": "45",
        "bbox": [
            489,
            954,
            507,
            964
        ],
        "page_idx": 49
    },
    {
        "type": "header",
        "text": "5.1 线性空间",
        "bbox": [
            806,
            55,
            905,
            68
        ],
        "page_idx": 49
    },
    {
        "type": "text",
        "text": "对应 $V_{1} \\cap V_{2}$ 的基础解系为：",
        "bbox": [
            90,
            85,
            300,
            99
        ],
        "page_idx": 50
    },
    {
        "type": "equation",
        "text": "$$\n- \\alpha_ {1} - \\alpha_ {2} + \\alpha_ {3} = (0, 0, 0, 0) ^ {T}, \\alpha_ {1} - 4 \\alpha_ {2} = (5, - 2, - 3, - 4) ^ {T}\n$$",
        "text_format": "latex",
        "bbox": [
            282,
            108,
            712,
            123
        ],
        "page_idx": 50
    },
    {
        "type": "text",
        "text": "只取非零的，故只能为 $k(5, -2, -3, -4)$ ，维数为1，基即 $\\alpha_{1} - 4\\alpha_{2}$ 。",
        "bbox": [
            90,
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        "page_idx": 50
    },
    {
        "type": "text",
        "text": "练习 5.9 和与并的区别 (1) 设 $V_{1}, V_{2}$ 是 P 上线性空间 V 的两个非平凡子空间，证明： $\\exists \\alpha \\in V$ 使得 $\\alpha \\notin V_{1}, \\alpha \\notin V_{2}$ (2) $V_{1}, \\cdots, V_{s}$ 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 $V_{1}, \\cdots, V_{s}$ 中的任意一个。",
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    },
    {
        "type": "text",
        "text": "(3) $V_{1}, V_{2}$ 是两个子空间，证明： $V_{1} + V_{2} = V_{1} \\cup V_{2}$ 当且仅当 $V_{1} \\subseteq V_{2}$ 或者 $V_{2} \\subseteq V_{1}$",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "证明 (1) 由于 $V_{1}$ 非平凡， $\\exists \\alpha \\notin V_{1}$ ，若 $\\alpha \\notin V_{2}$ ，则结论成立；否则 $\\alpha \\in V_{2}$ ，由于 $V_{2}$ 非平凡，因此 $\\exists \\beta \\notin V_{2}$ ，若 $\\beta \\notin V_{1}$ ，则结论成立；否则 $\\beta \\in V_{1}$ ，此时有",
        "bbox": [
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    },
    {
        "type": "equation",
        "text": "$$\n\\alpha \\notin V _ {1}, \\beta \\in V _ {1}, \\alpha \\in V _ {2}, \\beta \\notin V _ {2}\n$$",
        "text_format": "latex",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "因此 $\\gamma = \\alpha +\\beta \\notin V_1,\\gamma \\notin V_2$",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "text",
        "text": "(2) 用归纳法",
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        "page_idx": 50
    },
    {
        "type": "text",
        "text": "(3) 右推左显然。左推右：设 $V_{1} + V_{2} = V_{1} \\cup V_{2}$ ，但 $V_{1} \\notin V_{2}$ ，则 $\\exists \\alpha_{1} \\in V_{1}, \\alpha_{1} \\notin V_{2}$ ，但",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "equation",
        "text": "$$\n\\forall \\alpha_ {2} \\in V _ {2}, \\alpha = \\alpha_ {1} + \\alpha_ {2} \\in V _ {1} + V _ {2}\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "text",
        "text": "由于 $V_{1} + V_{2} = V_{1}\\cup V_{2}$ ，因此 $\\alpha$ 至少属于 $V_{1},V_{2}$ 中的一个，显然 $\\alpha \\notin V_2$ (根据加法封闭)，因此 $\\alpha \\in V_{1}$ ，故",
        "bbox": [
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        ],
        "page_idx": 50
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha_ {2} = \\alpha - \\alpha_ {1} \\in V _ {1}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 50
    },
    {
        "type": "text",
        "text": "综上得到 $V_{2} \\subseteq V_{1}$",
        "bbox": [
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        ],
        "page_idx": 50
    },
    {
        "type": "text",
        "text": "定理 5.6 (维数公式)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 50
    },
    {
        "type": "text",
        "text": "设 $V_{1}, V_{2}$ 是有限维线性空间 $V$ 的两个子空间，则",
        "bbox": [
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        ],
        "page_idx": 50
    },
    {
        "type": "equation",
        "text": "$$\n\\dim V _ {1} + \\dim V _ {2} = \\dim (V _ {1} + V _ {2}) + \\dim (V _ {1} \\cap V _ {2})\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "text",
        "text": "证明 设小扩大。设 $V_{1} \\cap V_{2}$ 的基为 $\\alpha_{1}, \\cdots, \\alpha_{s}$ ， $V_{1}, V_{2}$ 的基分别为：",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} \\alpha_ {1}, \\dots , \\alpha_ {s}, \\beta_ {s + 1}, \\dots , \\beta_ {m} \\\\ \\alpha_ {1}, \\dots , \\alpha_ {s}, \\gamma_ {s + 1}, \\dots , \\gamma_ {n} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "text",
        "text": "则 $V_{1} + V_{2} = L(\\alpha_{1}, \\cdots, \\alpha_{s}, \\beta_{s+1}, \\cdots, \\beta_{m}, \\gamma_{s+1}, \\cdots, \\gamma_{n})$ ，下面证明上面的向量全部线性无关。不妨设",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "equation",
        "text": "$$\nk _ {1} \\alpha_ {1} + \\dots + k _ {s} \\alpha_ {s} + k _ {s + 1} \\beta_ {s + 1} + \\dots + k _ {n} \\beta_ {n} + l _ {s + 1} \\gamma_ {s + 1} + \\dots + l _ {n} \\gamma_ {n} = 0\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "text",
        "text": "将 $l_{i}\\gamma_{i}$ 的部分移到右侧，此时左侧线性组合属于 $V_{1}$ ，右侧属于 $V_{2}$ ，因此都属于 $V_{1} \\cap V_{2}$ 。根据左侧得到：",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "equation",
        "text": "$$\nk _ {1} \\alpha_ {1} + \\dots + k _ {s} \\alpha_ {s} + k _ {s + 1} \\beta_ {s + 1} + \\dots + k _ {n} \\beta_ {n} \\in V _ {1} \\cap V _ {2} \\Rightarrow k _ {s + 1} = k _ {s + 2} = \\dots = k _ {n} = 0 \\Rightarrow k _ {1} = \\dots = k _ {s} = 0\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "text",
        "text": "因此左侧系数全为0，同理右侧系数也全为0",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "text",
        "text": "练习5.10维数公式相关(1)经典： $V_{1}, V_{2}$ 是有限维线性空间 $V$ 的子空间， $\\dim (V_1 + V_2) = \\dim (V_1 \\cap V_2) + 1$ ，证明：要么 $V_{1} \\subseteq V_{2}$ ，要么 $V_{2} \\subseteq V_{1}$",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "text",
        "text": "证明 (1) 根据维数公式得到",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "equation",
        "text": "$$\n\\dim (V _ {1}) + \\dim (V _ {2}) = \\dim (V _ {1} + V _ {2}) + \\dim (V _ {1} \\cap V _ {2}) = 2 \\dim (V _ {1} \\cap V _ {2}) + 1\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "text",
        "text": "等价于",
        "bbox": [
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        ],
        "page_idx": 50
    },
    {
        "type": "equation",
        "text": "$$\n\\left(\\dim V _ {1} - \\dim (V _ {1} \\cap V _ {2})\\right) + \\left(\\dim V _ {2} - \\dim (V _ {1} \\cap V _ {2})\\right) = 1\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 50
    },
    {
        "type": "text",
        "text": "由于都是整数，只能一个0，一个1，因此结论成立。",
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        ],
        "page_idx": 50
    },
    {
        "type": "header",
        "text": "5.1 线性空间",
        "bbox": [
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        ],
        "page_idx": 50
    },
    {
        "type": "page_number",
        "text": "46",
        "bbox": [
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        ],
        "page_idx": 50
    },
    {
        "type": "image",
        "img_path": "images/cee4c1078050101154e1beeff5719a1fa75de396fb1f2dfef978054d78e0b5d3.jpg",
        "image_caption": [],
        "image_footnote": [],
        "content": "",
        "bbox": [
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        "page_idx": 51
    },
    {
        "type": "text",
        "text": "5.1.7 直和理论",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "定义 5.8 (直和)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "$V_{1}, V_{2}$ 是线性空间 $V$ 的子空间，若 $V_{1} \\cap V_{2} = \\{0\\}$ ，则记 $V_{1} \\oplus V_{2} := V_{1} + V_{2}$",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "定义 5.9 (补空间)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "若 $V = V_{1} \\oplus V_{2}$ ，则称 $V_{2}$ 是 $V_{1}$ 的补空间。",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "笔记 补空间一般是不唯一的，因为 $\\alpha_{1},\\cdots,\\alpha_{m}$ 扩充为 $\\alpha_{1},\\cdots,\\alpha_{n}$ 的方式不唯一。",
        "bbox": [
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        "page_idx": 51
    },
    {
        "type": "text",
        "text": "定理 5.7 (直和等价条件)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "$W = V_{1} \\oplus V_{2}$ 当且仅当",
        "bbox": [
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            292
        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "\\- $W$ 中任意向量 $\\alpha = \\alpha_{1} + \\alpha_{2},\\alpha_{1}\\in V_{1},\\alpha_{2}\\in V_{2}$ 唯一表出",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "\\- $W$ 中零向量被 $V_{1}, V_{2}$ 中向量唯一表出: $0 = 0 + 0$",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "\\- $\\dim W = \\dim V_1 + \\dim V_2$",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "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 $V = M_{n}(P)$ ， $V_{1}$ 表示所有对称矩阵组成的集合， $V_{2}$ 表示所有反对称矩阵组成的集合，证明 $V_{1}, V_{2}$ 是子空间且 $V = V_{1} \\oplus V_{2}$",
        "bbox": [
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        "page_idx": 51
    },
    {
        "type": "text",
        "text": "证明 (1) 显然关于加法和数乘封闭，因此是子空间。首先证明 $V = V_{1} + V_{2}$ ，由于矩阵 $A$ 都可以写为：",
        "bbox": [
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        "page_idx": 51
    },
    {
        "type": "equation",
        "text": "$$\nA = \\frac {A + A ^ {T}}{2} + \\frac {A ^ {T} - A}{2}\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 51
    },
    {
        "type": "text",
        "text": "从而 $V = V_{1} + V_{2}$ 。",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "证明 $V = V_{1} \\oplus V_{2}$ ：可用 $V_{1} \\cap V_{2} = \\{\\}$ ，此时取 $A \\in V_{1} \\cap V_{2}$ ，则 $A = A^{T} = -A^{T}$ ，得到 $A^{T} = A = O$ ，因此为直和。也可以用 $O$ 分解唯一，设 $O = B + C = B^{T} - C^{T}$ ，而 $O^{T} = B^{T} + C^{T}$ ，因此 $B = C = O$ 。",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "定理 5.8 (多空间直和等价条件)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "$V_{1},\\cdots,V_{s}$ 是 W 的子空间，且 $W=V_{1}+V_{2}+\\cdots+V_{s}$ ，则 $W=V_{1}\\oplus V_{2}\\oplus\\cdots\\oplus V_{s}$ 当且仅当",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "\\- $W$ 中零向量分解唯一",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "\\- $W$ 中任意向量 $\\alpha$ 分解唯一",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "equation",
        "text": "$$\nV _ {i} \\cap \\sum_ {j \\neq i} V _ {j} = \\{0 \\}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "equation",
        "text": "$$\n\\bullet \\dim W = \\dim V _ {1} + \\dots + \\dim V _ {s}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "证明 (3) 只证明第三条。第三条推直和：若",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "equation",
        "text": "$$\n0 = \\alpha_ {1} + \\dots + \\alpha_ {n} \\Rightarrow \\alpha_ {i} = \\alpha_ {1} + \\dots + \\alpha_ {i - 1} + \\alpha_ {i + 1} + \\dots + \\alpha_ {n}\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 51
    },
    {
        "type": "text",
        "text": "由于 $V_{i} \\cap \\sum_{j \\neq i} V_{j} = \\{0\\}$ ，因此 $\\alpha_{k} = 0, \\forall k = 1, 2, \\cdots, s$ ，从而是直和",
        "bbox": [
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        "page_idx": 51
    },
    {
        "type": "text",
        "text": "直和推第三条：若 $\\alpha \\in V_{i} \\cap \\sum_{j \\neq i} V_{j}$ ，则 $\\alpha \\in V_{i}$ ，且",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha = \\alpha_ {1} + \\dots + \\alpha_ {i - 1} + \\alpha_ {i + 1} + \\dots + \\alpha_ {n} \\Rightarrow \\alpha_ {1} + \\dots + \\alpha_ {i - 1} + (- \\alpha) + \\alpha_ {i + 1} + \\dots + \\alpha_ {n} = 0\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "由于 $\\sum_{i=1}^{n} V_i$ 为直和，因此 $\\alpha = 0$ ，即第三条结论成立。",
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        "page_idx": 51
    },
    {
        "type": "header",
        "text": "5.1 线性空间",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "page_number",
        "text": "47",
        "bbox": [
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        ],
        "page_idx": 51
    },
    {
        "type": "text",
        "text": "定理 5.9 (直和合并定理)",
        "text_level": 2,
        "bbox": [
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            101
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "若 $V = V_{1} \\oplus V_{2}, V_{1} = V_{11} \\oplus V_{12} \\oplus \\cdots \\oplus V_{1s}, V_{2} = V_{21} \\oplus \\cdots \\oplus V_{2t}$ ，则",
        "bbox": [
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        ],
        "page_idx": 52
    },
    {
        "type": "equation",
        "text": "$$\nV = V _ {1 1} \\oplus \\dots V _ {1 s} \\oplus V _ {2 1} \\oplus \\dots \\oplus V _ {2 t}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "证明 只证明 $V = V_{1} \\oplus V_{2}$ ， $V_{1} = V_{11} \\oplus V_{12}$ 时， $V = V_{11} \\oplus V_{12} \\oplus V_{2}$ 。",
        "bbox": [
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        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "首先显然有 $V = V_{11} + V_{12} + V_2$ 。设",
        "bbox": [
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        ],
        "page_idx": 52
    },
    {
        "type": "equation",
        "text": "$$\n0 = \\alpha_ {1 1} + \\alpha_ {1 2} + \\alpha_ {2} = (\\alpha_ {1 1} + \\alpha_ {1 2}) + \\alpha_ {2}\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 52
    },
    {
        "type": "text",
        "text": "由于 $V = V_{1} \\oplus V_{2}$ ，因此得到 $\\alpha_{11} + \\alpha_{12} = \\alpha_{2} = 0$ ，同理根据 $V_{1} = V_{11} \\oplus V_{12}$ 得到 $\\alpha_{11} = \\alpha_{12} = 0$ ，综上0元素分解唯一，为直和。",
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        "page_idx": 52
    },
    {
        "type": "text",
        "text": "5.1.8 商空间",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "定义 5.10 (商空间的元素)",
        "text_level": 2,
        "bbox": [
            122,
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            312,
            330
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "W 是线性空间 V 的子空间， $\\alpha \\in V$ ，定义",
        "bbox": [
            115,
            336,
            425,
            350
        ],
        "page_idx": 52
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha + W := \\{\\alpha + \\omega : \\omega \\in W \\}\n$$",
        "text_format": "latex",
        "bbox": [
            394,
            359,
            600,
            374
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "定义 5.11 (商空间)",
        "text_level": 2,
        "bbox": [
            122,
            399,
            260,
            414
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "定义 $V / W$ 为 $\\{\\alpha + W : \\alpha \\in V\\}$ ，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": [
            114,
            418,
            781,
            435
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "定义 5.12 (自然映射)",
        "text_level": 2,
        "bbox": [
            122,
            458,
            278,
            474
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "定义 $\\eta : V \\to V / W$ 为自然映射",
        "bbox": [
            115,
            480,
            346,
            495
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "5.2 线性映射",
        "text_level": 2,
        "bbox": [
            95,
            539,
            240,
            558
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "5.2.1 线性映射的概念",
        "text_level": 2,
        "bbox": [
            94,
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            292,
            595
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "定义 5.13 (线性映射)",
        "text_level": 2,
        "bbox": [
            122,
            609,
            280,
            625
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "$V, W$ 是 $\\mathbb{P}$ 上的线性空间， $\\varphi: V \\to W$ 若满足：",
        "bbox": [
            115,
            630,
            458,
            645
        ],
        "page_idx": 52
    },
    {
        "type": "equation",
        "text": "$$\n\\bullet \\varphi (\\alpha + \\beta) = \\varphi (\\alpha) + \\varphi (\\beta)\n$$",
        "text_format": "latex",
        "bbox": [
            139,
            650,
            339,
            664
        ],
        "page_idx": 52
    },
    {
        "type": "equation",
        "text": "$$\n\\bullet \\varphi (c \\alpha) = c \\varphi (\\alpha)\n$$",
        "text_format": "latex",
        "bbox": [
            141,
            669,
            268,
            683
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "则称 $\\varphi$ 为 $V$ 到 $W$ 的线性映射，将 $V$ 到 $W$ 的线性映射全体记为集合 $\\mathrm{Hom}_{\\mathbb{P}}(V,W)$",
        "bbox": [
            115,
            686,
            717,
            701
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "5.2.2 核与像",
        "text_level": 2,
        "bbox": [
            95,
            740,
            211,
            757
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "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。",
        "bbox": [
            122,
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            823,
            784
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "定义 5.14 (核与像)",
        "text_level": 2,
        "bbox": [
            122,
            793,
            262,
            808
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "$U, V$ 是 $\\mathbb{P}$ 上的线性空间， $f: U \\to V$ ，定义 $\\operatorname{Ker}(f) = \\{\\alpha \\in U: f(\\alpha) = 0\\}$ ， $\\operatorname{Im}(f) = \\{f(\\alpha): \\alpha \\in U\\}$",
        "bbox": [
            114,
            812,
            842,
            829
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "笔记像的维度对应着 $(\\alpha_{1},\\cdots,\\alpha_{n})A$ 的秩，核的维度对应着线性方程组AX=0的解空间维数。",
        "bbox": [
            89,
            848,
            788,
            863
        ],
        "page_idx": 52
    },
    {
        "type": "header",
        "text": "5.2 线性映射",
        "bbox": [
            805,
            55,
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        ],
        "page_idx": 52
    },
    {
        "type": "page_number",
        "text": "48",
        "bbox": [
            489,
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            507,
            964
        ],
        "page_idx": 52
    },
    {
        "type": "text",
        "text": "定理 5.10 (核像维数公式)",
        "text_level": 2,
        "bbox": [
            122,
            86,
            312,
            101
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "$f \\in \\text{Hom}(U, V)$ 其核与像满足：",
        "bbox": [
            114,
            107,
            342,
            121
        ],
        "page_idx": 53
    },
    {
        "type": "equation",
        "text": "$$\n\\dim \\operatorname{Ker} (f) + \\dim \\operatorname{Im} (f) = \\dim (U)\n$$",
        "text_format": "latex",
        "bbox": [
            366,
            130,
            628,
            145
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "证明 设 $\\alpha_{1},\\cdots,\\alpha_{r}$ 是 Kerf 的基, $\\alpha_{1},\\cdots,\\alpha_{n}$ 是 U 的基, 而 $\\mathrm{Im}f=L(f(\\alpha_{1}),\\cdots,f(\\alpha_{n}))=L(f(\\alpha_{r+1}),\\cdots,f(\\alpha_{n}))$ , 下面证明线性无关。假设 $k_{r+1}f(\\alpha_{r+1})+\\cdots+k_{n}f(\\alpha_{n})=0$ , 根据线性性可知 $f(k_{r+1}\\alpha_{r+1}+\\cdots+k_{n}\\alpha_{n})=0$ , 即 $k_{r+1}\\alpha_{r+1}+\\cdots+k_{n}\\alpha_{n}\\in\\mathrm{Ker}(f)$ , 根据基的线性无关性, 显然 $k_{r+1},\\cdots,k_{n}$ 均为 0",
        "bbox": [
            90,
            164,
            905,
            218
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "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": [
            87,
            219,
            905,
            254
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "练习 5.12 核像维数公式推广证明 证明以下核像命题",
        "text_level": 2,
        "bbox": [
            58,
            275,
            477,
            291
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "(1) 定义域为子空间: $\\mathcal{A}$ 是有限维线性空间 $V$ 的线性变换, $W$ 是 $V$ 的子空间, 证明:",
        "bbox": [
            121,
            294,
            746,
            309
        ],
        "page_idx": 53
    },
    {
        "type": "equation",
        "text": "$$\n\\dim (\\mathcal {A} W) + \\dim (\\operatorname{Ker} \\mathcal {A} \\cap W) = \\dim (W)\n$$",
        "text_format": "latex",
        "bbox": [
            347,
            318,
            647,
            332
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "(2) 复合映射： $\\sigma,\\tau$ 是有限维空间 $U\\to V,V\\to W$ 的线性映射，证明：",
        "bbox": [
            122,
            341,
            631,
            356
        ],
        "page_idx": 53
    },
    {
        "type": "equation",
        "text": "$$\n\\dim (\\operatorname{Ker} (\\sigma)) + \\dim (\\operatorname{Im} (\\sigma) \\cap \\operatorname{Ker} (\\tau)) = \\dim (\\operatorname{Ker} (\\tau \\sigma))\n$$",
        "text_format": "latex",
        "bbox": [
            300,
            363,
            694,
            380
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "证明 (1) 设 $\\alpha_{1},\\cdots,\\alpha_{r}$ 是 $\\operatorname{Ker}\\mathcal{A}\\cap W$ 的一组基， $\\alpha_{1},\\cdots,\\alpha_{n}$ 是 W 的一组基，则 $\\mathcal{A}W=L(\\mathcal{A}\\alpha_{r+1},\\cdots,\\mathcal{A}\\alpha_{n})$ 。考虑 AW 的维数，设 $k_{r+1}\\mathcal{A}\\alpha_{r+1}+\\cdots+k_{n}\\mathcal{A}\\alpha_{n}=0$ ，则有 $k_{r+1}\\alpha_{r+1}+\\cdots+k_{n}\\alpha_{n}\\in\\operatorname{Ker}\\mathcal{A}\\cap W$ ，因此得到 $k_{r+1}=\\cdots=k_{n}=0$ 。",
        "bbox": [
            89,
            386,
            905,
            439
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "(2) 先得找原空间最小的空间，可知 $\\mathrm{Ker}(\\sigma) \\subseteq \\mathrm{Ker}(\\tau\\sigma)$ ，因此设 $\\alpha_{1}, \\cdots, \\alpha_{r}$ 是 $\\mathrm{Ker}(\\sigma)$ 的基， $\\alpha_{1}, \\cdots, \\alpha_{n}$ 是 $\\mathrm{Ker}(\\tau\\sigma)$ 的基。设 $V_{1} = L(\\sigma\\alpha_{r+1}, \\cdots, \\sigma\\alpha_{n})$ 。显然 $V_{1} \\subseteq \\mathrm{Im}(\\sigma) \\cap \\mathrm{Ker}(\\tau)$ 。",
        "bbox": [
            87,
            442,
            902,
            476
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "另一侧 $\\forall \\alpha \\in \\mathrm{Im}(\\sigma) \\cap \\mathrm{Ker}(\\tau)$ ， $\\exists \\beta$ 使得 $\\sigma(\\beta) = \\alpha, \\tau(\\alpha) = 0$ ，这说明 $\\tau(\\sigma(\\beta)) = 0$ ，因此 $\\beta \\in \\mathrm{Ker}(\\tau\\sigma)$ ，即 $\\alpha \\in V_1$ ，最终得出 $V_1 = \\mathrm{Im}(\\sigma) \\cap \\mathrm{Ker}(\\tau)$ 。",
        "bbox": [
            87,
            479,
            905,
            514
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "定理 5.11 (核像与单射、满射)",
        "text_level": 2,
        "bbox": [
            124,
            539,
            344,
            554
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "$\\sigma$ 是 $U\\to W$ 的线性映射，则：",
        "bbox": [
            114,
            558,
            342,
            573
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "\\- $\\sigma$ 是满射当且仅当 $\\operatorname{Im}(\\sigma) = W$",
        "bbox": [
            139,
            577,
            381,
            592
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "\\- $\\sigma$ 是单射当且仅当 $\\operatorname{Ker}(\\sigma) = \\{\\mathbf{0}\\}$",
        "bbox": [
            139,
            596,
            396,
            609
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "\\- 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": [
            139,
            615,
            801,
            630
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "定理 5.12 (核、像任意性)",
        "text_level": 2,
        "bbox": [
            124,
            653,
            312,
            669
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "$V$ 是 $P$ 上 $n$ 维线性空间， $V_{1}, V_{2}$ 是其子空间， $\\dim V_{1} + \\dim V_{2} = n$ ，则存在 $V$ 上的线性变换 $\\mathcal{A}$ 使得 $\\operatorname{Ker} \\mathcal{A} = V_{1}, \\operatorname{Im} \\mathcal{A} = V_{2}$",
        "bbox": [
            112,
            674,
            890,
            709
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "证明 设 $V_{1}$ 的一组基为 $\\alpha_{1},\\cdots,\\alpha_{r}$ ，扩充为 V 的一组基 $\\alpha_{1},\\cdots,\\alpha_{n}$ ，设 $\\beta_{r+1},\\cdots,\\beta_{n}$ 是 $V_{2}$ 的一组基，构造映射 $\\mathcal{A}(k_{1}\\alpha_{1}+\\cdots+k_{n}\\alpha_{n})=k_{r+1}\\beta_{r+1}+\\cdots+k_{n}\\beta_{n}$ 。可证明 A 是线性变换，且核、像满足条件。",
        "bbox": [
            89,
            727,
            902,
            762
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "练习 5.13 计算核与像 (1) $A: P^{4} \\rightarrow P^{3}$ ，求 KerA, ImA",
        "bbox": [
            58,
            783,
            487,
            797
        ],
        "page_idx": 53
    },
    {
        "type": "equation",
        "text": "$$\n\\mathcal {A} \\left( \\begin{array}{c} x _ {1} \\\\ x _ {2} \\\\ x _ {3} \\\\ x _ {4} \\end{array} \\right) = \\left( \\begin{array}{c} x _ {1} - 3 x _ {2} + x _ {3} - 2 x _ {4} \\\\ - x _ {1} - 1 1 x _ {2} + 2 x _ {3} - 5 x _ {4} \\\\ 3 x _ {1} + 5 x _ {2} + x _ {4} \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            332,
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            873
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "解 (1) $\\operatorname{Ker} \\mathcal{A}$ 相当于 $AX = 0$ 的解空间， $\\operatorname{Im} \\mathcal{A}$ 相当于 $A$ 的列空间",
        "bbox": [
            87,
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            895
        ],
        "page_idx": 53
    },
    {
        "type": "header",
        "text": "5.2 线性映射",
        "bbox": [
            806,
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        ],
        "page_idx": 53
    },
    {
        "type": "page_number",
        "text": "49",
        "bbox": [
            489,
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            964
        ],
        "page_idx": 53
    },
    {
        "type": "text",
        "text": "5.2.3 幂等变换的核与像",
        "text_level": 2,
        "bbox": [
            92,
            86,
            314,
            102
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "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": [
            90,
            114,
            905,
            147
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "定义 5.15 (幂等变换)",
        "text_level": 2,
        "bbox": [
            121,
            156,
            278,
            172
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "若线性变换 $\\mathcal{A}$ 满足 $\\mathcal{A}^2 = \\mathcal{A}$ ，则称其为幂等变换。",
        "bbox": [
            115,
            177,
            478,
            193
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "定理 5.13 (幂等变换核像空间分解)",
        "text_level": 2,
        "bbox": [
            121,
            217,
            379,
            231
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "$\\mathcal{A}$ 是 $V$ 上的幂等变换，则",
        "bbox": [
            114,
            237,
            310,
            250
        ],
        "page_idx": 54
    },
    {
        "type": "equation",
        "text": "$$\nV = \\operatorname{Ker} \\mathcal {A} \\oplus \\operatorname{Im} \\mathcal {A}\n$$",
        "text_format": "latex",
        "bbox": [
            428,
            256,
            566,
            269
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "证明 (1) 先证明 $V$ 是核与像的和: $\\forall \\alpha \\in V$ , 有 $\\alpha = \\alpha - \\mathcal{A}\\alpha + \\mathcal{A}\\alpha$ , 其中 $\\mathcal{A}\\alpha \\in \\operatorname{Im}\\mathcal{A}$ 。而 $\\mathcal{A}(\\alpha - \\mathcal{A}\\alpha) = \\mathcal{A}\\alpha - \\mathcal{A}^2\\alpha = 0$ , 因此 $\\alpha - \\mathcal{A}\\alpha \\in \\operatorname{Ker}\\mathcal{A}$",
        "bbox": [
            90,
            288,
            904,
            322
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "(2) 证明直和: $\\forall \\alpha \\in \\operatorname{Ker} \\mathcal{A} \\cap \\operatorname{Im} \\mathcal{A}$ , 则 $\\mathcal{A} \\alpha = 0$ 且 $\\exists \\beta, \\mathcal{A} \\beta = \\alpha$ , 两侧同时作用 $\\mathcal{A}$ , 左侧得到 $\\mathcal{A}^2 \\beta = \\mathcal{A} \\beta = \\alpha$ , 右侧 $\\mathcal{A} \\alpha = 0$ , 因此 $\\alpha = 0$",
        "bbox": [
            90,
            326,
            905,
            359
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "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": [
            58,
            382,
            638,
            395
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "(1) $\\mathcal{A},\\mathcal{B}$ 有相同的像当且仅当 $\\mathcal{AB} = \\mathcal{B},\\mathcal{BA} = \\mathcal{A}$",
        "bbox": [
            122,
            400,
            470,
            414
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "(2) $\\mathcal{A},\\mathcal{B}$ 有相同的核当且仅当 $\\mathcal{AB} = \\mathcal{A},\\mathcal{BA} = \\mathcal{B}$",
        "bbox": [
            122,
            419,
            468,
            432
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "证明 (1) 左推右: $\\forall \\alpha \\in V$ 有 $\\mathcal{B} \\alpha = \\in \\operatorname{Im}(\\mathcal{B})$ , 根据条件即 $\\mathcal{B} \\alpha \\in \\operatorname{Im} \\mathcal{A}$ , 因此根据幂等变换投影的性质 ( $\\mathcal{A}$ 不改变 $\\operatorname{Im} \\mathcal{A}$ 中的向量) 得到",
        "bbox": [
            87,
            436,
            905,
            470
        ],
        "page_idx": 54
    },
    {
        "type": "equation",
        "text": "$$\n\\mathcal {A} (\\mathcal {B} \\alpha) = \\mathcal {B} \\alpha \\Rightarrow \\mathcal {A B} = \\mathcal {B}\n$$",
        "text_format": "latex",
        "bbox": [
            405,
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            591,
            489
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "同理得到 $\\mathcal{BA} = \\mathcal{A}$",
        "bbox": [
            90,
            502,
            230,
            516
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "右推左： $\\forall \\gamma \\in \\operatorname{Im}(\\mathcal{A})$ ，有 $\\mathcal{A}\\gamma = \\gamma$ ，而 $\\mathcal{B}\\mathcal{A}\\gamma = \\mathcal{A}\\gamma = \\gamma$ ，因此 $\\mathcal{B}\\gamma = \\gamma$ ，即 $\\gamma \\in \\operatorname{Im}(\\mathcal{B})$ ，另一侧同理",
        "bbox": [
            122,
            520,
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            536
        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "(2) 左推右：由于 $V = \\operatorname{Im}(\\mathcal{A}) \\oplus \\operatorname{Ker}(\\mathcal{A})$ ，因此 $\\forall \\alpha \\in V$ 有",
        "bbox": [
            122,
            539,
            542,
            552
        ],
        "page_idx": 54
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha = \\alpha_ {1} + \\alpha_ {2}, \\alpha_ {1} \\in \\operatorname{Im} (\\mathcal {A}), \\alpha_ {2} \\in \\operatorname{Ker} \\mathcal {A}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "$\\alpha - \\alpha_{1} \\in \\operatorname{Ker} \\mathcal{A}$ ，因此 $\\alpha - \\alpha_{1} \\in \\operatorname{Ker} \\mathcal{B}$ ，故 $\\mathcal{B}(\\alpha - \\alpha_{1}) = 0$ ，即 $\\mathcal{B} \\alpha = \\mathcal{B} \\alpha_{1}$ 。而 $\\mathcal{A} \\alpha = \\alpha_{1}$ ，得到",
        "bbox": [
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        ],
        "page_idx": 54
    },
    {
        "type": "equation",
        "text": "$$\n(\\mathcal {B A}) \\alpha = \\mathcal {B} \\alpha_ {1} = \\mathcal {B} \\alpha\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 54
    },
    {
        "type": "text",
        "text": "因此得出 $\\mathcal{BA} = \\mathcal{B}$",
        "bbox": [
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        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "右推左",
        "bbox": [
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        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "5.2.4 一般线性变换的核像分解",
        "text_level": 2,
        "bbox": [
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        "page_idx": 54
    },
    {
        "type": "text",
        "text": "定理 5.14 (线性变换核像空间分解)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 54
    },
    {
        "type": "text",
        "text": "假设 $\\mathcal{A}$ 为线性空间 $V$ 中的线性变换， $\\dim V = n$ ，则 $V$ 可进行如下分解：",
        "bbox": [
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        ],
        "page_idx": 54
    },
    {
        "type": "equation",
        "text": "$$\nV = \\operatorname{Ker} (\\mathcal {A} ^ {n}) \\oplus \\operatorname{Im} (\\mathcal {A} ^ {n})\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 54
    },
    {
        "type": "image",
        "img_path": "images/0c75a68015f2cdf632b969cf7c46095894a2cd05bcb7d257bbf03652b7bd42b8.jpg",
        "image_caption": [],
        "image_footnote": [],
        "content": "",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "笔记 该结论在后续证明空间分解法中会用到",
        "text_level": 2,
        "bbox": [
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    },
    {
        "type": "header",
        "text": "5.3 线性变换",
        "bbox": [
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        "page_idx": 54
    },
    {
        "type": "page_number",
        "text": "50",
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    },
    {
        "type": "image",
        "img_path": "images/b7f542caab526a8bf3277323ee7deae5a41b0c8eeb8714d2af1c4ee924c40700.jpg",
        "image_caption": [],
        "image_footnote": [],
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    },
    {
        "type": "text",
        "text": "5.3 线性变换",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "5.3.1 线性变换基本概念",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "定义 5.16 (线性变换)",
        "text_level": 2,
        "bbox": [
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    },
    {
        "type": "text",
        "text": "$V$ 是数域 $\\mathbb{P}$ 上的线性空间， $\\mathcal{A}$ 是 $V$ 到自身的线性映射，则称 $\\mathcal{A}$ 是 $V$ 内的线性变换。 $V$ 内全体线性变换组成集合 $\\operatorname{End}(V)$ 。",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "练习 5.15 线性变换基本练习 (1) 若 $T_{1}, \\cdots, T_{m}$ 是 m 个不同的线性变换，证明存在 $\\alpha \\in V$ ，使得 $T_{1}\\alpha, \\cdots, T_{m}\\alpha$ 互不相同",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "证明 (1) 令 $V_{ij} = \\{x \\in V : T_i x = T_j x\\}$ ，则显然 $V_{ij}$ 是 $V$ 的子空间。因为 $T_1, \\cdots, T_m$ 两两不同，故 $\\forall T_i, T_j, \\exists \\beta$ 使得 $T_i \\beta \\neq T_j \\beta$ ，从而 $V_{ij} \\neq V$ 。设 $V_{ij}$ 中有 $s$ 个是零空间，则其余均为非平凡子空间，而根据线性空间习题 (和与交一节) 可知存在 $\\alpha$ 不属于任一非平凡子空间，故 $\\exists \\alpha$ 不属于 $V_{ij}, \\forall i, j$ ，因此 $T_1 \\alpha, \\cdots, T_m \\alpha$ 两两不同。",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "定理 5.15 (线性变换的可逆性)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "线性变换 T 可逆当且仅当",
        "bbox": [
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        ],
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    },
    {
        "type": "text",
        "text": "\\- $T\\epsilon_{1}, \\cdots, T\\epsilon_{n}$ 线性无关",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "\\- $T$ 是单射",
        "bbox": [
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        ],
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    },
    {
        "type": "text",
        "text": "\\- $T$ 是满射",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "证明 (1) 从矩阵角度看即可",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "(2) 左推右：若 $T\\alpha_{1} = T\\alpha_{2}$ ，则两侧作用 $T^{-1}$ 得到 $\\alpha_{1} = \\alpha_{2}$",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "右推左：下证 $T\\epsilon_{1},\\cdots,T\\epsilon_{n}$ 线性无关，若 $k_{1}T\\epsilon_{1}+\\cdots+k_{n}T\\epsilon_{n}=0$ ，则",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "equation",
        "text": "$$\nT (k _ {1} \\epsilon_ {1} + \\dots + k _ {n} \\epsilon_ {n}) = 0\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "由于 T 是单射，得到 $k_{1}\\epsilon_{1}+\\cdots+k_{n}\\epsilon_{n}=0$ ，因此 $k_{1}=\\cdots=k_{n}=0$ ，根据命题 (1) 知可逆",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "(3) 左推右：设 $T$ 可逆，则 $\\forall \\alpha \\in V$ ，存在 $T^{-1}\\alpha$ ，而 $\\alpha = T(T^{-1}\\alpha)$ ，因此是满射",
        "bbox": [
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        ],
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    },
    {
        "type": "text",
        "text": "右推左：由于满射，对任一一组基 $\\eta_{1},\\cdots,\\eta_{n}$ ，总存在 $\\epsilon_{1},\\cdots,\\epsilon_{n}$ ，使得",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "equation",
        "text": "$$\nT \\epsilon_ {1} = \\eta_ {1}, \\dots , T \\epsilon_ {n} = \\eta_ {n}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "下面证明 $\\epsilon_{1},\\cdots,\\epsilon_{n}$ 线性无关，设 $k_{1}\\epsilon_{1}+\\cdots+k_{n}\\epsilon_{n}=0$ ，则",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "equation",
        "text": "$$\nT (k _ {1} \\epsilon_ {1} + \\dots + k _ {n} \\epsilon_ {n}) = k _ {1} \\eta_ {1} + \\dots + k _ {n} \\eta_ {n} = 0\n$$",
        "text_format": "latex",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "这说明 $k_{1}=\\cdots=k_{n}=0$ ，因此可逆。",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "5.3.2 线性变换在基下的矩阵、变换后的坐标",
        "text_level": 2,
        "bbox": [
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    },
    {
        "type": "text",
        "text": "定义 5.17 (线性变换的矩阵)",
        "text_level": 2,
        "bbox": [
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    },
    {
        "type": "text",
        "text": "$\\mathcal{A}$ 是 $V$ 内的线性变换， $V = L(\\epsilon_1, \\epsilon_2, \\dots, \\epsilon_n)$ ，若 $(\\mathcal{A}(\\epsilon_1), \\dots, \\mathcal{A}(\\epsilon_n)) = (\\epsilon_1, \\dots, \\epsilon_n)A$ ，则 $A$ 称为 $\\mathcal{A}$ 在 $\\epsilon_1, \\dots, \\epsilon_n$ 基下的矩阵。",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "笔记 计算线性变换在基下的矩阵：根据 $(\\mathcal{A}\\epsilon_{1},\\cdots,\\mathcal{A}\\epsilon_{n})=(\\epsilon_{1},\\cdots,\\epsilon_{n})A$ 推出",
        "bbox": [
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        ],
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    },
    {
        "type": "equation",
        "text": "$$\nA = (\\epsilon_ {1}, \\dots , \\epsilon_ {n}) ^ {- 1} (\\mathcal {A} \\epsilon_ {1}, \\dots , \\mathcal {A} \\epsilon_ {n})\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "text",
        "text": "可以用矩阵求逆方式求解。",
        "bbox": [
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        ],
        "page_idx": 55
    },
    {
        "type": "header",
        "text": "5.3 线性变换",
        "bbox": [
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        "page_idx": 55
    },
    {
        "type": "page_number",
        "text": "51",
        "bbox": [
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        "page_idx": 55
    },
    {
        "type": "text",
        "text": "练习5.16线性变换在基下的矩阵已知 $A = (a_{ij}),i,j = 1,2,3$ 是 $\\mathcal{A}$ 在 $\\epsilon_1,\\epsilon_2,\\epsilon_3$ 下的矩阵，计算",
        "bbox": [
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        "page_idx": 56
    },
    {
        "type": "text",
        "text": "(1) $\\mathcal{A}$ 在 $\\epsilon_3, \\epsilon_2, \\epsilon_1$ 下的矩阵",
        "bbox": [
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        ],
        "page_idx": 56
    },
    {
        "type": "text",
        "text": "(2) $\\mathcal{A}$ 在 $\\epsilon_1, k\\epsilon_2, \\epsilon_3$ 下的矩阵",
        "bbox": [
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        ],
        "page_idx": 56
    },
    {
        "type": "text",
        "text": "(3) $\\mathcal{A}$ 在 $\\epsilon_1 + \\epsilon_2, \\epsilon_2, \\epsilon_3$ 下的矩阵",
        "bbox": [
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        ],
        "page_idx": 56
    },
    {
        "type": "text",
        "text": "解 根据 $T(\\epsilon_1, \\epsilon_2, \\epsilon_3) = (\\epsilon_1, \\epsilon_2, \\epsilon_3)A$ ，得到",
        "bbox": [
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        "page_idx": 56
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} T \\epsilon_ {1} = a _ {1 1} \\epsilon_ {1} + a _ {2 1} \\epsilon_ {2} + a _ {3 1} \\epsilon_ {3} \\\\ T \\epsilon_ {2} = a _ {1 2} \\epsilon_ {1} + a _ {2 2} \\epsilon_ {2} + a _ {3 2} \\epsilon_ {3} \\\\ T \\epsilon_ {3} = a _ {1 3} \\epsilon_ {1} + a _ {2 3} \\epsilon_ {2} + a _ {3 3} \\epsilon_ {3} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "(1) 重新排列得到:",
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        ],
        "page_idx": 56
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} T \\epsilon_ {3} = a _ {3 3} \\epsilon_ {3} + a _ {2 3} \\epsilon_ {2} + a _ {1 3} \\epsilon_ {1} \\\\ T \\epsilon_ {2} = a _ {3 2} \\epsilon_ {3} + a _ {2 2} \\epsilon_ {2} + a _ {1 2} \\epsilon_ {1} \\\\ T \\epsilon_ {1} = a _ {3 1} \\epsilon_ {3} + a _ {2 1} \\epsilon_ {2} + a _ {1 1} \\epsilon_ {1} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 56
    },
    {
        "type": "text",
        "text": "得到矩阵为",
        "bbox": [
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        ],
        "page_idx": 56
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{l l l} a _ {3 3} & a _ {3 2} & a _ {3 1} \\\\ a _ {2 3} & a _ {2 2} & a _ {2 1} \\\\ a _ {1 3} & a _ {1 2} & a _ {1 1} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 56
    },
    {
        "type": "text",
        "text": "(2)(3) 同理列开即可",
        "bbox": [
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        ],
        "page_idx": 56
    },
    {
        "type": "text",
        "text": "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， $T^{n-1} \\neq O, T^n = O$ ，证明：T 在某组基下的矩阵为",
        "bbox": [
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        ],
        "page_idx": 56
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c c c c} 0 & 0 & \\dots & 0 & 0 \\\\ 1 & 0 & \\dots & 0 & 0 \\\\ \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 0 & 0 & \\dots & 1 & 0 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 56
    },
    {
        "type": "text",
        "text": "证明 首先 $\\alpha, T\\alpha, \\cdots, T^{n-1}\\alpha$ 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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        ],
        "page_idx": 56
    },
    {
        "type": "text",
        "text": "定理 5.16 (线性变换后的坐标)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 56
    },
    {
        "type": "text",
        "text": "若 $\\alpha$ 在 $(\\epsilon_{1},\\cdots,\\epsilon_{n})$ 基下的坐标为 X，则 $A\\alpha$ 在这组基下的坐标为 AX。",
        "bbox": [
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        "page_idx": 56
    },
    {
        "type": "text",
        "text": "证明 $\\mathcal{A}\\alpha=\\mathcal{A}(x_{1}\\epsilon_{1}+\\cdots+x_{n}\\epsilon_{n})=\\mathcal{A}(\\epsilon_{1},\\cdots,\\epsilon_{n})X=(\\epsilon_{1},\\cdots,\\epsilon_{n})AX,$",
        "bbox": [
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        "page_idx": 56
    },
    {
        "type": "text",
        "text": "5.3.3 线性变换在不同基下的矩阵：相似关系",
        "text_level": 2,
        "bbox": [
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        "page_idx": 56
    },
    {
        "type": "text",
        "text": "定理 5.17 (线性变换在不同基下的矩阵)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 56
    },
    {
        "type": "text",
        "text": "$\\epsilon_{i},\\eta_{i}(i=1,\\cdots,n)$ 是 V 两组基，且 $(\\eta_{1},\\cdots,\\eta_{n})=(\\epsilon_{1},\\cdots,\\epsilon_{n})T$ 。若 A 在 $\\epsilon_{i},\\eta_{i}$ 基下的矩阵分别为 A,B，则有 $B=T^{-1}AT$ 。",
        "bbox": [
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        ],
        "page_idx": 56
    },
    {
        "type": "text",
        "text": "证明 由线性变换矩阵定义， $\\mathcal{A}(\\epsilon_{1},\\cdots,\\epsilon_{n})=(\\epsilon_{1},\\cdots,\\epsilon_{n})A,\\mathcal{A}(\\eta_{1},\\cdots,\\eta_{n})=(\\eta_{1},\\cdots,\\eta_{n})B$ 。而 $\\mathcal{A}(\\eta_{1},\\cdots,\\eta_{n})=\\mathcal{A}(\\epsilon_{1},\\cdots,\\epsilon_{n})T=(\\epsilon_{1},\\cdots,\\epsilon_{n})AT$ ，且 $\\mathcal{A}(\\eta_{1},\\cdots,\\eta_{n})=(\\epsilon_{1},\\cdots,\\epsilon_{n})TB$ 上面两式取等号即可。",
        "bbox": [
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        "page_idx": 56
    },
    {
        "type": "text",
        "text": "笔记 记忆时根据 $(\\eta_{1},\\cdots,\\eta_{n})=(\\epsilon_{1},\\cdots,\\epsilon_{n})T$ ，将 $\\eta,\\epsilon$ 基直接换成对应的矩阵 B,A，有 T 的一边添上 $T^{-1}$ 即可，即 $B=T^{-1}AT$",
        "bbox": [
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    },
    {
        "type": "header",
        "text": "5.3 线性变换",
        "bbox": [
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        "page_idx": 56
    },
    {
        "type": "page_number",
        "text": "52",
        "bbox": [
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        "page_idx": 56
    },
    {
        "type": "text",
        "text": "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 $T \\epsilon_{i} = \\eta_{i}$ ,",
        "bbox": [
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        ],
        "page_idx": 57
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} \\epsilon_ {1} = (1, 0, 1) ^ {T}, \\epsilon_ {2} = (2, 1, 0) ^ {T}, \\epsilon_ {3} = (1, 1, 1) ^ {T} \\\\ \\eta_ {1} = (1, 2, - 1) ^ {T}, \\eta_ {2} = (2, 2, - 1) ^ {T}, \\eta_ {3} = (2, - 1, - 1) ^ {T} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 57
    },
    {
        "type": "text",
        "text": "(1) 写出 $\\epsilon_{1}, \\epsilon_{2}, \\epsilon_{3}$ 到 $\\eta_{1}, \\eta_{2}, \\eta_{3}$ 的过渡矩阵",
        "bbox": [
            122,
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "(2) 写出 $T$ 在 $\\epsilon_1, \\epsilon_2, \\epsilon_3$ 下的矩阵",
        "bbox": [
            122,
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            186
        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "(3) 写出 $T$ 在 $\\eta_1, \\eta_2, \\eta_3$ 下的矩阵",
        "bbox": [
            122,
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "解 (1) $e_{1}, e_{2}, e_{3}$ 下的矩阵为",
        "bbox": [
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        ],
        "page_idx": 57
    },
    {
        "type": "equation",
        "text": "$$\n(\\eta_ {1}, \\eta_ {2}, \\eta_ {3}) =\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "定理 5.18 (线性变换构造的等价类)",
        "text_level": 2,
        "bbox": [
            122,
            277,
            379,
            292
        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "矩阵 $A, B$ 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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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "5.3.4 相似矩阵的性质",
        "text_level": 2,
        "bbox": [
            94,
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            292,
            368
        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "定理 5.19 (相似矩阵的基本性质)",
        "text_level": 2,
        "bbox": [
            122,
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            398
        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "$K$ 上矩阵 $A, B$ 相似，且满足 $B = P^{-1}AP$ ，则",
        "bbox": [
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "• $r(A) = r(B)$",
        "bbox": [
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "• $A^{m} \\sim B^{m}$",
        "bbox": [
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "\\- $f(x)$ 为多项式，则 $f(A) \\sim f(B)$",
        "bbox": [
            139,
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "若 $A_{1} \\sim B_{1}, A_{2} \\sim B_{2}$ ，不一定有 $A_{1} + A_{2} \\sim B_{1} + B_{2}$ ，要求过渡矩阵相同才行",
        "bbox": [
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "证明 (1) $r(A)=r(B)$ ：因为 $B=P^{-1}AP$ ，而 P 满秩，等价于一系列初等变换，故不改变秩。",
        "bbox": [
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "5.4 特征值、相似对角化",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "5.4.1 特征值与特征向量",
        "text_level": 2,
        "bbox": [
            94,
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "定义 5.18 (特征值与特征向量)",
        "text_level": 2,
        "bbox": [
            122,
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            640
        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "$\\mathcal{A}$ 是数域 $\\mathbb{P}$ 上线性空间 $V$ 内的一个线性变换，若对于 $\\lambda \\in \\mathbb{P}$ ， $\\exists \\xi \\neq 0$ ，使得 $\\mathcal{A}\\xi = \\lambda \\xi$ ，则称 $\\lambda$ 是 $\\mathcal{A}$ 的一个特征值， $\\xi$ 是 $\\mathcal{A}$ 属于 $\\lambda$ 的特征向量。",
        "bbox": [
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "定义 5.19 (特征子空间)",
        "text_level": 2,
        "bbox": [
            122,
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            295,
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "对于 $\\lambda\\in P$ ，定义其特征子空间为所有满足 $A\\xi=\\lambda\\xi$ 的特征向量 $\\xi$ 所组成的集合 $V_{\\lambda}$ 。",
        "bbox": [
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "引理 5.1 (特征子空间的性质)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "特征子空间 $V_{\\lambda}$ 有如下性质：",
        "bbox": [
            115,
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            322,
            799
        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "\\- $V_{\\lambda}$ 是 $V$ 的子空间。",
        "bbox": [
            139,
            804,
            299,
            816
        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "\\- 对于 $\\lambda \\in \\mathbb{P}$ , $V_{\\lambda} \\neq \\{\\mathbf{0}\\}$ 的充要条件为 $|\\lambda E - A| = 0$ , 其中 $A$ 是 $\\mathcal{A}$ 在任意基下的矩阵。",
        "bbox": [
            139,
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            837
        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "证明 (1) 只需要证明 $V_{\\lambda}$ 对 $V$ 内的数乘和加法封闭即可。",
        "bbox": [
            90,
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "(2) 对于 $\\lambda \\alpha = \\mathcal{A}\\alpha$ ，可以取 $V$ 一组基，若 $\\alpha$ 在这组基下的坐标为 $X$ 。则等价于求解 $\\lambda X = AX$ ，其中 $A$ 是 $\\mathcal{A}$ 在这组基下的矩阵。等价于求解 $(\\lambda E - A)X = 0$ ，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。",
        "bbox": [
            87,
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        ],
        "page_idx": 57
    },
    {
        "type": "page_number",
        "text": "53",
        "bbox": [
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        ],
        "page_idx": 57
    },
    {
        "type": "header",
        "text": "5.4 特征值、相似对角化",
        "bbox": [
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        ],
        "page_idx": 57
    },
    {
        "type": "text",
        "text": "定义 5.20 (特征多项式)",
        "text_level": 2,
        "bbox": [
            122,
            86,
            295,
            101
        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "对于 P 上的矩阵 A，特征多项式定义为 $f(\\lambda) = |\\lambda E - A|$ 。",
        "bbox": [
            114,
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            122
        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "定理 5.20 (相似矩阵的特征多项式)",
        "text_level": 2,
        "bbox": [
            121,
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            379,
            162
        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "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": [
            114,
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        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "证明 若 $B = T^{-1}AT$ ，则 $|\\lambda E - B| = |\\lambda E - T^{-1}AT| = |T^{-1}||\\lambda E - A||T| = |\\lambda E - A|$ 。",
        "bbox": [
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        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "定理 5.21 (特征值基本性质)",
        "text_level": 2,
        "bbox": [
            122,
            228,
            329,
            243
        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "$\\lambda_{1},\\cdots,\\lambda_{n}$ 是矩阵 A 的特征值。则 $\\sum\\lambda_{i}=\\mathrm{tr}(A),\\quad\\prod\\lambda_{i}=|A|$",
        "bbox": [
            114,
            249,
            581,
            268
        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "证明 用行列式展开与韦达定理证明。",
        "bbox": [
            89,
            290,
            364,
            304
        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "练习 5.19 特征值与特征向量的基本练习 (1) $\\lambda_{1}, \\lambda_{2}$ 是 T 两个不同的特征值， $\\alpha_{1}, \\alpha_{2}$ 分别是对应的特征向量，证明： $\\alpha_{1} + \\alpha_{2}$ 不是 T 的特征向量",
        "bbox": [
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        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "(2) $T$ 是可逆线性变换，若 $\\lambda$ 是 $T$ 的特征值，则 $\\lambda^{-1}$ 是 $T^{-1}$ 的特征值证明 (1) $T\\alpha_{1} = \\lambda_{1}\\alpha_{1}, T\\alpha_{2} = \\lambda_{2}\\alpha_{2}$ ，因此",
        "bbox": [
            87,
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        ],
        "page_idx": 58
    },
    {
        "type": "equation",
        "text": "$$\nT (\\alpha_ {1} + \\alpha_ {2}) = \\lambda_ {1} \\alpha_ {1} + \\lambda_ {2} \\alpha_ {2}\n$$",
        "text_format": "latex",
        "bbox": [
            394,
            406,
            600,
            420
        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "若 $\\alpha_{1} + \\alpha_{2}$ 是特征向量， $\\lambda$ 是特征值，则 $T(\\alpha_{1} + \\alpha_{2}) = \\lambda (\\alpha_{1} + \\alpha_{2})$ ，得到",
        "bbox": [
            90,
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            631,
            444
        ],
        "page_idx": 58
    },
    {
        "type": "equation",
        "text": "$$\n(\\lambda - \\lambda_ {1}) \\alpha_ {1} + (\\lambda - \\lambda_ {2}) \\alpha_ {2} = 0\n$$",
        "text_format": "latex",
        "bbox": [
            389,
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        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "因此 $\\lambda - \\lambda_{1} = \\lambda - \\lambda_{2} = 0$ ，这与假设矛盾。",
        "bbox": [
            90,
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            489
        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "(2) 由于 $T\\alpha = \\lambda \\alpha$ ，因此 $\\alpha = \\lambda T^{-1}(\\alpha)$ ，即 $T^{-1}(\\alpha) = \\lambda^{-1}\\alpha$ ，因此 $\\lambda^{-1}$ 是 $T^{-1}$ 的特征值",
        "bbox": [
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        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "5.4.2 特征值与特征向量进一步结论",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "定理 5.22 (AB 与 BA 的特征值)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "$V, V'$ 是 K 上的线性空间， $\\mathcal{A} \\in \\mathrm{Hom}(V, V')$ ， $\\mathcal{B} \\in \\mathrm{Hom}(V', V)$ （矩阵同理），则\n- AB 与 BA 有相同非零特征值",
        "bbox": [
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        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "\\- 若 $\\xi$ 是 $AB$ 属于 $\\lambda_0$ 的特征向量，则 $B\\xi$ 是 $BA$ 属于 $\\lambda_0$ 的一个特征向量",
        "bbox": [
            139,
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        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "证明 设 $\\lambda_{0}$ 是 AB 非零特征值， $\\exists\\xi\\in V^{\\prime}$ 使得 $(\\mathcal{AB})\\xi=\\lambda_{0}\\xi$ ，两侧同时作用 B 得到：",
        "bbox": [
            90,
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            688
        ],
        "page_idx": 58
    },
    {
        "type": "equation",
        "text": "$$\n\\mathcal {B} (\\mathcal {A B}) \\xi = \\mathcal {B} (\\lambda_ {0} \\xi) \\Rightarrow (\\mathcal {B A}) (\\mathcal {B} \\xi) = \\lambda_ {0} (\\mathcal {B} \\xi)\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "定理 5.23 (Sylvester 公式)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "$A_{m\\times n},B_{n\\times m}$ ， $f_{AB}(\\lambda),f_{BA}(\\lambda)$ 是 $AB,BA$ 的特征多项式，则",
        "bbox": [
            115,
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        ],
        "page_idx": 58
    },
    {
        "type": "equation",
        "text": "$$\n\\lambda^ {n} f _ {A B} (\\lambda) = \\lambda^ {m} f _ {B A} (\\lambda)\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "证明 证明：设 $\\lambda \\neq 0$ ，根据",
        "bbox": [
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        ],
        "page_idx": 58
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} E _ {n} & 0 \\\\ - A & E _ {m} \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {n} & \\frac {1}{\\lambda} B \\\\ A & E _ {m} \\end{array} \\right] = \\left[ \\begin{array}{c c} E _ {n} & \\frac {1}{\\lambda} B \\\\ O & E _ {m} - \\frac {1}{\\lambda} A B \\end{array} \\right] \\Rightarrow | H | = \\left| \\begin{array}{c c} E _ {n} & \\frac {1}{\\lambda} B \\\\ O & E _ {m} - \\frac {1}{\\lambda} A B \\end{array} \\right| = \\left(\\frac {1}{\\lambda}\\right) ^ {m} | \\lambda E _ {m} - A B |\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 58
    },
    {
        "type": "header",
        "text": "5.4 特征值、相似对角化",
        "bbox": [
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        ],
        "page_idx": 58
    },
    {
        "type": "page_number",
        "text": "54",
        "bbox": [
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        ],
        "page_idx": 58
    },
    {
        "type": "text",
        "text": "同理根据",
        "bbox": [
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        ],
        "page_idx": 59
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} E _ {n} & \\frac {1}{\\lambda} B \\\\ A & E _ {m} \\end{array} \\right] \\cdot \\left[ \\begin{array}{c c} E _ {n} & 0 \\\\ - A & E _ {m} \\end{array} \\right] = \\left[ \\begin{array}{c c} E _ {n} - \\frac {1}{\\lambda} A B & \\frac {1}{\\lambda} B \\\\ 0 & E _ {m} \\end{array} \\right] \\Rightarrow | H | = \\left(\\frac {1}{\\lambda}\\right) ^ {n} | \\lambda E _ {n} - B A |\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "根据 $|H|$ 的值即可推出结论。",
        "bbox": [
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        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "练习5.20相关结论(1) $A, B$ 是两个 $n$ 阶方阵，证明： $AB + A$ 和 $BA + A$ 有相同的特征多项式",
        "bbox": [
            58,
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        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "(2) 计算下面矩阵 $C$ 的特征值",
        "bbox": [
            122,
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            219
        ],
        "page_idx": 59
    },
    {
        "type": "equation",
        "text": "$$\nC = \\left[ \\begin{array}{c c c c} a _ {1} b _ {1} & a _ {1} b _ {2} & \\dots & a _ {1} b _ {n} \\\\ a _ {2} b _ {1} & a _ {2} b _ {2} & \\dots & a _ {2} b _ {n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a _ {n} b _ {1} & a _ {n} b _ {2} & \\dots & a _ {n} b _ {n} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
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            300
        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "(3) 特例：已知 $A = (a_{1}, \\cdots, a_{n})$ ，证明 $A^{T}A$ 的特征值为 0 和 $a_{1}^{2} + \\cdots + a_{n}^{2}$ ，这里 0 是 n - 1 重根证明 (1) 由于 $AB + A = A(B + E)$ ， $BA + A = (B + E)A$ ，因此根据前面定理可知。",
        "bbox": [
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        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "(2) 令 $A = (a_{1}, \\cdots, a_{n})^{T}$ , $B = (b_{1}, \\cdots, b_{n})$ , 则 $C = AB$ , 而 $BA = \\sum_{i=1}^{n} a_{i} b_{i}$ 为一阶矩阵, 其特征值为 $\\sum_{i=1}^{n} a_{i} b_{i}$ , $AB$ 和 $BA$ 特征多项式相差 $\\lambda^{n-1}$ , 因此 $C = AB$ 的特征多项式为",
        "bbox": [
            89,
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        ],
        "page_idx": 59
    },
    {
        "type": "equation",
        "text": "$$\n\\lambda^ {n - 1} (\\lambda - \\sum_ {i = 1} ^ {n} a _ {i} b _ {i})\n$$",
        "text_format": "latex",
        "bbox": [
            428,
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        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "(3) 即 (2) 的特例",
        "bbox": [
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        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "定理 5.24 (可逆的特征值表达)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "$n$ 阶方阵 $A$ 可逆当且仅当 $A$ 的特征值全不为0",
        "bbox": [
            114,
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        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "证明 用 $\\left|A\\right|=\\lambda_{1}\\cdots\\lambda_{n}$ 即可",
        "bbox": [
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        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "定理 5.25 (逆矩阵的特征值)",
        "text_level": 2,
        "bbox": [
            122,
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            329,
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        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "n 阶可逆矩阵 A 的特征值为 $\\lambda_{1},\\cdots,\\lambda_{n}$ ，则 $A^{-1}$ 的全部特征值为 $\\lambda_{1}^{-1},\\cdots,\\lambda_{n}^{-1}$",
        "bbox": [
            112,
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            593
        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "证明 用 A 相似于 Jordan 阵 J， $J^{-1}$ 为上三角且对角为 $\\lambda_{1}^{-1},\\cdots,\\lambda_{n}^{-1}$ ，因此 $A^{-1}$ 的特征值也为 $\\lambda_{1}^{-1},\\cdots,\\lambda_{n}^{-1}$",
        "bbox": [
            90,
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        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "5.4.3 相似的进一步研究",
        "text_level": 2,
        "bbox": [
            94,
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            312,
            665
        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "定理 5.26 (相似的矩阵分解)",
        "text_level": 2,
        "bbox": [
            122,
            682,
            329,
            697
        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "$A, B$ 相似当且仅当存在方阵 $P, Q$ ，使得",
        "bbox": [
            114,
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            411,
            715
        ],
        "page_idx": 59
    },
    {
        "type": "equation",
        "text": "$$\nA = P Q, B = Q P\n$$",
        "text_format": "latex",
        "bbox": [
            431,
            725,
            566,
            740
        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "这里 $P, Q$ 至少有一个可逆",
        "bbox": [
            115,
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            763
        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "证明 由于 $A, B$ 相似，故存在可逆 $P$ 使得 $P^{-1}AP = B$ ，令 $Q = P^{-1}A$ ，则得到 $A = PQ, B = QP$",
        "bbox": [
            90,
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            818,
            797
        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "定理 5.27 (一些相似结论)",
        "text_level": 2,
        "bbox": [
            122,
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            312,
            824
        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "$A, B$ 为 $n$ 阶方阵，则",
        "bbox": [
            115,
            829,
            278,
            843
        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "\\- $AB$ 与 $BA$ : 若方阵 $A$ 可逆, 则 $AB$ 与 $BA$ 相似",
        "bbox": [
            139,
            848,
            515,
            860
        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "\\- $A^{-1}$ 与 $B^{-1}$ : 若 $A, B$ 可逆, 则 $A^{-1}$ 与 $B^{-1}$ 相似",
        "bbox": [
            139,
            865,
            521,
            879
        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "\\- 多项式：任一多项式 $f(x)$ ，有 $f(A), f(B)$ 相似",
        "bbox": [
            139,
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            507,
            898
        ],
        "page_idx": 59
    },
    {
        "type": "header",
        "text": "5.4 特征值、相似对角化",
        "bbox": [
            721,
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        ],
        "page_idx": 59
    },
    {
        "type": "page_number",
        "text": "55",
        "bbox": [
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        ],
        "page_idx": 59
    },
    {
        "type": "text",
        "text": "\\- 伴随矩阵：若 $A, B$ 相似，则 $A^{*}$ 与 $B^{*}$ 相似",
        "bbox": [
            139,
            96,
            480,
            110
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "证明 (1) $BA = (A^{-1}A)(BA) = A^{-1}(AB)A$",
        "bbox": [
            90,
            130,
            406,
            146
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "(2) $(P^{-1}AP)^{-1} = B^{-1}$ , 得到 $P^{-1}A^{-1}P = B^{-1}$",
        "bbox": [
            122,
            148,
            472,
            164
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "(3) 显然",
        "bbox": [
            124,
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            188,
            181
        ],
        "page_idx": 60
    },
    {
        "type": "equation",
        "text": "$$\n(4) B ^ {*} = (P ^ {- 1} A P) ^ {*} = P ^ {*} A ^ {*} (P ^ {- 1}) ^ {*} = P ^ {*} A ^ {*} (P ^ {*}) ^ {- 1}\n$$",
        "text_format": "latex",
        "bbox": [
            124,
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        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "推论5.2(伴随矩阵的特征值)",
        "text_level": 2,
        "bbox": [
            124,
            227,
            336,
            243
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "若矩阵 A 的特征值为 $\\lambda_{1},\\cdots,\\lambda_{m}$ ，则 $A^{*}$ 的特征值为 $\\prod_{i\\neq j}\\lambda_{i}$",
        "bbox": [
            114,
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            281
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "证明 考虑 $A$ 在 $\\mathbb{C}$ 上相似于 Jordan 阵 $J$ ，而 $J^{*}$ 的特征值易算，相似时特征值不变，因此可算出 $A^{*}$ 的特征值。",
        "bbox": [
            90,
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        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "5.4.4 相似对角化",
        "text_level": 2,
        "bbox": [
            94,
            338,
            253,
            354
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "定义 5.21 (相似对角化)",
        "text_level": 2,
        "bbox": [
            122,
            370,
            295,
            385
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "$\\mathcal{A}$ 是 $V$ 内的线性变换，如果存在 $V$ 的一组基使得 $\\mathcal{A}$ 在该基下的矩阵为对角阵，则称 $\\mathcal{A}$ 可对角化。",
        "bbox": [
            114,
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            836,
            405
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "定义 5.22 (代数重数、几何重数)",
        "bbox": [
            122,
            429,
            361,
            444
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "代数重数指特征值 $\\lambda_{i}$ 在特征多项式 $f(\\lambda)$ 中的重数。几何重数指 $\\lambda_{i}$ 对应特征子空间的维数。",
        "bbox": [
            115,
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            464
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "引理 5.2 (代数重数与几何重数关系)",
        "text_level": 2,
        "bbox": [
            124,
            489,
            386,
            505
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "对每个特征值 $\\lambda_{i}$ ，其代数重数大于等于几何重数。",
        "bbox": [
            115,
            510,
            482,
            524
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "引理 5.3 (不同特征值的特征向量)",
        "text_level": 2,
        "bbox": [
            122,
            549,
            371,
            564
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "A 属于不同特征值的特征向量线性无关。",
        "bbox": [
            114,
            569,
            413,
            583
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "证明 设 $\\lambda_{1},\\cdots,\\lambda_{s}$ 是 A 的 s 个不同的特征值，对应特征向量 $\\xi_{1},\\cdots,\\xi_{s}$ 。用归纳法，假设 $\\xi_{1},\\cdots,\\xi_{k-1}$ 线性无关，若",
        "bbox": [
            90,
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            907,
            634
        ],
        "page_idx": 60
    },
    {
        "type": "equation",
        "text": "$$\nc _ {1} \\xi_ {1} + \\dots + c _ {k} \\xi_ {k} = 0\\tag{5.1}\n$$",
        "text_format": "latex",
        "bbox": [
            418,
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            656
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "两侧作用 A 得到 $c_{1}\\lambda_{1}\\xi_{1}+\\cdots+c_{k}\\lambda_{k}\\xi_{k}=0$ ，用该式子减去 $\\lambda_{k}$ 倍的 (5.1) 得到",
        "bbox": [
            90,
            668,
            662,
            683
        ],
        "page_idx": 60
    },
    {
        "type": "equation",
        "text": "$$\nc _ {1} \\left(\\lambda_ {1} - \\lambda_ {k}\\right) \\xi_ {1} + \\dots + c _ {k - 1} \\left(\\lambda_ {k - 1} - \\lambda_ {k}\\right) \\xi_ {k - 1} = 0\n$$",
        "text_format": "latex",
        "bbox": [
            324,
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            706
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "由于 $\\lambda_{i}$ 两两不相等，因此 $c_{1}=\\cdots=c_{k-1}=0$ ，即特征向量彼此线性无关。",
        "bbox": [
            92,
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        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "定理 5.28 (可对角化条件)",
        "text_level": 2,
        "bbox": [
            124,
            740,
            312,
            755
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "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": [
            115,
            760,
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            774
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "\\- 充要条件 1: $\\mathcal{A}$ 有 $n$ 个线性无关特征向量。",
        "bbox": [
            139,
            778,
            470,
            793
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "\\- 充分条件：若 $\\mathcal{A}$ 有 $n$ 个不同特征值，则 $\\mathcal{A}$ 可对角化",
        "bbox": [
            139,
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            542,
            810
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "\\- 充要条件2： $V$ 是所有特征子空间的直和 $V = \\bigoplus_{i=1}^{m} V_i$ (特征子空间维数可能没有 $\\lambda_i$ 重数高)",
        "bbox": [
            139,
            814,
            810,
            834
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "\\- 充要条件3：所有特征值的代数重数等于几何重数",
        "bbox": [
            139,
            837,
            527,
            850
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "\\- 充要条件4： $\\dim V = \\sum_{i=1}^{m} \\dim(V_i)$",
        "bbox": [
            139,
            852,
            405,
            884
        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "\\- 充要条件 5(最小多项式): 最小多项式都是一次幂的乘积",
        "bbox": [
            139,
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            571,
            898
        ],
        "page_idx": 60
    },
    {
        "type": "header",
        "text": "5.4 特征值、相似对角化",
        "bbox": [
            721,
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        ],
        "page_idx": 60
    },
    {
        "type": "page_number",
        "text": "56",
        "bbox": [
            489,
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        ],
        "page_idx": 60
    },
    {
        "type": "text",
        "text": "证明 (1) 左推右：由对角化形式，右侧乘开 $n$ 个基向量均是特征向量，显然基向量都线性无关。",
        "bbox": [
            87,
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            99
        ],
        "page_idx": 61
    },
    {
        "type": "equation",
        "text": "$$\n(A \\eta_ {1}, A \\eta_ {2}, \\dots , A \\eta_ {n}) = (\\eta_ {1}, \\eta_ {2}, \\dots , \\eta_ {n}) \\left[ \\begin{array}{c c c} \\lambda_ {1} & & \\\\ & \\lambda_ {2} & \\\\ & \\ddots & \\lambda_ {n} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            290,
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        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "右推左：n 个线性无关特征向量当一组基即可。",
        "bbox": [
            122,
            168,
            470,
            183
        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "(2) 根据引理5.3可知",
        "bbox": [
            122,
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            277,
            200
        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "后面三个都根据第一个可以推出",
        "bbox": [
            124,
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            219
        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "练习5.21对角化计算(1)计算 $A = \\left( \\begin{array}{ll}3 & 4\\\\ 5 & 2 \\end{array} \\right)$ 的特征值，并计算 $Q$ 使得 $Q^{-1}AQ$ 是对角矩阵",
        "bbox": [
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        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "定理 5.29 (相关矩阵的可对角化性质)",
        "text_level": 2,
        "bbox": [
            122,
            288,
            396,
            303
        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "若矩阵 $A$ 可对角化，则",
        "bbox": [
            115,
            307,
            292,
            322
        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "1. $A^{*}$ 可对角化",
        "bbox": [
            136,
            326,
            255,
            341
        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "2. $f(x)$ 为多项式，则 $f(A)$ 可对角化",
        "bbox": [
            134,
            344,
            410,
            360
        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "3. 若 $A$ 可逆，则 $A^{-1}$ 可对角化",
        "bbox": [
            134,
            363,
            374,
            378
        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "证明 (1) 设 $P^{-1}AP = \\text{diag}\\{\\lambda_1, \\cdots, \\lambda_n\\}$ ，两侧取伴随得到 $P^*A^*(P^*)^{-1} = P^{-1}A^*P = \\text{diag}\\{\\prod_{i \\neq 1} \\lambda_i, \\prod_{i \\neq 2} \\lambda_i, \\cdots, \\prod_{i \\neq n} \\lambda_i\\}$ 。其中伴随转逆用了 $A^{-1} = \\frac{1}{|A|}A^*$",
        "bbox": [
            90,
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            932,
            450
        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "(2,3) 显然",
        "bbox": [
            122,
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            200,
            469
        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "5.4.5 相似上三角型",
        "text_level": 2,
        "bbox": [
            94,
            489,
            273,
            507
        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "定理 5.30 (相似上三角型)",
        "text_level": 2,
        "bbox": [
            122,
            523,
            312,
            538
        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "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。",
        "bbox": [
            112,
            542,
            658,
            558
        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "证明 n=1 时显然成立。假设 n-1 时成立，则取 $x_{1}$ 为 A 对应 $\\lambda_{1}$ 的特征向量，构造矩阵 $P_{1}=[x_{1},\\cdots,x_{n}](x_{2}\\sim x_{n})$ 无要求)，可以得到：",
        "bbox": [
            90,
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            904,
            611
        ],
        "page_idx": 61
    },
    {
        "type": "equation",
        "text": "$$\nA P _ {1} = (x _ {1}, \\dots , x _ {n}) \\left[ \\begin{array}{c c c c} \\lambda_ {1} & b _ {1 2} & \\dots & b _ {1 n} \\\\ 0 & b _ {2 2} & \\dots & b _ {2 n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ 0 & b _ {n 1} & \\dots & b _ {n n} \\end{array} \\right] \\Rightarrow P _ {1} ^ {- 1} A P _ {1} = \\left[ \\begin{array}{c c c c} \\lambda_ {1} & b _ {1 2} & \\dots & b _ {1 n} \\\\ 0 & b _ {2 2} & \\dots & b _ {2 n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ 0 & b _ {n 1} & \\dots & b _ {n n} \\end{array} \\right]. = \\left[ \\begin{array}{c c} \\lambda_ {1} & * \\\\ 0 & A _ {1} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            137,
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        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "根据归纳假设可知对 $A_{1}$ 有可逆矩阵 Q 使得 $Q^{-1}A_{1}Q$ 为上三角且对角为特征值，取 $P_{2}=\\left[\\begin{array}{cc}1 & \\mathbf{0}^{T}\\\\ \\mathbf{0} & Q\\end{array}\\right]$ ，取 S=P₁P₂， $S^{-1}AS$ 即所求上三角",
        "bbox": [
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        ],
        "page_idx": 61
    },
    {
        "type": "header",
        "text": "5.5 最小多项式",
        "bbox": [
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        ],
        "page_idx": 61
    },
    {
        "type": "page_number",
        "text": "57",
        "bbox": [
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        ],
        "page_idx": 61
    },
    {
        "type": "text",
        "text": "5.5 最小多项式",
        "text_level": 2,
        "bbox": [
            92,
            87,
            265,
            107
        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "5.5.1 最小多项式与基本性质",
        "text_level": 2,
        "bbox": [
            92,
            126,
            352,
            143
        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "定义 5.23 (零化多项式与最小多项式)",
        "text_level": 2,
        "bbox": [
            122,
            159,
            396,
            174
        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "$A$ 是数域 $K$ 上的 $n$ 阶方阵， $g(x)$ 是 $K$ 上的多项式，",
        "bbox": [
            114,
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            193
        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "\\- 零化多项式：若 $g(A) = 0$ ，则称 $g(x)$ 是 $A$ 的一个零化多项式。",
        "bbox": [
            139,
            198,
            618,
            212
        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "\\- 最小多项式：将 $A$ 的次数最低首一零化多项式称为 $A$ 的最小多项式",
        "bbox": [
            139,
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            231
        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "定理 5.31 (最小多项式的性质)",
        "text_level": 2,
        "bbox": [
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        "page_idx": 62
    },
    {
        "type": "text",
        "text": "最小多项式有以下性质：",
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "\\- 唯一性：最小多项式是唯一的",
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "\\- 整除性：最小多项式整除任意零化多项式",
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "\\- 分块对角： $A = \\text{diag}\\{A_1, \\cdots, A_s\\}$ ，则 $m_A(\\lambda) = [m_1(\\lambda), \\cdots, m_s(\\lambda)]$ (最小公倍数)",
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "相似：相似矩阵有相同最小多项式",
        "bbox": [
            139,
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        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "证明 (1) 假设有两个不同最小多项式 $m_{1}(\\lambda), m_{2}(\\lambda)$ ，则令 $h(\\lambda) = m_{1}(\\lambda) - m_{2}(\\lambda)$ ，有 $h(A) = m_{1}(A) - m_{2}(A) = 0$ ，显然 $h(\\lambda)$ 是最小多项式，矛盾。",
        "bbox": [
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        "page_idx": 62
    },
    {
        "type": "text",
        "text": "(2) 对任意零化多项式 $f(x)$ ，其可以表示为 $f(x) = q(x)m(x) + r(x)$ ，代入 $x = A$ 得到 $r(A) = 0$ ，其只能为 $r(x) = 0$ ，否则与最小多项式矛盾，因此整除。",
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "(3) 由于 $m(A)=\\mathrm{diag}\\{m(A_{1}),\\cdots,m(A_{s})\\}$ ，最大公因式即可满足所有分块均为 0。",
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "(4) 若 $B = P^{-1}AP$ ，则 $f(B) = f(P^{-1}AP) = P^{-1}f(A)P$ ，因此可知最小多项式相同。",
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "5.5.2 Hamilton-Cayley 定理",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "定理 5.32 (Hamilton-Cayley 定理)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "$A$ 是 $K$ 上的 $n$ 阶矩阵， $f(\\lambda) = |\\lambda E - A|$ 即 $A$ 的特征多项式，则 $f(x)$ 是 $A$ 的零化多项式。",
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "证明 设 $B(\\lambda)$ 是 $\\lambda E - A$ 的伴随矩阵，则 $B(\\lambda)(\\lambda E - A) = f(\\lambda)E$ ，由于 $B(\\lambda)$ 元素含 $1 \\sim \\lambda^{n-1}$ ，因此可以写为 $B(\\lambda) = \\lambda^{n-1}B_{0} + \\cdots + B_{n-1}$ ，从而得到：",
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "equation",
        "text": "$$\nB (\\lambda) (\\lambda E - A) = \\lambda^ {n} B _ {0} + \\lambda^ {n - 1} (B _ {1} - B _ {0} A) + \\dots + \\lambda (B _ {n - 1} - B _ {n - 2} A) - B _ {n - 1} A\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 62
    },
    {
        "type": "text",
        "text": "根据 $B(\\lambda)(\\lambda E - A) = f(\\lambda)E = \\lambda^{n}E + a_{1}\\lambda^{n-1}E + \\cdots + a_{n-1}\\lambda E + a_{n}E$ 进行比较得到：",
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} B _ {0} = E \\\\ B _ {1} - B _ {0} A = a _ {1} E \\\\ \\vdots \\\\ B _ {n - 1} - B _ {n - 2} A = a _ {n - 1} E \\\\ - B _ {n - 1} A = a _ {n} E \\end{array} \\right. \\Rightarrow \\left\\{ \\begin{array}{l} B _ {0} A ^ {n} = A ^ {n} \\\\ B _ {1} A ^ {n - 1} - B _ {0} A ^ {n} = a _ {1} A ^ {n - 1} \\\\ \\vdots \\\\ B _ {n - 1} A - B _ {n - 2} A ^ {2} = a _ {n - 2} A \\\\ - B _ {n - 1} A = a _ {n} E \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 62
    },
    {
        "type": "text",
        "text": "两端分别相加，左侧为 0，因此右侧为 0",
        "bbox": [
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        "page_idx": 62
    },
    {
        "type": "text",
        "text": "证明 使用 Jordan 阵证明：分块看，显然每个 $\\lambda_{i}$ 对应的块的阶数小于 $\\lambda_{i}$ 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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        "page_idx": 62
    },
    {
        "type": "header",
        "text": "5.5 最小多项式",
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "page_number",
        "text": "58",
        "bbox": [
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        ],
        "page_idx": 62
    },
    {
        "type": "text",
        "text": "推论 5.3 (Hamilton-Cayley 定理推论)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 63
    },
    {
        "type": "text",
        "text": "Hamilton-Cayley 定理有以下推论:",
        "bbox": [
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        ],
        "page_idx": 63
    },
    {
        "type": "text",
        "text": "\\- 最小多项式与特征多项式同根(重数可以不同)。",
        "bbox": [
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        ],
        "page_idx": 63
    },
    {
        "type": "text",
        "text": "5.5.3 最小多项式与相似对角化",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 63
    },
    {
        "type": "text",
        "text": "定理 5.33 (最小多项式与相似对角化)",
        "bbox": [
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        ],
        "page_idx": 63
    },
    {
        "type": "text",
        "text": "$\\mathbb{P}$ 上的 $n$ 阶矩阵 $A$ 可对角化当且仅当 $A$ 的最小多项式在 $\\mathbb{P}$ 上可分解为互素一次因子乘积。",
        "bbox": [
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        ],
        "page_idx": 63
    },
    {
        "type": "text",
        "text": "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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        ],
        "page_idx": 63
    },
    {
        "type": "text",
        "text": "(2) 对合矩阵 $A^2 = E$ 一定可对角化",
        "bbox": [
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        ],
        "page_idx": 63
    },
    {
        "type": "text",
        "text": "(3) 幂零矩阵 $A^l = O$ 不一定可对角化",
        "bbox": [
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        ],
        "page_idx": 63
    },
    {
        "type": "text",
        "text": "证明 (1) $A(A-E)=O$ ，一定为一次因子",
        "bbox": [
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        ],
        "page_idx": 63
    },
    {
        "type": "text",
        "text": "(2) $(A + E)(A - E) = O$ ，一定可对角化",
        "bbox": [
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        ],
        "page_idx": 63
    },
    {
        "type": "text",
        "text": "(3) 例如 Jordan 块",
        "bbox": [
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        ],
        "page_idx": 63
    },
    {
        "type": "header",
        "text": "5.5 最小多项式",
        "bbox": [
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        ],
        "page_idx": 63
    },
    {
        "type": "page_number",
        "text": "59",
        "bbox": [
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        ],
        "page_idx": 63
    },
    {
        "type": "text",
        "text": "第 6 章 Jordan 标准型",
        "text_level": 1,
        "bbox": [
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "6.1 Jordan 标准形：空间分解法",
        "text_level": 2,
        "bbox": [
            92,
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "6.1.1 空间第二分解定理：循环不变子空间分解",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "定义 6.1 (幂零线性变换)",
        "text_level": 2,
        "bbox": [
            121,
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            304,
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "A 是 V 内的线性变换，若 $\\exists m$ 使得 $A^{m}=0$ ，则 A 是幂零线性变换。",
        "bbox": [
            112,
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "引理6.1(幂零基)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "A 是幂零变换， $\\forall\\alpha\\in V$ 若 $A^{k-1}\\alpha\\neq0,A^{k}\\alpha=0$ ，则 $\\alpha,A\\alpha,\\cdots,A^{k-1}\\alpha$ 线性无关。",
        "bbox": [
            112,
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "证明 设 $a_{0}\\alpha + a_{1}A\\alpha + \\cdots + a_{k-1}A^{k-1}\\alpha = 0$ ，若从左至右第一个非零的是 $a_{i}$ ，即 $a_{i}A^{i}\\alpha + \\cdots + a_{k}A^{k-1}\\alpha = 0$ ，两侧同时作用 $A^{k-1-i}$ ，上式变为 $a_{i}A^{k-1}\\alpha = 0$ ，而 $A^{k-1}\\alpha \\neq 0$ ，故 $a_{i} = 0$ ，与假设矛盾。",
        "bbox": [
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "定义 6.2 (循环不变子空间)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "对 $\\forall\\alpha\\in V,\\exists k$ 使得 $A^{k}=0,A^{k-1}\\neq0$ ，则 $I(\\alpha):=L(\\alpha,A\\alpha,\\cdots,A^{k-1}\\alpha)$ 称为 $\\alpha$ 生成的 A 的循环不变子空间。",
        "bbox": [
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "定理 6.1 (循环不变子空间分解)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "对于 V 内的幂零变换 A， $\\exists v_{1},\\cdots,v_{n}\\in V,\\ m_{1},\\cdots,m_{n}\\in N$ 使得 $V=I(v_{1})\\oplus I(v_{2})\\oplus\\cdots\\oplus I(v_{n})$ 。其中 $I(v_{i})=L(v_{i},\\mathcal{A}v_{i}\\cdots,\\mathcal{A}^{m_{i}}v_{i})$ ，且 $A^{m_{i}+1}v_{i}=0$ 。",
        "bbox": [
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "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 $A^{m_{1}}v_{1}, \\cdots, Av_{1}, v_{1}, \\cdots, A^{m_{n}}v_{n}, \\cdots, v_{n}$ ，由于上述向量组是 ImA 的一组基，对 $\\forall j$ 存在 V 中向量 $u_{j}$ 使得 $v_{j} = Au_{j}$ 。",
        "bbox": [
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "下面说明下面的向量组线性无关:",
        "bbox": [
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        "page_idx": 64
    },
    {
        "type": "equation",
        "text": "$$\n\\mathcal {A} ^ {m _ {1} + 1} u _ {1}, \\dots , \\mathcal {A} u _ {1}, u _ {1}, \\dots , \\mathcal {A} ^ {m _ {n} + 1} u _ {n}, \\dots , u _ {n}\\tag{6.1}\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 64
    },
    {
        "type": "text",
        "text": "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 $A^{m_{1}+1}u_{1},\\cdots,A^{m_{n}+1}u_{n}$ 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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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "于是将 (6.1) 扩成 V 的一组基 $A^{m_{1}+1}u_{1},\\cdots,u_{1},\\cdots,A^{m_{n}+1}u_{n},\\cdots,u_{n},w_{1},\\cdots,w_{p}$ ，由于 $Aw_{j}\\in ImA$ ，因此存在 (6.1) 张成空间中元素 $x_{j}$ ，使得 $Aw_{j}=Ax_{j}$ 。令 $u_{n+j}=w_{j}-x_{j}$ ，则有 $Au_{n+j}=0$ ，从而 (6.1) 配上 $u_{n+1},\\cdots,u_{n+p}$ 是所需 V 的一组基。",
        "bbox": [
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "6.1.2 核像升链与根子空间",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "设 $\\mathcal{A}$ 是 $V$ 上的线性变换，定义 $M_0 = \\{0\\}, M_i = \\mathrm{Ker}\\mathcal{A}^i$ ， $N_0 = V, N_i = \\mathrm{Im}(\\mathcal{A}^i)$ ，下面来研究它们的性质。",
        "bbox": [
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        "page_idx": 64
    },
    {
        "type": "text",
        "text": "引理 6.2 (核像空间升链)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 64
    },
    {
        "type": "text",
        "text": "上述空间有： $\\{0\\}=M_{0}\\subseteq M_{1}\\subseteq M_{2}\\subseteq\\cdots,\\quad V=N_{0}\\supseteq N_{1}\\supseteq N_{2}\\supseteq\\cdots$",
        "bbox": [
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        "page_idx": 64
    },
    {
        "type": "text",
        "text": "证明 (1) 对 $\\forall \\alpha \\in M_i$ ，由定义有 $\\mathcal{A}^i\\alpha = 0$ ，显然 $\\mathcal{A}^{i+1}\\alpha = 0$ ，因此 $M_i \\subseteq M_{i+1}$ 。",
        "bbox": [
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        "page_idx": 64
    },
    {
        "type": "text",
        "text": "(2) 对于 $\\alpha \\in N_i$ ，由定义 $\\exists \\beta \\in V$ 使得 $\\alpha = \\mathcal{A}^i\\beta$ ，从而 $\\alpha = \\mathcal{A}^{i-1}(\\mathcal{A}\\beta) \\in N_{i-1}$ 。由 $\\alpha$ 任意性可知 $N_i \\subseteq N_{i-1}$ 。",
        "bbox": [
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        "page_idx": 65
    },
    {
        "type": "text",
        "text": "引理 6.3 (核空间增长)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "核空间 $M_{i}$ 的增长满足以下性质：",
        "bbox": [
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "若 $M_{k} = M_{k + 1}$ ，则 $M_{k} = M_{k + 1} = M_{k + 2} = \\dots$",
        "bbox": [
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "若 dimV = n，则 $M_{n} = M_{n+1} = M_{n+2} = \\cdots$ 。",
        "bbox": [
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "证明 (1) 显然有 $M_{k+1} \\subseteq M_{k+2}$ , 只需要证明 $M_{k+2} \\subseteq M_{k+1}$ 。对于 $\\forall \\alpha \\in M_{k+2}$ , 有 $\\mathcal{A}^{k+2} \\alpha = 0$ , 即 $\\mathcal{A}^{k+1}(\\mathcal{A} \\alpha) = 0$ , 因此 $\\mathcal{A} \\alpha \\in M_{k+1} = M_k$ , 即 $\\mathcal{A}^{k+1} \\alpha = 0 \\to \\alpha \\in M_{k+1}$ , 由任意性可知 $M_{k+2} \\subseteq M_{k+1}$ , 故 $M_{k+1} = M_{k+2}$ , 后续类似。",
        "bbox": [
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "(2) 只需要证明 $M_{n} = M_{n+1}$ ，假设不然，则 $\\{0\\} = M_{0} \\subset M_{1} \\subset \\cdots \\subset M_{n} \\subset M_{n+1}, dimM_{n+1} > n$ ，与 $M_{n+1} \\subseteq V$ 矛盾。",
        "bbox": [
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "定理 6.2 (幂次变换核像分解)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "$V = M_{n} \\oplus N_{n}$ , 其中 $n = dimV$ 。",
        "bbox": [
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            323,
            361,
            338
        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "证明 显然由核像性质有 $dimM_{n} + dimN_{n} = n$ ，只需证 $M_{n} \\cap N_{n} = \\{0\\}$ 。 $\\forall v \\in M_{n} \\cap N_{n}$ ，有 $A^{n}v = 0$ ，且 $\\exists u \\in V$ 使得 $v = A^{n}u$ ，从而 $A^{2n}u = 0$ ，而 $M_{2n} = M_{2n-1} = \\cdots = M_{n}$ ，故 $A^{n}u = 0$ ，从而 v = 0。",
        "bbox": [
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "定义 6.3 (广义特征向量与根子空间 (广义特征子空间))",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "\\- 广义特征向量：线性变换 $\\mathcal{A}$ 关于特征值 $\\lambda$ 的广义特征向量定义为满足 $\\exists j \\in \\mathbb{N}^{+}, (\\mathcal{A} - \\lambda E)^{j}v = 0$ 的所有向量。",
        "bbox": [
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "\\- 根子空间：线性变换 $\\mathcal{A}$ 关于 $\\lambda$ 的所有广义特征向量加上零向量组成的空间称为 $\\mathcal{A}$ 关于特征值 $\\lambda$ 的根子空间，记作 $G(\\lambda, \\mathcal{A})$ 。",
        "bbox": [
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "定理 6.3 (根子空间的具体表达)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "根子空间 $G(\\lambda, \\mathcal{A})$ 满足 $G(\\lambda, \\mathcal{A}) = \\operatorname{Ker}(\\mathcal{A} - \\lambda E)^{dimV}$ 。",
        "bbox": [
            114,
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "证明 右推左：若 $v \\in Ker(\\mathcal{A} - \\lambda E)^{dimV}$ ，则由 $G(\\lambda, \\mathcal{A})$ 定义可知 $v \\in G(\\lambda, \\mathcal{A})$ 。左推右： $v \\in G(\\lambda, \\mathcal{A})$ 即 $\\exists j \\in \\mathbb{N}$ s.t. $(\\mathcal{A} - \\lambda E)^j v = 0$ ，而前面已证明这样的 $v$ 必然满足 $v \\in Ker(\\mathcal{A} - \\lambda E)^{dimV}$ 。",
        "bbox": [
            90,
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "笔记 为了知识体系的完整，这里补充一下： $\\lambda_{i}$ 根子空间的次数最低压缩为 $\\lambda_{i}$ 对应 Jordan 块的最高阶数，即最小多项式中的 $\\lambda_{i}$ 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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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "6.1.3 空间第一分解定理：根子空间分解",
        "text_level": 2,
        "bbox": [
            92,
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            453,
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "定理 6.4 (根子空间的不变性)",
        "text_level": 2,
        "bbox": [
            121,
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            337,
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "根子空间 $G(\\lambda, \\mathcal{A})$ 是 A 不变的。",
        "bbox": [
            114,
            738,
            349,
            753
        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "证明 由定理6.3可知 $G(\\lambda, \\mathcal{A}) = Ker(\\mathcal{A} - \\lambda E)^n$ ， $\\forall \\alpha \\in G(\\lambda, \\mathcal{A})$ 有 $(\\mathcal{A} - \\lambda E)^n \\alpha = 0$ ，显然 $(\\mathcal{A} - \\lambda E)^n \\mathcal{A} \\alpha = \\mathcal{A}((\\mathcal{A} - \\lambda E)^n \\alpha) = 0$ ，故 $\\mathcal{A} \\alpha \\in G(\\lambda, \\mathcal{A})$ 。",
        "bbox": [
            89,
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "定理 6.5 (空间第一分解定理：根子空间分解)",
        "text_level": 2,
        "bbox": [
            121,
            819,
            453,
            834
        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "若 V 是复线性空间，A 是线性变换， $\\lambda_{1},\\cdots,\\lambda_{m}$ 是其特征值，则 $V=G(\\lambda_{1},\\mathcal{A})\\oplus\\cdots\\oplus G(\\lambda_{m},\\mathcal{A})$ 。",
        "bbox": [
            112,
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            835,
            854
        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "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 $V=G(\\lambda_{1},\\mathcal{A})\\oplus Im(\\mathcal{A}-\\lambda_{1}E)^{n}$ ，若 $U=Im(\\mathcal{A}-\\lambda_{1}E)^{n}$ 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 $(\\mathcal{A}-\\lambda_{1}E)$ 可交换)，去除 $\\lambda_{1}$ 所有对应特征向量后， $A|_{U}$ 在 U 有 $\\lambda_{2},\\cdots,\\lambda_{m}$ 特征值，对 U 用归纳法， $U=G(\\lambda_{2},\\mathcal{A}|_{U})\\oplus\\cdots\\oplus G(\\lambda_{m},\\mathcal{A}|_{U})$ ，由于 $\\forall j\\geq2,G(\\lambda_{j},\\mathcal{A}|_{U})=G(\\lambda_{j},\\mathcal{A})$ ，从而得证。",
        "bbox": [
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        ],
        "page_idx": 65
    },
    {
        "type": "page_number",
        "text": "61",
        "bbox": [
            489,
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        ],
        "page_idx": 65
    },
    {
        "type": "header",
        "text": "6.1 Jordan 标准形：空间分解法",
        "bbox": [
            672,
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        ],
        "page_idx": 65
    },
    {
        "type": "text",
        "text": "",
        "bbox": [
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        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "6.1.4 Jordan 标准形：空间分解法",
        "text_level": 2,
        "bbox": [
            92,
            140,
            400,
            156
        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "定理 6.6 (幂零变换的 Jordan 标准型)",
        "text_level": 2,
        "bbox": [
            121,
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            396,
            187
        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "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。",
        "bbox": [
            115,
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        ],
        "page_idx": 66
    },
    {
        "type": "equation",
        "text": "$$\nJ = \\left[ \\begin{array}{c c c c} J _ {1} & & & \\\\ & J _ {2} & & 0 \\\\ 0 & & \\ddots & \\\\ & & & J _ {s} \\end{array} \\right], \\quad J _ {i} = \\left[ \\begin{array}{c c c c} 0 & 1 & & 0 \\\\ & 0 & \\ddots & \\\\ & & \\ddots & 1 \\\\ 0 & & & 0 \\end{array} \\right] _ {n _ {i} \\times n _ {i}}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "证明 代入计算显然可得。",
        "bbox": [
            90,
            332,
            280,
            348
        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "定理 6.7 (核限制变换是幂零变换)",
        "text_level": 2,
        "bbox": [
            122,
            357,
            369,
            373
        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "对于 $\\forall$ 线性变换 $\\mathcal{A}$ ，令 $\\mathcal{B} = \\mathcal{A} - \\lambda E$ ，则 $\\mathcal{B}$ 在 $M_{k} = Ker\\mathcal{B}^{k}$ 中的限制变换 $\\left.\\mathcal{B}\\right|_{M_k}$ 是幂零变换。",
        "bbox": [
            112,
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            394
        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "证明 由于 $M_{k} = \\mathrm{Ker}\\mathcal{B}^{k}$ ，因此 $\\forall \\alpha \\in M_{k}$ 有 $\\mathcal{B}^{k}\\alpha = 0$ ，故 $(\\mathcal{B}|_{M_k})^k = 0$ 。",
        "bbox": [
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            430
        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "定理 6.8 (Jordan 标准型：空间分解法)",
        "text_level": 2,
        "bbox": [
            122,
            441,
            408,
            456
        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "对于 $\\mathbb{C}$ 上线性变换 $\\mathcal{A}$ 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。",
        "bbox": [
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        ],
        "page_idx": 66
    },
    {
        "type": "equation",
        "text": "$$\nJ = \\left[ \\begin{array}{c c c c} J _ {1} & & & 0 \\\\ & J _ {2} & & \\\\ & & \\ddots & \\\\ 0 & & & J _ {s} \\end{array} \\right], \\quad J _ {i} = \\left[ \\begin{array}{c c c c} \\lambda_ {i} & 1 & & 0 \\\\ & \\lambda_ {i} & \\ddots & \\\\ & & \\ddots & 1 \\\\ 0 & & & \\lambda_ {i} \\end{array} \\right] _ {n _ {i} \\times n _ {i}}\n$$",
        "text_format": "latex",
        "bbox": [
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            612
        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "证明 先由空间分解第一定理将 $V$ 分解为根子空间直和，由于 $\\mathcal{A} - \\lambda E$ 在 $G(\\lambda, \\mathcal{A})$ 中的限制变换是幂零变换，因此再对 $G(\\lambda, \\mathcal{A})$ 进行循环不变子空间分解，再由幂零变换的 Jordan 标准形可证。",
        "bbox": [
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        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "6.2 空间分解法：零化多项式角度",
        "text_level": 2,
        "bbox": [
            92,
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        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "6.2.1 根子空间分解：矩阵角度",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "定理 6.9 (解空间分解)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "$A$ 是 $n$ 阶方阵， $f(x), f_1(x), f_2(x) \\in P[x]$ ，且 $f(x) = f_1(x)f_2(x), (f_1(x), f_2(x)) = 1$ ，设 $f(A)X = 0, f_1(A)X = 0, f_2(A)X = 0$ 的解空间分别为 $V, V_1, V_2$ ，则",
        "bbox": [
            112,
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        ],
        "page_idx": 66
    },
    {
        "type": "equation",
        "text": "$$\nV = V _ {1} \\oplus V _ {2}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "证明 由于互素，得到 $\\exists u(x), v(x)$ ，使得 $E = u(A)f_1(A) + v(A)f_2(A)$ ，从而 $\\forall \\alpha \\in V$ 有",
        "bbox": [
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        ],
        "page_idx": 66
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha = u (A) f _ {1} (A) \\alpha + v (A) f _ {2} (A) \\alpha := \\alpha_ {1} + \\alpha_ {2}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 66
    },
    {
        "type": "page_number",
        "text": "62",
        "bbox": [
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        ],
        "page_idx": 66
    },
    {
        "type": "header",
        "text": "6.2 空间分解法：零化多项式角度",
        "bbox": [
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        ],
        "page_idx": 66
    },
    {
        "type": "text",
        "text": "且满足 $f(A)\\alpha = f_{1}(A)f_{2}(A)\\alpha = 0$ 。故考察 $f_{2}(A)\\alpha_{1}$",
        "bbox": [
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        ],
        "page_idx": 67
    },
    {
        "type": "equation",
        "text": "$$\nf _ {2} (A) \\alpha_ {1} = f _ {2} (A) u (A) f _ {1} (A) \\alpha = u (A) f _ {1} (A) f _ {2} (A) = 0 \\Rightarrow \\alpha_ {1} \\in V _ {2}\n$$",
        "text_format": "latex",
        "bbox": [
            263,
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "同理得到 $f_{1}(A)\\alpha_{2} = 0, \\alpha_{2} \\in V_{1}$ ，从而得到 $V = V_{1} + V_{2}$ 。",
        "bbox": [
            92,
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "下证直和， $\\forall \\alpha \\in V_1\\cap V_2$ ，则 $f_{1}(A)\\alpha = f_{2}(A)\\alpha = 0$ ，因此",
        "bbox": [
            124,
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            164
        ],
        "page_idx": 67
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha = u (A) f _ {1} (A) \\alpha + v (A) f _ {2} (A) \\alpha = 0\n$$",
        "text_format": "latex",
        "bbox": [
            364,
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "因此结论成立。",
        "bbox": [
            92,
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            203,
            209
        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "定理 6.10 (多空间分解)",
        "text_level": 2,
        "bbox": [
            122,
            219,
            295,
            235
        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "A 是数域 P 上 n 阶方阵， $f(x), f_{1}(x), \\cdots, f_{s}(x) \\in P[x]$ ， $f(x) = f_{1}(x)f_{2}(x)\\cdots f_{s}(x)$ ，且 $f_{1}(x), \\cdots, f_{s}(x)$ 两两互素，设 $f(A)X = 0, f_{1}(A)X = 0, \\cdots, f_{s}(A)X = 0$ 的解空间为 $V, V_{1}, \\cdots, V_{s}$ ，则",
        "bbox": [
            112,
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            273
        ],
        "page_idx": 67
    },
    {
        "type": "equation",
        "text": "$$\nV = V _ {1} \\oplus V _ {2} \\oplus \\dots \\oplus V _ {s}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "证明 用数学归纳法即可。",
        "bbox": [
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            330
        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "定理 6.11 (空间分解研究矩阵秩)",
        "text_level": 2,
        "bbox": [
            122,
            357,
            361,
            372
        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "考虑 $f(A)=f_{1}(A)\\cdots f_{s}(A)$ ，则",
        "bbox": [
            115,
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            357,
            393
        ],
        "page_idx": 67
    },
    {
        "type": "equation",
        "text": "$$\nn - \\mathrm{r} (f (A)) = n - \\mathrm{r} (f _ {1} (A)) + \\dots + n - \\mathrm{r} (f _ {s} (A))\n$$",
        "text_format": "latex",
        "bbox": [
            319,
            401,
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            416
        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "证明 设 $f(A)X = 0, f_{1}(A)X = 0, \\cdots, f_{s}(A)X = 0$ 的解空间为 $V, V_{1}, \\cdots, V_{s}$ ，根据解空间分解得到 $V = V_{1} \\oplus \\cdots \\oplus V_{s}$ ，因此根据解空间的维数可得到结论。",
        "bbox": [
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "练习 6.1 空间分解研究矩阵的秩 证明以下命题",
        "bbox": [
            58,
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "(1) $A^{2}=A$ 当且仅当 $r(A)+r(A-E)=n$",
        "bbox": [
            122,
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            433,
            524
        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "(2) $A^2 = E$ 当且仅当 $r(A + E) + r(A - E) = n$",
        "bbox": [
            122,
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            468,
            543
        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "(3) $A^{3}=E$ 当且仅当 $r(A-E)+r(A^{2}+A+E)=n$",
        "bbox": [
            122,
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "证明 (1) $f(A)=A^{2}-A=A(A-E)$ ，因此 $n-r(A^{2}-A)=n-r(A)+n-r(A-E)$ ，左侧等于n，右侧等于 $2n-r(A)-r(A-E)$ ，因此 $r(A)+r(A-E)=n$",
        "bbox": [
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "6.2.2 根子空间分解：线性变换角度",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "引理 6.4 (零变换的核空间)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "V 是 P 上的线性空间，O 是零变换，则其核空间为整个空间：",
        "bbox": [
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        ],
        "page_idx": 67
    },
    {
        "type": "equation",
        "text": "$$\n\\operatorname{Ker} (\\mathcal {O}) = V\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "定理 6.12 (核空间分解)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "$\\mathcal{A}$ 为线性变换， $f(x) = f_{1}(x)f_{2}(x)$ 为多项式，且 $(f_{1}(x),f_{2}(x)) = 1$ ，则有如下核空间分解：",
        "bbox": [
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        ],
        "page_idx": 67
    },
    {
        "type": "equation",
        "text": "$$\n\\operatorname{Ker} (f (\\mathcal {A})) = \\operatorname{Ker} (f _ {1} (\\mathcal {A})) \\oplus \\operatorname{Ker} (f _ {2} (\\mathcal {A}))\n$$",
        "text_format": "latex",
        "bbox": [
            352,
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "证明 根据 $(f_{1}(x), f_{2}(x)) = 1$ 得到 $\\exists u(x), v(x)$ 使得 $u(x)f_{1}(x) + v(x)f_{2}(x) = 1$ ，因此",
        "bbox": [
            90,
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            709,
            848
        ],
        "page_idx": 67
    },
    {
        "type": "equation",
        "text": "$$\n\\mathcal {E} = u (\\mathcal {A}) f _ {1} (\\mathcal {A}) + v (\\mathcal {A}) f _ {2} (\\mathcal {A})\n$$",
        "text_format": "latex",
        "bbox": [
            389,
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        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "对 $\\forall\\alpha\\in\\mathrm{Ker}(f(\\mathcal{A}))$",
        "bbox": [
            90,
            881,
            233,
            895
        ],
        "page_idx": 67
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha = \\alpha_ {1} + \\alpha_ {2} := u (\\mathcal {A}) f _ {1} (\\mathcal {A}) \\alpha + v (\\mathcal {A}) f _ {2} (\\mathcal {A}) \\alpha\n$$",
        "text_format": "latex",
        "bbox": [
            337,
            898,
            658,
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        ],
        "page_idx": 67
    },
    {
        "type": "page_number",
        "text": "63",
        "bbox": [
            489,
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        ],
        "page_idx": 67
    },
    {
        "type": "header",
        "text": "6.2 空间分解法：零化多项式角度",
        "bbox": [
            653,
            55,
            902,
            68
        ],
        "page_idx": 67
    },
    {
        "type": "text",
        "text": "显然 $f_{2}(\\mathcal{A})\\alpha_{1} = u(\\mathcal{A})f(\\mathcal{A})\\alpha = 0$ ，得到 $\\alpha_{1}\\in \\mathrm{Ker}(f_{1}(\\mathcal{A}))$ ，同理有 $\\alpha_{2}\\in \\mathrm{Ker}(f_{2}(\\mathcal{A}))$ ，综上得到",
        "bbox": [
            90,
            84,
            781,
            99
        ],
        "page_idx": 68
    },
    {
        "type": "equation",
        "text": "$$\n\\operatorname{Ker} (f (\\mathcal {A})) = \\operatorname{Ker} (f _ {1} (\\mathcal {A})) + \\operatorname{Ker} (f _ {2} (\\mathcal {A}))\n$$",
        "text_format": "latex",
        "bbox": [
            352,
            108,
            643,
            123
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "下证直和：对 $\\forall \\alpha \\in \\mathrm{Ker}(f_1(\\mathcal{A}))\\cap \\mathrm{Ker}(f_2(\\mathcal{A}))$ ，有",
        "bbox": [
            124,
            131,
            490,
            146
        ],
        "page_idx": 68
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha = u (\\mathcal {A}) f _ {1} (\\mathcal {A}) \\alpha + v (\\mathcal {A}) f _ {2} (\\mathcal {A}) \\alpha = 0\n$$",
        "text_format": "latex",
        "bbox": [
            363,
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        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "因此为直和",
        "bbox": [
            92,
            177,
            179,
            191
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "定理 6.13 (全空间分解：零化多项式角度)",
        "text_level": 2,
        "bbox": [
            121,
            200,
            430,
            216
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "V 是 P 上的线性空间, A 是 V 上的线性变换, $f(x) = f_{1}(x)f_{2}(x)\\cdots f_{s}(x)$ 为零化多项式, 且 $f_{1}(x), \\cdots, f_{s}(x)$ 两两互素, 则",
        "bbox": [
            112,
            221,
            880,
            254
        ],
        "page_idx": 68
    },
    {
        "type": "equation",
        "text": "$$\nV = \\operatorname{Ker} (f _ {1} (\\mathcal {A})) \\oplus \\dots \\oplus \\operatorname{Ker} (f _ {s} (\\mathcal {A}))\n$$",
        "text_format": "latex",
        "bbox": [
            364,
            259,
            630,
            275
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "证明 由于 $f(A) = \\mathcal{O}$ ，根据零变换的核空间以及核空间分解定理可知。",
        "bbox": [
            90,
            294,
            606,
            309
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "引理 6.5 (Hamilton-Cayley)",
        "text_level": 2,
        "bbox": [
            122,
            321,
            324,
            335
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "若 $f(\\lambda)$ 是线性变换 $\\mathcal{A}$ 的特征多项式，则 $f(\\lambda)$ 是 $\\mathcal{A}$ 的零化多项式",
        "bbox": [
            114,
            340,
            600,
            356
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "证明 该定理在最小多项式一章已经证明过了",
        "bbox": [
            90,
            376,
            421,
            391
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "定理 6.14 (根子空间分解：零化多项式角度)",
        "text_level": 2,
        "bbox": [
            121,
            401,
            447,
            416
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "若 $\\mathbb{C}$ 上线性变换 $\\mathcal{A}$ 的特征多项式为 $f(\\lambda) = (\\lambda - \\lambda_1)^{k_1} \\cdots (\\lambda - \\lambda_s)^{k_s}$ ，最小多项式为 $m(\\lambda) = (\\lambda - \\lambda_1)^{l_1} \\cdots (\\lambda - \\lambda_s)^{l_s}$ ，则有以下根子空间分解",
        "bbox": [
            112,
            420,
            882,
            456
        ],
        "page_idx": 68
    },
    {
        "type": "equation",
        "text": "$$\nV = \\operatorname{Ker} (\\mathcal {A} - \\lambda_ {1} \\mathcal {E}) ^ {k _ {1}} \\oplus \\dots \\oplus \\operatorname{Ker} (\\mathcal {A} - \\lambda_ {s} \\mathcal {E}) ^ {k _ {s}} = \\operatorname{Ker} (\\mathcal {A} - \\lambda_ {1} \\mathcal {E}) ^ {l _ {1}} \\oplus \\dots \\oplus \\operatorname{Ker} (\\mathcal {A} - \\lambda_ {s} \\mathcal {E}) ^ {l _ {s}}\n$$",
        "text_format": "latex",
        "bbox": [
            176,
            462,
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            479
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "证明 根据 Hamilton-Cayley 定理以及零化多项式全空间分解可知。",
        "bbox": [
            87,
            499,
            571,
            514
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "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， $\\varphi$ 是 V 上线性变换，特征多项式 $f(\\lambda)=\\left(\\lambda-2\\right)^{6}(\\lambda+2)^{4}$ ，将 V 分解为两个非平凡不变子空间的直和，并证明结论。",
        "bbox": [
            58,
            536,
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        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "定理 6.15 (根子空间分解与 Jordan 标准型)",
        "text_level": 2,
        "bbox": [
            121,
            596,
            438,
            611
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "根子空间分解将Jordan阵按照特征值 $\\lambda_{i}$ 分为若干份，每份由 $\\lambda_{i}$ 为对角的若干个Jordan块组成，对角 $\\lambda_{i}$ 在Jordan阵中的出现次数等于其代数重数。",
        "bbox": [
            112,
            615,
            884,
            650
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "6.2.3 Jordan 阵与循环不变子空间",
        "text_level": 2,
        "bbox": [
            92,
            687,
            400,
            703
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "引理 6.6 (核限制变换是幂零变换)",
        "text_level": 2,
        "bbox": [
            122,
            720,
            371,
            734
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "$\\lambda_{i}$ 为 $\\mathcal{A}$ 的特征值，则 $(\\mathcal{A} - \\lambda_i\\mathcal{E})\\big|_{\\mathrm{Ker}(\\mathcal{A} - \\lambda_i\\mathcal{E})^k}$ 在 $\\mathrm{Ker}(\\mathcal{A} - \\lambda_i\\mathcal{E})^k$ 中为幂零变换",
        "bbox": [
            112,
            739,
            682,
            760
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "证明 由于 $\\forall \\alpha \\in \\mathrm{Ker}(\\mathcal{A} - \\lambda_i\\mathcal{E})^k$ 满足 $(\\mathcal{A} - \\lambda_i\\mathcal{E})^k\\alpha = 0$ ，因此限制变换满足 $(\\mathcal{A} - \\lambda_i\\mathcal{E})^k\\big|_{\\mathrm{Ker}(\\mathcal{A} - \\lambda_i\\mathcal{E})^k} = \\mathcal{O}$",
        "bbox": [
            89,
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        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "定理 6.16 (循环不变子空间分解)",
        "text_level": 2,
        "bbox": [
            122,
            810,
            361,
            825
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "对于 V 内的幂零变换 A， $\\exists v_{1},\\cdots,v_{n}\\in V,\\ m_{1},\\cdots,m_{n}\\in N$ 使得 $V=I(v_{1})\\oplus I(v_{2})\\oplus\\cdots\\oplus I(v_{n})$ 。其中 $I(v_{i})=L(v_{i},\\mathcal{A}v_{i}\\cdots,\\mathcal{A}^{m_{i}}v_{i})$ ，且 $A^{m_{i}+1}v_{i}=0$ 。",
        "bbox": [
            110,
            829,
            894,
            869
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "证明 在空间第二分解定理中已经证明过了。",
        "bbox": [
            89,
            884,
            413,
            898
        ],
        "page_idx": 68
    },
    {
        "type": "page_number",
        "text": "64",
        "bbox": [
            489,
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            965
        ],
        "page_idx": 68
    },
    {
        "type": "header",
        "text": "6.2 空间分解法：零化多项式角度",
        "bbox": [
            653,
            55,
            904,
            68
        ],
        "page_idx": 68
    },
    {
        "type": "text",
        "text": "定理 6.17 (Jordan 阵与循环不变子空间)",
        "text_level": 2,
        "bbox": [
            122,
            86,
            416,
            102
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "线性变换 A 在根子空间 $\\mathrm{Ker}(\\mathcal{A}-\\lambda_{i}\\mathcal{E})^{k}$ 下的限制变换 $A|_{\\mathrm{Ker}(\\mathcal{A}-\\lambda_{i}\\mathcal{E})^{k}}$ 在一组循环不变基 $((\\mathcal{A}-\\lambda_{i}\\mathcal{E})^{s}\\alpha, (\\mathcal{A}-\\lambda_{i}\\mathcal{E})^{s-1}\\alpha,\\cdots,\\alpha)$ 下的矩阵为一个 Jordan 块：",
        "bbox": [
            114,
            107,
            878,
            141
        ],
        "page_idx": 69
    },
    {
        "type": "equation",
        "text": "$$\nJ _ {s} (\\lambda_ {i}) = \\left[ \\begin{array}{c c c c c} \\lambda_ {i} & 1 & & & \\\\ & \\lambda_ {i} & 1 & & \\\\ & & \\ddots & \\ddots & \\\\ & & & \\lambda_ {i} & 1 \\\\ & & & & \\lambda_ {i} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            366,
            146,
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            244
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "证明 直接验证即可。",
        "bbox": [
            90,
            263,
            245,
            279
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "定理 6.18 (Jordan 阵与几何重数)",
        "text_level": 2,
        "bbox": [
            122,
            288,
            368,
            304
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "特征值 $\\lambda_{i}$ 的代数重数是特征多项式 $f(\\lambda)$ 的 $\\lambda = \\lambda_{i}$ 根的重数，几何重数 $(\\lambda_{i}E - A = 0$ 解空间维数) 是 $\\lambda_{i}$ 对应以 $\\lambda_{i}$ 为对角的 Jordan 块个数。",
        "bbox": [
            114,
            309,
            894,
            344
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "证明 根据 $\\lambda_{i}E - A = 0$ 的解空间维数等于 $\\lambda_{i}E - J = 0$ 解空间维数，而每有一个以 $\\lambda_{i}$ 为对角的Jordan块， $\\lambda_{i}E - J$ 上就有一行全为0，因此秩减1，从而块数等于几何重数。",
        "bbox": [
            90,
            362,
            907,
            395
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "笔记 即以 $\\lambda_{i}$ 为对角的 Jordan 块个数为 $n - r(\\lambda E - A)$ ，反过来写也要分得清。",
        "bbox": [
            89,
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            414
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "6.3 Jordan 标准形： $\\lambda$ 矩阵法",
        "text_level": 2,
        "bbox": [
            92,
            441,
            411,
            461
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "6.3.1 $\\lambda$ 矩阵与相抵",
        "text_level": 2,
        "bbox": [
            94,
            480,
            268,
            497
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "定义 6.4 ( $\\lambda$ 矩阵)",
        "bbox": [
            122,
            512,
            250,
            527
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "$A(\\lambda)$ 是元素为 $\\lambda$ 多项式的矩阵，则将 $A(\\lambda)$ 称为 $\\lambda$ 矩阵。",
        "bbox": [
            114,
            533,
            532,
            548
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "定义 6.5 ( $\\lambda$ 矩阵初等变换)",
        "text_level": 2,
        "bbox": [
            122,
            573,
            317,
            588
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "$\\lambda$ 矩阵有三种初等变换：",
        "bbox": [
            115,
            594,
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            608
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "\\- 互换两行/列",
        "bbox": [
            139,
            613,
            250,
            626
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "\\- 用常数 $c$ 乘一行/列 (不能乘 $\\lambda$ 多项式！)",
        "bbox": [
            139,
            630,
            452,
            645
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "\\- 将一行/列乘以一个 $\\lambda$ 多项式加到另一行/列上",
        "bbox": [
            139,
            648,
            490,
            663
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "定义 6.6 (相抵)",
        "text_level": 2,
        "bbox": [
            122,
            688,
            236,
            703
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "若可以用初等变换将一个 $\\lambda$ 矩阵转换为另一个 $\\lambda$ 矩阵，则称两个 $\\lambda$ 矩阵相抵。",
        "bbox": [
            115,
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            722
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "定理 6.19 (相抵标准形：法式)",
        "text_level": 2,
        "bbox": [
            122,
            747,
            344,
            763
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "对于数值矩阵 A， $\\lambda E - A$ 总是相抵于 $\\mathrm{diag}\\{1,\\cdots,1,d_{1}(\\lambda),\\cdots,d_{m}(\\lambda)\\}$ ，其中 $d_{i}(\\lambda)$ 是首一多项式，且 $d_{i}(\\lambda)|d_{i+1}(\\lambda)$ 。",
        "bbox": [
            112,
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            803
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "引理 6.7 (对角整除化)",
        "text_level": 2,
        "bbox": [
            124,
            826,
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        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "$A(\\lambda)$ 为非零 $\\lambda$ 矩阵，则 $A(\\lambda)$ 一定相抵于 $B(\\lambda)$ ，满足 $b_{11}(\\lambda) \\neq 0$ 且 $b_{11}(\\lambda)$ 可整除 $B(\\lambda)$ 中的所有元素",
        "bbox": [
            114,
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            863
        ],
        "page_idx": 69
    },
    {
        "type": "text",
        "text": "证明 先将 $A(\\lambda)$ 中次数最低的元素放置于对角 $a_{11}(\\lambda)$",
        "bbox": [
            90,
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        ],
        "page_idx": 69
    },
    {
        "type": "page_number",
        "text": "65",
        "bbox": [
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        ],
        "page_idx": 69
    },
    {
        "type": "header",
        "text": "6.3 Jordan 标准形： $\\lambda$ 矩阵法",
        "bbox": [
            692,
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            68
        ],
        "page_idx": 69
    },
    {
        "type": "image",
        "img_path": "images/f82ca568e8c477de45bbe1b6a8fb8599b6d29b00dd1fab12ae185a1b2156e3f9.jpg",
        "image_caption": [],
        "image_footnote": [],
        "content": "",
        "bbox": [
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        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "(1) 消去第一行、第一列：不妨设 $a_{i1}(\\lambda)$ 不被 $a_{11}(\\lambda)$ 整除，则做带余除法 $a_{i1}(\\lambda) = q(\\lambda)a_{11}(\\lambda) + r(\\lambda)$ ，再将 $r(\\lambda)$ 放置于 $a_{11}$ ，直至全部整除，可消去第一列和第一行。",
        "bbox": [
            87,
            85,
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            118
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "(2) 考虑第一列、第一行外的其他元素，若存在 $a_{ij}(\\lambda)$ 不被 $a_{11}$ 整除，则把第 $j$ 列加到第一列上，做带余除法，把余项放到 $a_{11}$ ，以此类推。",
        "bbox": [
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        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "定理 6.20 (法式的计算：初等变换法)",
        "text_level": 2,
        "bbox": [
            122,
            165,
            396,
            180
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "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": [
            114,
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            199
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "\\- 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 $\\lambda$ 多项式",
        "bbox": [
            139,
            204,
            682,
            218
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "\\- 将矩阵对角整除化：具体见引理的过程",
        "bbox": [
            139,
            222,
            448,
            236
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "\\- 消去一列，再消去一行，重复对角整除化和消去",
        "bbox": [
            139,
            240,
            515,
            255
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "定理 6.21 (相抵与相似)",
        "text_level": 2,
        "bbox": [
            122,
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            295,
            294
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "A 与 B 相似当且仅当 $\\lambda E - A$ 与 $\\lambda E - B$ 相抵。",
        "bbox": [
            115,
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            463,
            313
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "证明 (1) 左推右：若 $B = P^{-1}AP$ ，则 $\\lambda E - B = P^{-1}(\\lambda E - A)P$ ，显然右侧两个 $\\lambda$ 矩阵相抵。",
        "bbox": [
            89,
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        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "(2) 右推左：若 $\\lambda E - B = P(\\lambda)(\\lambda E - A)Q(\\lambda)$ ，设 $Q(\\lambda) = Q_k \\lambda^k + \\cdots + Q_1 \\lambda + Q_0$ ， $Q(\\lambda)^{-1} = R_m \\lambda^m + \\cdots + R_1 \\lambda + R_0$ ，下记 $W = Q(B)$ 。由 $Q(\\lambda)Q^{-1}(\\lambda) = E$ 得 $R_m Q(\\lambda) \\lambda^m + \\cdots + R_1 Q(\\lambda) \\lambda + R_0 Q(\\lambda) = E$ ，代入 B 得到 $R_m WB^m + \\cdots + R_1 WB + R_0 W = E$ 。",
        "bbox": [
            90,
            351,
            904,
            404
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "由于 $P(\\lambda)^{-1}(\\lambda E - B) = (\\lambda E - A)Q(\\lambda) = Q(\\lambda)\\lambda - AQ(\\lambda)$ ，代入 $\\lambda = B$ 得到 $Q(B)B = AQ(B)$ (前面一式子左 $= 0$ )，即 $WB = AW$ ，从而 $WB^{l} = A^{l}W$ (多作用一个 $A$ 相当于多右乘一个 $B$ )。",
        "bbox": [
            90,
            407,
            905,
            441
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "再由前面的式子得到 $(R_{m}A^{m}+\\cdots+R_{0})W=E$ ，则表明W可逆，从而 $B=W^{-1}AW$ 。",
        "bbox": [
            122,
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            771,
            458
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "6.3.2 行列式因子、不变因子",
        "text_level": 2,
        "bbox": [
            94,
            481,
            352,
            498
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "定义 6.7 (行列式 k 阶子式)",
        "bbox": [
            122,
            513,
            322,
            527
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "$|A|$ 的 k 阶子式是其任选 k 行、k 列组成的矩阵的行列式。",
        "bbox": [
            112,
            533,
            537,
            548
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "定义 6.8 (行列式因子)",
        "text_level": 2,
        "bbox": [
            122,
            574,
            285,
            589
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "$A(\\lambda)$ 是 $\\lambda$ 矩阵，若其所有 $k$ 阶子式(行列式)均为0，则定义其 $k$ 阶行列式因子 $D_{k}(\\lambda) = 0$ ，否则定义为所有非零 $k$ 阶子式的首一最大公因式。",
        "bbox": [
            112,
            594,
            880,
            627
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "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": [
            87,
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            678
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "引理 6.8 (行列式因子的相除性)",
        "text_level": 2,
        "bbox": [
            122,
            690,
            354,
            703
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "设 $D_{1}(\\lambda),\\cdots,D_{r}(\\lambda)$ 是 $A(\\lambda)$ 非零行列式因子，则 $D_{i}(\\lambda)|D_{i+1}(\\lambda)$ 。",
        "bbox": [
            115,
            709,
            600,
            725
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "证明 设 $A_{i+1}$ 是 $A(\\lambda)$ 任意 $i + 1$ 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 $i$ 阶子式乘积，而 $D_i(\\lambda)$ 是所有 $i$ 阶子式公因子，因此 $D_i(\\lambda)|_{A_{i+1}(\\forall i)}$ ，而 $D_{i+1}(\\lambda)$ 是所有 $A_{i+1}$ 的最大公因式，因此证毕。",
        "bbox": [
            89,
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            797
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "定义 6.9 (不变因子)",
        "text_level": 2,
        "bbox": [
            122,
            807,
            268,
            822
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "定义 $d_{1}(\\lambda) = D_{1}(\\lambda)$ ， $d_{i}(\\lambda) = \\frac{D_{i}(\\lambda)}{D_{i - 1}(\\lambda)} (\\forall i\\neq 1)$ 为 $A(\\lambda)$ 的不变因子。",
        "bbox": [
            114,
            826,
            618,
            858
        ],
        "page_idx": 70
    },
    {
        "type": "page_number",
        "text": "66",
        "bbox": [
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        ],
        "page_idx": 70
    },
    {
        "type": "header",
        "text": "6.3 Jordan 标准形： $\\lambda$ 矩阵法",
        "bbox": [
            692,
            55,
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            68
        ],
        "page_idx": 70
    },
    {
        "type": "text",
        "text": "定理 6.22 (行列式因子、不变因子的性质)",
        "text_level": 2,
        "bbox": [
            122,
            86,
            428,
            102
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "\\- 相抵矩阵的因子：相抵的 $\\lambda$ 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": [
            139,
            105,
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            139
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "\\- 法式与不变因子： $\\lambda E - A$ 法式的所有对角元素即 $\\lambda E - A$ 的不变因子。",
        "bbox": [
            139,
            143,
            680,
            158
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "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": [
            90,
            177,
            773,
            193
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "(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 $\\lambda E - A$ 的不变因子即其法式对角元素。",
        "bbox": [
            90,
            196,
            904,
            229
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "定理 6.23 (不变因子与最小多项式)",
        "text_level": 2,
        "bbox": [
            122,
            237,
            379,
            253
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "$A$ 的最小多项式即 $\\lambda E - A$ 的最后一个不变因子",
        "bbox": [
            114,
            259,
            468,
            274
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "证明 设 A 的互异特征值为 $\\lambda_{1},\\cdots,\\lambda_{s}$ ，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": [
            89,
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            326
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "6.3.3 初等因子与 Jordan 标准型",
        "text_level": 2,
        "bbox": [
            94,
            349,
            384,
            366
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "定义 6.10 (初等因子)",
        "text_level": 2,
        "bbox": [
            122,
            381,
            278,
            395
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "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": [
            114,
            401,
            697,
            417
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "定理 6.24 (对角化计算初等因子)",
        "text_level": 2,
        "bbox": [
            122,
            441,
            361,
            456
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "$\\lambda E-A$ 经过初等变化变为对角阵 (不一定要法式)",
        "bbox": [
            114,
            460,
            480,
            475
        ],
        "page_idx": 71
    },
    {
        "type": "equation",
        "text": "$$\n\\left( \\begin{array}{c c c c} f _ {1} (\\lambda) & & & \\\\ & f _ {2} (\\lambda) & & \\\\ & & \\ddots & \\\\ & & & f _ {n} (\\lambda) \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            381,
            479,
            613,
            558
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "其中 $f_{i}(\\lambda)$ 为非零首一多项式，则 $A$ 的初等因子组是所有 $f_{i}(\\lambda)$ 做不可约分解得到的因式组合。",
        "bbox": [
            115,
            562,
            808,
            577
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "引理 6.9 (Jordan 块、Jordan 阵的初等因子)",
        "text_level": 2,
        "bbox": [
            124,
            602,
            447,
            618
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "r 阶 $\\lambda_{0}$ 对角的 Jordan 块的初等因子组为 $(\\lambda - \\lambda_{0})^{r}$ 。对角为 $J_{1}, J_{2}, \\cdots, J_{k}$ 的 Jordan 阵的初等因子组为 $(\\lambda - \\lambda_{1})^{r_{1}}, \\cdots, (\\lambda - \\lambda_{k})^{r_{k}}$ 。",
        "bbox": [
            114,
            623,
            880,
            658
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "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 $r$ 阶非 1，因此不变因子好算)",
        "bbox": [
            90,
            677,
            788,
            692
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "(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。",
        "bbox": [
            122,
            696,
            823,
            711
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "定理 6.25 (使用初等因子计算 Jordan 标准型)",
        "text_level": 2,
        "bbox": [
            122,
            722,
            455,
            736
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "若 K 上的矩阵 A 有初等因子组 $(\\lambda - \\lambda_{1})^{r_{1}}, \\cdots, (\\lambda - \\lambda_{k})^{r_{k}}$ ，则 A 相似于 Jordan 阵 $J = \\text{diag}\\{J_{1}, \\cdots, J_{k}\\}$ ，其中 $J_{i}$ 是以 $\\lambda_{i}$ 为对角的 $r_{i}$ 阶 Jordan 块。",
        "bbox": [
            114,
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            885,
            775
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "定理 6.26 (Jordan 标准型存在充要条件)",
        "text_level": 2,
        "bbox": [
            122,
            799,
            418,
            814
        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "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。",
        "bbox": [
            114,
            819,
            657,
            834
        ],
        "page_idx": 71
    },
    {
        "type": "header",
        "text": "6.4 $\\lambda$ 矩阵的进一步研究",
        "bbox": [
            724,
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        ],
        "page_idx": 71
    },
    {
        "type": "page_number",
        "text": "67",
        "bbox": [
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        ],
        "page_idx": 71
    },
    {
        "type": "text",
        "text": "6.4 $\\lambda$ 矩阵的进一步研究",
        "text_level": 2,
        "bbox": [
            92,
            87,
            359,
            108
        ],
        "page_idx": 72
    },
    {
        "type": "text",
        "text": "6.4.1 Jordan 标准型的直接计算",
        "text_level": 2,
        "bbox": [
            92,
            127,
            381,
            143
        ],
        "page_idx": 72
    },
    {
        "type": "text",
        "text": "一、初等变换方法：只需要将 $\\lambda E - A$ 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",
        "bbox": [
            87,
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        ],
        "page_idx": 72
    },
    {
        "type": "text",
        "text": "练习 6.3 无法直接整除的情况 (1) 计算下面矩阵 A 的法式:",
        "bbox": [
            58,
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            519,
            225
        ],
        "page_idx": 72
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c c} - 1 & 1 & 3 \\\\ 3 & 0 & - 4 \\\\ - 2 & 1 & 4 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            415,
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        ],
        "page_idx": 72
    },
    {
        "type": "text",
        "text": "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",
        "bbox": [
            87,
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            744,
            306
        ],
        "page_idx": 72
    },
    {
        "type": "equation",
        "text": "$$\n\\left[\\begin{array}{c c c}1&0&0\\\\0&\\lambda - 1&- \\lambda + 1\\\\0&\\lambda^ {2} + \\lambda - 3&- 3 \\lambda + 4\\end{array}\\right]\\rightarrow \\left[\\begin{array}{c c c}1&0&0\\\\0&\\lambda - 1&- \\lambda + 1\\\\0&(\\lambda^ {2} + \\lambda - 3) - (\\lambda - 1) (\\lambda + 2)&(- 3 \\lambda + 4) - (\\lambda + 2) (- \\lambda + 1)\\end{array}\\right] = \\left[\\begin{array}{c c c}1&0&\\\\0&\\lambda - 1&- \\lambda\\\\0&- 1&\\lambda^ {2} - \\lambda^ {2}\\end{array}\\right]\n$$",
        "text_format": "latex",
        "bbox": [
            92,
            309,
            1000,
            368
        ],
        "page_idx": 72
    },
    {
        "type": "text",
        "text": "因此最终可以化简为：",
        "bbox": [
            92,
            370,
            255,
            385
        ],
        "page_idx": 72
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c c c} 1 & & & \\\\ & 1 & & \\\\ & & (\\lambda - 1) ^ {3} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            423,
            382,
            573,
            444
        ],
        "page_idx": 72
    },
    {
        "type": "text",
        "text": "练习6.4初等变换方法计算Jordan标准型 计算 $A = \\left[ \\begin{array}{ccc}1 & -1 & 2\\\\ 3 & -3 & 6\\\\ 2 & -2 & 4 \\end{array} \\right]$ 的Jordan标准型解通过初等变换得到 $\\lambda E - A$ 的法式为 $\\left[ \\begin{array}{lll}1 & 0 & 0\\\\ 0 & \\lambda & 0\\\\ 0 & 0 & -\\lambda^2 +2\\lambda \\end{array} \\right]$ ，因此初等因子为 $\\lambda ,\\lambda ,( \\lambda -2)$ ，对应Jordan阵为： $J = \\left[ \\begin{array}{ccccc}0 &  &  &  \\\\  & 0 &  &  \\\\  &  & 2 \\end{array} \\right]$",
        "bbox": [
            58,
            451,
            910,
            634
        ],
        "page_idx": 72
    },
    {
        "type": "text",
        "text": "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。",
        "bbox": [
            87,
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        ],
        "page_idx": 72
    },
    {
        "type": "text",
        "text": "练习 6.5 经典而重要的题目 $a, b \\in C$ ，求下面矩阵 A 的 Jordan 标准型",
        "bbox": [
            58,
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        ],
        "page_idx": 72
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c c c c} a & b & \\dots & b & b \\\\ & a & \\dots & b & b \\\\ & & \\ddots & \\vdots & \\vdots \\\\ & & & a & b \\\\ & & & & a \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            396,
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            816
        ],
        "page_idx": 72
    },
    {
        "type": "header",
        "text": "6.4 $\\lambda$ 矩阵的进一步研究",
        "bbox": [
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        ],
        "page_idx": 72
    },
    {
        "type": "page_number",
        "text": "68",
        "bbox": [
            489,
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        ],
        "page_idx": 72
    },
    {
        "type": "text",
        "text": "解 若 b=0，则 Jordan 标准型即 aE。若 $b\\neq0$ ，则考虑 $\\lambda E-A$ ：",
        "bbox": [
            89,
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            566,
            99
        ],
        "page_idx": 73
    },
    {
        "type": "equation",
        "text": "$$\n\\lambda E - A = \\left[ \\begin{array}{c c c c c} \\lambda - a & - b & \\dots & - b & - b \\\\ & \\lambda - a & \\dots & - b & - b \\\\ & & \\ddots & \\vdots & \\vdots \\\\ & & & \\lambda - a & - b \\\\ & & & & \\lambda - a \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            312,
            104,
            678,
            200
        ],
        "page_idx": 73
    },
    {
        "type": "text",
        "text": "考虑 $n - 1$ 阶子式，右下子式行列式为 $(\\lambda - a)^{n - 1}$ ，右上子式在 $\\lambda = a$ 时显然不是0(根据余数定理)，因此互素，即初等因子为 $(\\lambda - a)^n$ ，因此Jordan阵为一个 $n$ 阶的以 $a$ 为对角的Jordan块",
        "bbox": [
            89,
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            240
        ],
        "page_idx": 73
    },
    {
        "type": "text",
        "text": "练习6.6行列式因子应用相关练习 求不变因子:(1) $A=\\left[\\begin{array}{cccc} & & 1 & \\lambda+2 \\\\ & 1 & \\lambda+2 \\\\ 1 & \\lambda+2 & & \\\\ \\lambda+2 & & & \\end{array}\\right]$ (2) $A=\\left[\\begin{array}{cccc} \\lambda & -1 & & \\\\ & \\lambda & -1 & \\\\ & & \\lambda & -1 \\\\ 5 & 4 & 3 & \\lambda+2 \\end{array}\\right]$",
        "bbox": [
            58,
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            329
        ],
        "page_idx": 73
    },
    {
        "type": "text",
        "text": "解(1)左上行列式因子为-1，因此不变因子为 $1,1,1,(\\lambda+2)^{4}$",
        "bbox": [
            87,
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            546,
            351
        ],
        "page_idx": 73
    },
    {
        "type": "text",
        "text": "(2) 右上行列式因子为 -1，因此不变因子 1,1,1, $|A|$",
        "bbox": [
            122,
            356,
            505,
            370
        ],
        "page_idx": 73
    },
    {
        "type": "text",
        "text": "定理 6.27 (重要 $\\lambda$ 矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            397,
            297,
            412
        ],
        "page_idx": 73
    },
    {
        "type": "text",
        "text": "下面 $\\lambda$ 矩阵 A 的不变因子为 $1,1,\\cdots,\\lambda^{n}+a_{1}\\lambda^{n-1}+a_{2}\\lambda^{n-2}+\\cdots+a_{n-1}\\lambda+a_{n}$",
        "bbox": [
            117,
            418,
            704,
            432
        ],
        "page_idx": 73
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c c c c c} \\lambda & & & & & a _ {n} \\\\ - 1 & \\lambda & & & & a _ {n - 1} \\\\ & - 1 & \\lambda & & & a _ {n - 2} \\\\ & & \\ddots & \\ddots & & \\vdots \\\\ & & & - 1 & \\lambda & a _ {2} \\\\ & & & & - 1 & \\lambda + a _ {1} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            341,
            438,
            650,
            555
        ],
        "page_idx": 73
    },
    {
        "type": "text",
        "text": "证明 考虑左下方行列式因子，其值为 1，因此不变因子为 $1, 1, \\cdots, 1, |A|$ ，而计算出 $|A| = \\lambda^{n} + a_{1}\\lambda^{n-1} + \\cdots + a_{n}$ ，因此结论成立",
        "bbox": [
            90,
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        ],
        "page_idx": 73
    },
    {
        "type": "text",
        "text": "推论6.1",
        "text_level": 2,
        "bbox": [
            122,
            618,
            188,
            633
        ],
        "page_idx": 73
    },
    {
        "type": "text",
        "text": "下列矩阵 A 在 C 可对角化当且仅当 $\\lambda^{n} + a_{1}\\lambda^{n-1} + a_{2}\\lambda^{n-2} + \\cdots + a_{n-1}\\lambda + a_{n}$ 在 C 上无重根",
        "bbox": [
            115,
            636,
            801,
            653
        ],
        "page_idx": 73
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c c c c c} 0 & & & & & a _ {n} \\\\ - 1 & 0 & & & & a _ {n - 1} \\\\ & - 1 & 0 & & & a _ {n - 2} \\\\ & & \\ddots & \\ddots & & \\vdots \\\\ & & & - 1 & 0 & a _ {2} \\\\ & & & & - 1 & a _ {1} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            346,
            657,
            645,
            774
        ],
        "page_idx": 73
    },
    {
        "type": "text",
        "text": "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) $A = \\left[ \\begin{array}{cccc}1 & 2 & 3 & 4\\\\ & 1 & 2 & 3\\\\ & & 1 & 2\\\\ & & & 1 \\end{array} \\right]$ (2) $A = \\left[ \\begin{array}{cccc}a & & & \\\\ a & a & & \\\\ \\vdots & \\vdots & \\ddots & \\\\ a & a & \\cdots & a \\end{array} \\right]$",
        "bbox": [
            58,
            804,
            946,
            887
        ],
        "page_idx": 73
    },
    {
        "type": "page_number",
        "text": "69",
        "bbox": [
            489,
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            964
        ],
        "page_idx": 73
    },
    {
        "type": "header",
        "text": "6.4 $\\lambda$ 矩阵的进一步研究",
        "bbox": [
            724,
            55,
            905,
            68
        ],
        "page_idx": 73
    },
    {
        "type": "equation",
        "text": "$$\n(3) A = \\left[ \\begin{array}{c c c c} a & & & \\\\ b _ {1} & a & & \\\\ & \\ddots & \\ddots & \\\\ & & b _ {n - 1} & a \\end{array} \\right] (4) A = \\left[ \\begin{array}{c c c c c} a & a _ {1 2} & a _ {1 3} & \\dots & a _ {1 n} \\\\ & a & a _ {2 3} & \\dots & a _ {2 n} \\\\ & & \\ddots & & \\vdots \\\\ & & & a & a _ {n - 1, n} \\\\ & & & & a \\end{array} \\right], a _ {1 2} a _ {2 3} \\dots a _ {n - 1, n} \\neq 0\n$$",
        "text_format": "latex",
        "bbox": [
            87,
            86,
            769,
            177
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "解 (1) $\\lambda E - A$ 左上行列式为 $(\\lambda - 1)^3$ ，右上在 $\\lambda = 1$ 是显然非零，因此不变因子 $1, 1, 1, |\\lambda E - A|$",
        "bbox": [
            87,
            181,
            794,
            197
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "(2) 左上行列式 $(\\lambda - a)^{n-1}$ ，左下在 $\\lambda = a$ 显然非零，因此不变因子 $1, 1, \\cdots, 1, |\\lambda E - A|$",
        "bbox": [
            121,
            200,
            771,
            215
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "(3)(4) 显然",
        "bbox": [
            122,
            218,
            208,
            233
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "笔记 如果对角相等，则用行列式因子计算可能会比较方便",
        "bbox": [
            55,
            235,
            519,
            252
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "6.4.2 $k$ 阶Jordan块的个数",
        "text_level": 2,
        "bbox": [
            92,
            273,
            341,
            291
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "定理 6.28 (k 阶 Jordan 块个数)",
        "text_level": 2,
        "bbox": [
            121,
            306,
            352,
            321
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "$n$ 阶复矩阵 $A$ 的Jordan标准型为 $J$ ，则 $J$ 中对角元 $\\lambda$ 的 $k$ 阶Jordan块 $J_{k}(\\lambda)$ 的个数如下。这里约定 $r((A - \\lambda E_n)^0) = n$",
        "bbox": [
            112,
            326,
            880,
            360
        ],
        "page_idx": 74
    },
    {
        "type": "equation",
        "text": "$$\nr ((A - \\lambda E _ {n}) ^ {k - 1}) - 2 r ((A - \\lambda E _ {n}) ^ {k}) + r ((A - \\lambda E _ {n}) ^ {k + 1})\n$$",
        "text_format": "latex",
        "bbox": [
            292,
            362,
            702,
            380
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "证明 考虑零对角 Jordan 块的幂次:",
        "bbox": [
            90,
            399,
            349,
            413
        ],
        "page_idx": 74
    },
    {
        "type": "equation",
        "text": "$$\nr (J _ {r} (0) ^ {k}) = \\left\\{ \\begin{array}{l l} r - k, & 0 \\leq k <   r \\\\ 0, & k \\geq r \\end{array} \\right. \\quad r (J _ {r} (0) ^ {k}) - r (J _ {r} (0) ^ {k + 1}) = \\left\\{ \\begin{array}{l l} 1, & 0 \\leq k <   r \\\\ 0, & k \\geq r \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
            208,
            417,
            784,
            461
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "而非零对角 Jordan 块的幂次满足 $r(J_{r}(a)^{k}) - r(J_{r}(a)^{k+1}) = 0$ 。",
        "bbox": [
            90,
            466,
            547,
            482
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "设 A 的 Jordan 标准型为 $J = \\text{diag}\\{J_{r_1}(\\lambda_1), \\cdots, J_{r_s}(\\lambda_s)\\}$ ， $(A - \\lambda E)^k, (J - \\lambda E)^k$ 相似。根据 J 是分块对角的性质可知 $r((A - \\lambda E)^k) = r((J - \\lambda E)^k) = \\sum_{i=1}^{s} r(J_{r_i}(\\lambda_i - \\lambda)^k)$ ，因此得到：",
        "bbox": [
            89,
            483,
            904,
            536
        ],
        "page_idx": 74
    },
    {
        "type": "equation",
        "text": "$$\nr ((A - \\lambda E) ^ {k}) - r ((A - \\lambda E) ^ {k + 1}) = \\sum_ {i = 1} ^ {s} [ r (J _ {r _ {i}} (\\lambda_ {i} - \\lambda) ^ {k}) - r (J _ {r _ {i}} (\\lambda_ {i} - \\lambda) ^ {k + 1}) ]\n$$",
        "text_format": "latex",
        "bbox": [
            220,
            539,
            774,
            577
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "根据前面的分析， $r((A-\\lambda E)^{k})-r((A-\\lambda E)^{k+1})$ 等于 $\\lambda_{i}=\\lambda$ 的且阶数大于k那些块个数，同理 $r((A-\\lambda E)^{k-1})-r((A-\\lambda E)^{k})$ 等于对角为 $\\lambda$ 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。",
        "bbox": [
            87,
            580,
            905,
            615
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "练习 6.8 上述定理的应用 计算 A 的 Jordan 标准型",
        "bbox": [
            58,
            636,
            458,
            651
        ],
        "page_idx": 74
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c c c c} a & 0 & 1 & & \\\\ & a & 0 & \\ddots & \\\\ & & a & \\ddots & 1 \\\\ & & & \\ddots & 0 \\\\ & & & & a \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            396,
            653,
            596,
            763
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "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 $\\forall \\lambda \\in C, k = 1, 2, \\cdots, n$ ，均有 $r((A - \\lambda E)^{k}) = r((B - \\lambda E)^{k})$",
        "bbox": [
            58,
            774,
            905,
            808
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "证明 直接用结论即可",
        "bbox": [
            89,
            810,
            255,
            826
        ],
        "page_idx": 74
    },
    {
        "type": "header",
        "text": "6.4 $\\lambda$ 矩阵的进一步研究",
        "bbox": [
            722,
            55,
            905,
            68
        ],
        "page_idx": 74
    },
    {
        "type": "page_number",
        "text": "70",
        "bbox": [
            489,
            953,
            509,
            965
        ],
        "page_idx": 74
    },
    {
        "type": "text",
        "text": "6.4.3 分块矩阵的 Jordan 阵",
        "text_level": 2,
        "bbox": [
            92,
            86,
            344,
            102
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "定理 6.29 (分块矩阵的 Jordan 标准型)",
        "text_level": 2,
        "bbox": [
            121,
            118,
            406,
            133
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "$P$ 上 $n$ 阶矩阵 $A, B$ 无公共复特征值，且 $A, B$ 的 Jordan 标准型分别为 $J_{1}, J_{2}$ ，则对任意 $C \\in P^{m \\times n}$ ，则：",
        "bbox": [
            114,
            137,
            878,
            154
        ],
        "page_idx": 75
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} A & C \\\\ O & B \\end{array} \\right] \\sim \\operatorname{diag} \\{J _ {1}, J _ {2} \\}\\tag{#}\n$$",
        "text_format": "latex",
        "bbox": [
            403,
            156,
            594,
            196
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "证明 当 $A, B$ 无公共特征值时，根据 $AX - XB$ 的矩阵方程结论可知，对任意矩阵 $C$ ， $AX - XB = C$ 都有唯一解 $X_0$ 使得 $AX_0 - X_0B = C$ ，因此：",
        "bbox": [
            89,
            217,
            905,
            250
        ],
        "page_idx": 75
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} E _ {m} & X _ {0} \\\\ O & E _ {n} \\end{array} \\right] \\left[ \\begin{array}{c c} A & C \\\\ O & B \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {m} & - X _ {0} \\\\ O & E _ {n} \\end{array} \\right] = \\left[ \\begin{array}{c c} A & C - (A X _ {0} - X _ {0} B) \\\\ O & B \\end{array} \\right] = \\left[ \\begin{array}{c c} A & O \\\\ O & B \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            184,
            252,
            810,
            291
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "因此得到：",
        "bbox": [
            90,
            294,
            169,
            309
        ],
        "page_idx": 75
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c} A & C \\\\ O & B \\end{array} \\right] \\sim \\left[ \\begin{array}{c c} A & O \\\\ O & B \\end{array} \\right] \\sim \\left[ \\begin{array}{c c} J _ {1} & O \\\\ O & J _ {2} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            351,
            306,
            643,
            347
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "证明 第二种证明：直接看 Jordan 标准型， $\\lambda_{i}$ 为对角的 k 阶 Jordan 块结论直接往上套即可。",
        "bbox": [
            87,
            354,
            757,
            369
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "推论6.2",
        "text_level": 2,
        "bbox": [
            122,
            395,
            189,
            411
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "对于分块上三角矩阵 $A, A_{11}, A_{22}, \\cdots, A_{ss}$ 两两无公共特征值， $A_{ii}$ 的 Jordan 标准型为 $J_i$ ，则 A 的 Jordan 标准型为 $\\mathrm{diag}\\{J_1, \\cdots, J_s\\}$",
        "bbox": [
            114,
            414,
            880,
            449
        ],
        "page_idx": 75
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c c c} A _ {1 1} & A _ {1 2} & \\dots & A _ {1 s} \\\\ & A _ {2 2} & \\dots & A _ {2 s} \\\\ & & \\ddots & \\vdots \\\\ & & & A _ {s s} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            383,
            448,
            608,
            529
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "练习 6.10 分块矩阵应用 求 A 的 Jordan 标准型:",
        "bbox": [
            58,
            568,
            440,
            582
        ],
        "page_idx": 75
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c c c} 1 & 5 & - 1 & 2 \\\\ 0 & 2 & 4 & 3 \\\\ 0 & 0 & - 3 & 0 \\\\ 0 & 0 & 4 & - 3 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            403,
            586,
            588,
            659
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "解 $\\mathrm{diag}\\{1,2,J_2(-3)\\}$",
        "bbox": [
            90,
            665,
            243,
            680
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "6.4.4 过渡矩阵的计算",
        "text_level": 2,
        "bbox": [
            92,
            701,
            292,
            718
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "定理 6.30 (过渡矩阵计算总结)",
        "text_level": 2,
        "bbox": [
            122,
            734,
            344,
            747
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "若已知 A 的 Jordan 标准型为 J，要计算 P 使得 $P^{-1}AP = J$ ，则可以按照以下步骤：",
        "bbox": [
            115,
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            734,
            768
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "1. 对于非一阶，以 $\\lambda$ 为对角的 Jordan 块，设对应的 P 位置为 $[X_{1},\\cdots,X_{k}]$ ，则满足",
        "bbox": [
            134,
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            752,
            787
        ],
        "page_idx": 75
    },
    {
        "type": "equation",
        "text": "$$\nA X _ {1} = \\lambda X _ {1}, A X _ {2} = \\lambda X _ {2} + X _ {1}, \\dots , A X _ {k} = \\lambda X _ {k} + X _ {k - 1}\n$$",
        "text_format": "latex",
        "bbox": [
            310,
            797,
            727,
            810
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "设右端项为 $b = [b_{1}, \\cdots, b_{n}]^{T}$ (未知量)，统一对增广矩阵做初等变换 $[A - \\lambda E, b] \\to [A', b']$ ，选取顺序为 $X_{1}, \\cdots, X_{k}$ 。 $X_{1}$ 要保证不能选 $[0, \\cdots, 0]^{T}$ ，且选的 $X_{k}$ 要保证 $X_{k+1}$ 有解。",
        "bbox": [
            156,
            818,
            880,
            853
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "2. 对于一阶：即 $AX = \\lambda X$ 特征向量，先求完非一阶再求，不要选 $[0, \\cdots, 0]^{T}$ ，保证与其他列向量线性无关。",
        "bbox": [
            134,
            854,
            880,
            889
        ],
        "page_idx": 75
    },
    {
        "type": "header",
        "text": "6.4 $\\lambda$ 矩阵的进一步研究",
        "bbox": [
            724,
            55,
            905,
            68
        ],
        "page_idx": 75
    },
    {
        "type": "page_number",
        "text": "71",
        "bbox": [
            489,
            954,
            507,
            965
        ],
        "page_idx": 75
    },
    {
        "type": "text",
        "text": "练习 6.11 已知矩阵 A 及其 Jordan 标准型 J，求过渡矩阵 P 使得 $P^{-1}AP = J$",
        "bbox": [
            58,
            84,
            657,
            99
        ],
        "page_idx": 76
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c c} 2 & 3 & 2 \\\\ 1 & 8 & 2 \\\\ - 2 & - 1 4 & - 3 \\end{array} \\right] \\quad J = \\left[ \\begin{array}{c c c} 1 & 0 & 0 \\\\ 0 & 3 & 1 \\\\ 0 & 0 & 3 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            329,
            101,
            662,
            161
        ],
        "page_idx": 76
    },
    {
        "type": "text",
        "text": "解设 $P = (X_{1}, X_{2}, X_{3})$ ，根据 $P^{-1}AP = J$ 得到 $(AX_{1}, AX_{2}, AX_{3}) = (X_{1}, X_{2}, X_{3})J$ ，因此 $AX_{1} = X_{1}, AX_{2} = 3X_{2}, AX_{3} = X_{2} + 3X_{3}$ ，即 $(A - E)X_{1} = 0, (A - 3E)X_{2} = 0, (A - 3E)X_{3} = X_{2}$ 。先算后面非一阶，做初等变换得到：",
        "bbox": [
            89,
            164,
            905,
            216
        ],
        "page_idx": 76
    },
    {
        "type": "equation",
        "text": "$$\n[ A - 3 E | b ] \\rightarrow \\left[\\begin{array}{c c c c}{- 1}&{3}&{2}&{b _ {1}}\\\\{0}&{8}&{4}&{b _ {1} + b _ {2}}\\\\{0}&{8}&{4}&{\\frac {1}{2} b _ {1} + \\frac {5}{2} b _ {2} + b _ {3}}\\end{array}\\right]\n$$",
        "text_format": "latex",
        "bbox": [
            324,
            216,
            668,
            278
        ],
        "page_idx": 76
    },
    {
        "type": "text",
        "text": "有解必须满足 $\\frac{1}{2} b_1 + \\frac{5}{2} b_2 + b_3 = 0$ ，因此 $AX_{2} = 0$ 等价于：",
        "bbox": [
            90,
            282,
            522,
            312
        ],
        "page_idx": 76
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c c} - 1 & 3 & 2 \\\\ 0 & 8 & 4 \\\\ \\frac {1}{2} & \\frac {5}{2} & 1 \\end{array} \\right] X _ {2} = 0\n$$",
        "text_format": "latex",
        "bbox": [
            413,
            315,
            586,
            376
        ],
        "page_idx": 76
    },
    {
        "type": "text",
        "text": "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 $X_{2} = [1, - 1,2]^{T}$ ，因此 $X_{3}$ 满足",
        "bbox": [
            90,
            381,
            796,
            397
        ],
        "page_idx": 76
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c c} - 1 & 3 & 2 \\\\ 0 & 8 & 4 \\\\ 0 & 0 & 0 \\end{array} \\right] X _ {3} = \\left[ \\begin{array}{c} 1 \\\\ 0 \\\\ 0 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            396,
            400,
            601,
            458
        ],
        "page_idx": 76
    },
    {
        "type": "text",
        "text": "$X_{3}$ 取法无所谓，因此取 $X_{3} = [-1,0,0]^{T}$ 。而 $X_{1}$ 取线性无关即可，如 $X_{1} = [-2,0,1]^{T}$",
        "bbox": [
            90,
            463,
            727,
            479
        ],
        "page_idx": 76
    },
    {
        "type": "text",
        "text": "练习 6.12 已知矩阵 A 及其 Jordan 标准型 J，求过渡矩阵 P 使得 $P^{-1}AP = J$",
        "bbox": [
            58,
            500,
            655,
            514
        ],
        "page_idx": 76
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{r r r} 0 & 1 & 0 \\\\ - 4 & 4 & 0 \\\\ - 2 & 1 & 2 \\end{array} \\right], J = \\left[ \\begin{array}{c c c} 2 & & \\\\ & 2 & 1 \\\\ & & 2 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            351,
            518,
            640,
            577
        ],
        "page_idx": 76
    },
    {
        "type": "text",
        "text": "解设 $P = [X_1, X_2, X_3]$ ，则 $(AX_1, AX_2, AX_3) = (2X_1, 2X_2, X_2 + 2X_3)$ ，得到方程组 $(A - 2E)X_1 = 0, (A - 2E)X_2 = 0, (A - 2E)X_3 = X_2$ 。做初等变换后：",
        "bbox": [
            87,
            580,
            905,
            613
        ],
        "page_idx": 76
    },
    {
        "type": "equation",
        "text": "$$\n[ A | b ] \\to \\left[ \\begin{array}{c c c c} - 2 & 1 & 0 & b _ {1} \\\\ 0 & 0 & 0 & b _ {2} - 2 b _ {1} \\\\ 0 & 0 & 0 & b _ {3} - b _ {1} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            369,
            617,
            621,
            676
        ],
        "page_idx": 76
    },
    {
        "type": "text",
        "text": "因此要使 $AX_{3} = X_{2}$ 有解， $X_{2}$ 必须满足 $b_{3} = b_{1}$ ，先选取 $X_{2} = [1,2,1]^{T}$ ，解出 $X_{3} = \\left[-\\frac{1}{2},0,0\\right]^{T}$ ， $X_{1}$ 要线性无关，取 $[0,0,1]^{T}$ 即可。",
        "bbox": [
            90,
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            905,
            721
        ],
        "page_idx": 76
    },
    {
        "type": "header",
        "text": "6.4 $\\lambda$ 矩阵的进一步研究",
        "bbox": [
            724,
            55,
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            68
        ],
        "page_idx": 76
    },
    {
        "type": "page_number",
        "text": "72",
        "bbox": [
            489,
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            964
        ],
        "page_idx": 76
    },
    {
        "type": "text",
        "text": "第 7 章 双线性函数与二次型",
        "text_level": 1,
        "bbox": [
            324,
            87,
            690,
            111
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "7.1 双线性函数",
        "text_level": 2,
        "bbox": [
            94,
            145,
            265,
            164
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "7.1.1 双线性函数概念",
        "text_level": 2,
        "bbox": [
            94,
            185,
            292,
            200
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "定义 7.1 (线性函数)",
        "text_level": 2,
        "bbox": [
            122,
            217,
            270,
            231
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "$V$ 是 $K$ 上的线性空间， $f: V \\to K$ 且满足 $f(\\alpha + \\beta) = f(\\alpha) + f(\\beta), f(k\\alpha) = kf(\\alpha)$ ，则称 $f$ 是 $V$ 上的线性函数。",
        "bbox": [
            112,
            236,
            882,
            269
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "定义 7.2 (双线性函数)",
        "text_level": 2,
        "bbox": [
            122,
            294,
            287,
            309
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "$V$ 是 $K$ 上的线性空间， $f: V \\times V \\to K$ ，且满足 $f(k_1\\alpha_1 + k_2\\alpha_2, \\beta) = k_1f(\\alpha_1, \\beta) + k_2f(\\alpha_2, \\beta), f(\\alpha, l_1\\beta_1 + l_2\\beta_2) = l_1f(\\alpha, \\beta_1) + l_2f(\\alpha, \\beta_2)$ ，则称 $f$ 是 $V$ 上的双线性函数",
        "bbox": [
            112,
            313,
            882,
            348
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "7.1.2 双线性函数度量矩阵",
        "text_level": 2,
        "bbox": [
            95,
            386,
            332,
            403
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "定义 7.3 (度量矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            418,
            268,
            432
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "V 是 K 上的 n 维线性空间， $\\epsilon_{1},\\cdots,\\epsilon_{n}$ 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": [
            112,
            439,
            880,
            470
        ],
        "page_idx": 77
    },
    {
        "type": "equation",
        "text": "$$\nG = \\left[ \\begin{array}{c c c c} f (\\epsilon_ {1}, \\epsilon_ {1}) & f (\\epsilon_ {1}, \\epsilon_ {2}) & \\dots & f (\\epsilon_ {1}, \\epsilon_ {n}) \\\\ f (\\epsilon_ {2}, \\epsilon_ {1}) & f (\\epsilon_ {2}, \\epsilon_ {2}) & \\dots & f (\\epsilon_ {2}, \\epsilon_ {n}) \\\\ \\vdots & \\vdots & & \\vdots \\\\ f (\\epsilon_ {n}, \\epsilon_ {1}) & f (\\epsilon_ {n}, \\epsilon_ {2}) & \\dots & f (\\epsilon_ {n}, \\epsilon_ {n}) \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            329,
            470,
            660,
            551
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "定理 7.1 (双线性函数度量矩阵唯一确定)",
        "text_level": 2,
        "bbox": [
            122,
            575,
            421,
            590
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "若 $x, y \\in V$ 在 $\\epsilon_{1}, \\cdots, \\epsilon_{n}$ 下的坐标为 X, Y，则 $f(\\alpha, \\beta) = X^{T}GY$",
        "bbox": [
            114,
            596,
            588,
            612
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "7.1.3 双线性函数不同基下的矩阵：合同",
        "text_level": 2,
        "bbox": [
            95,
            651,
            453,
            668
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "定义 7.4 (合同)",
        "text_level": 2,
        "bbox": [
            122,
            683,
            236,
            697
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "$A, B$ 是 $K$ 上的 $n$ 阶方阵，若存在 $K$ 上的可逆 $n$ 阶方阵 $T$ 使得 $B = T^T A T$ ，则称 $A, B$ 合同。",
        "bbox": [
            114,
            703,
            801,
            719
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "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": [
            87,
            738,
            642,
            753
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "定理 7.2 (不同基下度量矩阵关系)",
        "text_level": 2,
        "bbox": [
            122,
            764,
            371,
            778
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "矩阵 $A, B$ 合同当且仅当它们是双线性函数两组基下的矩阵。",
        "bbox": [
            115,
            784,
            554,
            799
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "定理 7.3 (分块矩阵合同)",
        "text_level": 2,
        "bbox": [
            122,
            824,
            302,
            839
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "这里只考虑 $A$ 为 $n$ 阶实对称矩阵，希望对 $A$ 整体或者一部分做合同变换：",
        "bbox": [
            115,
            844,
            657,
            859
        ],
        "page_idx": 77
    },
    {
        "type": "text",
        "text": "\\- $A = \\left( \\begin{array}{cc}A_{1} & \\alpha \\\\ \\alpha^{T} & a_{nn} \\end{array} \\right)$ ，若 $A_{1}$ 可逆，则取 $P = \\left( \\begin{array}{cc}E_{n - 1} & -A_{1}^{-1}\\alpha \\\\ 0 & 1 \\end{array} \\right)$ (用列变换记忆)，得到",
        "bbox": [
            139,
            95,
            805,
            135
        ],
        "page_idx": 78
    },
    {
        "type": "equation",
        "text": "$$\nP ^ {T} A P = \\left( \\begin{array}{c c} E _ {n - 1} & 0 \\\\ 0 & a _ {n n} - \\alpha^ {T} A _ {1} ^ {- 1} \\alpha \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            376,
            139,
            660,
            177
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "$\\bullet A = \\left( \\begin{array}{cc}a_{11} & \\alpha^T\\\\ \\alpha & A_{n - 1} \\end{array} \\right)$ ，若 $a_{11}\\neq 0$ ，取 $P = \\left( \\begin{array}{cc}1 & -\\frac{1}{a_{11}}\\alpha^{T}\\\\ 0 & E_{n - 1} \\end{array} \\right)$ ，则",
        "bbox": [
            139,
            181,
            650,
            225
        ],
        "page_idx": 78
    },
    {
        "type": "equation",
        "text": "$$\nP ^ {T} A P = \\left( \\begin{array}{c c} a _ {1 1} & 0 \\\\ 0 & A _ {n - 1} - \\frac {1}{a _ {1 1}} \\alpha \\alpha^ {T} \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            378,
            231,
            657,
            278
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "\\- 若想对上面的右下角分块 $A_{n - 1} - \\frac{1}{a_{11}}\\alpha \\alpha^T$ 做合同变换，变为 $P_{1}^{T}\\left(A_{n - 1} - \\frac{1}{a_{11}}\\alpha \\alpha^{T}\\right)P_{1}$ ，则 $Q = P\\left( \\begin{array}{ll}1 & 0\\\\ 0 & P_{1} \\end{array} \\right)$ (用列变换记忆)，得到",
        "bbox": [
            139,
            279,
            882,
            350
        ],
        "page_idx": 78
    },
    {
        "type": "equation",
        "text": "$$\nQ ^ {T} A Q = \\left( \\begin{array}{c c} a _ {1 1} & 0 \\\\ 0 & P _ {1} ^ {T} \\left(A _ {n - 1} - \\frac {1}{a _ {1 1}} \\alpha \\alpha^ {T}\\right) P _ {1} \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            339,
            354,
            692,
            403
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "7.2 二次型",
        "text_level": 2,
        "bbox": [
            94,
            445,
            216,
            464
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "7.2.1 二次型概念及其矩阵",
        "text_level": 2,
        "bbox": [
            94,
            485,
            332,
            502
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "定义 7.5 (二次型)",
        "text_level": 2,
        "bbox": [
            122,
            517,
            253,
            533
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "定义多元二次多项式 $f(x_{1},x_{2},\\cdots,x_{n})=\\sum_{i=1}^{n}a_{ii}x_{i}^{2}+\\sum_{1\\leq i<j\\leq n}2a_{ij}x_{i}x_{j}$ 为二次型",
        "bbox": [
            114,
            537,
            697,
            577
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "定义 7.6 (二次型的矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            602,
            302,
            618
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "对于任意二次型 $f(x_{1},x_{2},\\cdots,x_{n})$ ，总能找到一个对称矩阵 A，使得 $f(x_{1},\\cdots,x_{n})=x^{T}Ax$ ，称矩阵 A 为二次型的矩阵",
        "bbox": [
            114,
            621,
            882,
            657
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "笔记 对于一个二次型 $f(x_{1},\\cdots,x_{n})$ ，存在不止一个 A 使得 $f(x_{1},\\cdots,x_{n})=x^{T}Ax$ ，但是二次型的矩阵特指那个对称矩阵。",
        "bbox": [
            87,
            675,
            905,
            708
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "练习7.1二次型的矩阵(1)给定二次型 $f(x_{1},x_{2},x_{3}) = 2x_{1}x_{2} + 2x_{1}x_{3} - 6x_{2}x_{3}$ ，计算二次型的矩阵",
        "bbox": [
            58,
            731,
            808,
            747
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "(2) 给定二次型 $f(X)$ ，计算二次型的矩阵：",
        "bbox": [
            122,
            750,
            436,
            765
        ],
        "page_idx": 78
    },
    {
        "type": "equation",
        "text": "$$\nf (x _ {1}, x _ {2}) = \\left( \\begin{array}{c c} x _ {1} & x _ {2} \\end{array} \\right) \\left( \\begin{array}{c c} 2 & 1 \\\\ 3 & 1 \\end{array} \\right) \\binom{x _ {1}}{x _ {2}}\n$$",
        "text_format": "latex",
        "bbox": [
            334,
            769,
            657,
            807
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "(3) 重点：求 $f(x_{1}, \\cdots, x_{n}) = \\sum_{i=1}^{m} (a_{i1}x_{1} + \\cdots + a_{in}x_{n})^{2}$ 的矩阵",
        "bbox": [
            122,
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            584,
            848
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "解(1) $A = \\left( \\begin{array}{ccc}0 & 1 & 1\\\\ 1 & 0 & -3\\\\ 1 & -3 & 0 \\end{array} \\right)$",
        "bbox": [
            89,
            848,
            299,
            904
        ],
        "page_idx": 78
    },
    {
        "type": "header",
        "text": "7.2 二次型",
        "bbox": [
            821,
            55,
            905,
            68
        ],
        "page_idx": 78
    },
    {
        "type": "page_number",
        "text": "74",
        "bbox": [
            489,
            953,
            509,
            964
        ],
        "page_idx": 78
    },
    {
        "type": "text",
        "text": "(2) 展开得到 $f(x_{1}, x_{2}) = 2x_{1}^{2} + 4x_{1}x_{2} + x_{2}^{2}$ ，因此矩阵为",
        "bbox": [
            122,
            83,
            537,
            99
        ],
        "page_idx": 79
    },
    {
        "type": "equation",
        "text": "$$\n\\left( \\begin{array}{c c} 2 & 2 \\\\ 2 & 1 \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            458,
            104,
            536,
            141
        ],
        "page_idx": 79
    },
    {
        "type": "text",
        "text": "(3) 令 $A = (a_{ij})$ ，则",
        "bbox": [
            122,
            145,
            275,
            161
        ],
        "page_idx": 79
    },
    {
        "type": "equation",
        "text": "$$\nf = \\sum_ {i = 1} ^ {m} (a _ {i 1} x _ {1} + \\dots + a _ {i n} x _ {n}) ^ {2} = \\sum_ {i = 1} ^ {m} \\left[ \\left( \\begin{array}{c c c} x _ {1} & \\dots & x _ {n} \\end{array} \\right) \\left( \\begin{array}{c} a _ {i 1} \\\\ \\vdots \\\\ a _ {i n} \\end{array} \\right) \\right] ^ {2}\n$$",
        "text_format": "latex",
        "bbox": [
            196,
            165,
            695,
            228
        ],
        "page_idx": 79
    },
    {
        "type": "equation",
        "text": "$$\n= \\sum_ {i = 1} ^ {m} \\left[ \\left( \\begin{array}{c c c} x _ {1} & \\dots & x _ {n} \\end{array} \\right) \\left( \\begin{array}{c} a _ {i 1} \\\\ \\vdots \\\\ a _ {i n} \\end{array} \\right) \\left( \\begin{array}{c c c} a _ {i 1} & \\dots & a _ {i n} \\end{array} \\right) \\left( \\begin{array}{c} x _ {1} \\\\ \\vdots \\\\ x _ {n} \\end{array} \\right) \\right] = X ^ {T} (A ^ {T} A) X\n$$",
        "text_format": "latex",
        "bbox": [
            211,
            233,
            796,
            294
        ],
        "page_idx": 79
    },
    {
        "type": "text",
        "text": "其中计算出 $A^T A$ 的过程为",
        "bbox": [
            92,
            298,
            290,
            313
        ],
        "page_idx": 79
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c} A _ {1} \\\\ A _ {2} \\\\ \\vdots \\\\ A _ {m} \\end{array} \\right] \\Rightarrow A ^ {T} A = \\left[ \\begin{array}{c c c c} A _ {1} ^ {T} & A _ {2} ^ {T} & \\dots & A _ {m} ^ {T} \\end{array} \\right] \\left[ \\begin{array}{c} A _ {1} \\\\ A _ {2} \\\\ \\vdots \\\\ A _ {m} \\end{array} \\right] = \\sum_ {i = 1} ^ {m} A _ {i} ^ {T} A _ {i}\n$$",
        "text_format": "latex",
        "bbox": [
            236,
            315,
            757,
            397
        ],
        "page_idx": 79
    },
    {
        "type": "text",
        "text": "定义 7.7 (二次型等价)",
        "text_level": 2,
        "bbox": [
            122,
            413,
            285,
            428
        ],
        "page_idx": 79
    },
    {
        "type": "text",
        "text": "给定二次型 $f = x^{T}Ax, g = y^{T}By$ ，若存在可逆矩阵 $T$ ，使得 $x = Ty$ ，且 $f = y^{T}(T^{T}AT)y = y^{T}By = g$ ，则称二次型 $f, g$ 等价（即 $A, B$ 合同）",
        "bbox": [
            114,
            432,
            880,
            468
        ],
        "page_idx": 79
    },
    {
        "type": "text",
        "text": "定理 7.4 (合同的性质)",
        "text_level": 2,
        "bbox": [
            124,
            493,
            285,
            508
        ],
        "page_idx": 79
    },
    {
        "type": "text",
        "text": "若 $A = C^T BC$ ，则",
        "bbox": [
            115,
            513,
            260,
            527
        ],
        "page_idx": 79
    },
    {
        "type": "text",
        "text": "\\- 秩不变： $r(A)=r(B)$",
        "bbox": [
            139,
            533,
            319,
            546
        ],
        "page_idx": 79
    },
    {
        "type": "text",
        "text": "\\- $C$ 不一定唯一",
        "bbox": [
            141,
            551,
            265,
            564
        ],
        "page_idx": 79
    },
    {
        "type": "text",
        "text": "\\- 分块合同：若 $A_{1}, A_{2}$ 分别与 $B_{1}, B_{2}$ 合同，则 $\\operatorname{diag}\\{A_{1}, A_{2}\\}$ 与 $\\operatorname{diag}\\{B_{1}, B_{2}\\}$ 合同",
        "bbox": [
            139,
            568,
            749,
            585
        ],
        "page_idx": 79
    },
    {
        "type": "text",
        "text": "证明 (1) 由于 $C$ 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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    },
    {
        "type": "text",
        "text": "(2)例如 $\\left( \\begin{array}{cc}1 & 0\\\\ 0 & -1 \\end{array} \\right),\\left( \\begin{array}{cc}1 & 0\\\\ 0 & 1 \\end{array} \\right)$ 在 $\\mathbb{C}$ 上合同，但是",
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    },
    {
        "type": "equation",
        "text": "$$\n\\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & - 1 \\end{array} \\right) = \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & i \\end{array} \\right) \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & 1 \\end{array} \\right) \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & i \\end{array} \\right) = \\left( \\begin{array}{c c} - 1 & 0 \\\\ 0 & i \\end{array} \\right) \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & 1 \\end{array} \\right) \\left( \\begin{array}{c c} - 1 & 0 \\\\ 0 & i \\end{array} \\right)\n$$",
        "text_format": "latex",
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    },
    {
        "type": "text",
        "text": "练习7.2 反对称矩阵的二次型 证明： $A$ 是反对称矩阵当且仅当 $\\forall X, X^T A X = 0$ 证明 左推右显然。右推左根据 $X^T A X = X^T A^T X = -X^T A X = 0$ 可知",
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    },
    {
        "type": "text",
        "text": "7.2.2 二次型的标准型存在与配方法",
        "text_level": 2,
        "bbox": [
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    },
    {
        "type": "text",
        "text": "定理 7.5 (二次型标准型的存在性)",
        "text_level": 2,
        "bbox": [
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    },
    {
        "type": "text",
        "text": "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。",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "证明 $n = 1$ 则显然。设 $n - 1$ 时成立，则考虑 $n$ 阶矩阵 $A = \\left( \\begin{array}{ll}a_{11} & \\alpha^T\\\\ \\alpha & A_1 \\end{array} \\right)$ ，这里 $A_{1}$ 对称。若 $a_{11}\\neq 0$ ，则根据",
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    },
    {
        "type": "page_number",
        "text": "75",
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    },
    {
        "type": "header",
        "text": "7.2 二次型",
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    },
    {
        "type": "image",
        "img_path": "images/745302ee047e1008392d043c393193e07f26cf5103fa699b13fee41df18db0b2.jpg",
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    {
        "type": "text",
        "text": "前面分块矩阵合同结论，取 $P = \\left( \\begin{array}{cc}1 & -\\frac{1}{a_{11}}\\alpha^T\\\\ 0 & E_{n - 1} \\end{array} \\right)$ ，得到",
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    },
    {
        "type": "equation",
        "text": "$$\nP ^ {T} A P = \\left( \\begin{array}{c c} a _ {1 1} & 0 \\\\ 0 & A _ {1} - \\frac {1}{a _ {1 1}} \\alpha \\alpha^ {T} \\end{array} \\right) := \\left( \\begin{array}{c c} a _ {1 1} & 0 \\\\ 0 & B _ {1} \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 80
    },
    {
        "type": "text",
        "text": "根据归纳假设，存在 $Q_{1}$ 使得 $Q_{1}^{T}B_{1}Q_{1}$ 为对角阵，则取 $Q = \\left( \\begin{array}{cc}1 & 0\\\\ 0 & Q_{1} \\end{array} \\right)$ ，则",
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    },
    {
        "type": "equation",
        "text": "$$\nQ ^ {T} P ^ {T} A P Q = \\left( \\begin{array}{c c} a _ {1 1} & \\\\ & Q _ {1} ^ {T} B _ {1} Q _ {1} \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "若 $a_{11} = 0$ ，但 $a_{kk}\\neq 0$ ，则取 $P(1,k)$ 为 $E_{n}$ 第一列与第 $k$ 列互换的矩阵，则 $P^T (1,k)AP(1,k)$ 的 $a_{11}^{\\prime}\\neq 0$ 且两个矩阵合同，转换为第一种情况。",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "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 $a_{12} \\neq 0$ ，则 A 如下，做相同行列变换变为：",
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        ],
        "page_idx": 80
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left( \\begin{array}{c c c} 0 & a _ {1 2} & * \\\\ a _ {1 2} & 0 & * \\\\ * & * & * \\end{array} \\right) \\to \\left( \\begin{array}{c c c} 2 a _ {1 2} & a _ {1 2} & * \\\\ a _ {1 2} & a _ {2 2} & * \\\\ * & * & * \\end{array} \\right)\n$$",
        "text_format": "latex",
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    },
    {
        "type": "text",
        "text": "转换为第一种类型",
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    },
    {
        "type": "text",
        "text": "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": "text",
        "text": "定义 7.8 (二次型标准型)",
        "text_level": 2,
        "bbox": [
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        ],
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    },
    {
        "type": "text",
        "text": "对于二次型 $f = x^{T}Ax$ ，若 $A$ 与对角阵 $D$ 合同，则称 $g = z^{T}Dz$ 为 $f$ 的标准型",
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    },
    {
        "type": "text",
        "text": "定理 7.6 (配方法)",
        "text_level": 2,
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    },
    {
        "type": "text",
        "text": "配方法按照以下步骤进行：",
        "bbox": [
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        ],
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    },
    {
        "type": "text",
        "text": "1. 若无 $x_{1}$ 的平方项，但有 $x_{1}x_{2}$ 交叉项，则用 $x_{1} = y_{1} + y_{2}, x_{2} = y_{1} - y_{2}$ 换元。",
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        ],
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    },
    {
        "type": "text",
        "text": "2. 把 $x_{1}$ 的项放在一起，全部配方。",
        "bbox": [
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        ],
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    },
    {
        "type": "text",
        "text": "3. 把凑成的 $(x_{1}+\\cdots)^{2}$ 设为新变量",
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        ],
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    },
    {
        "type": "text",
        "text": "4. 以此类推",
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    },
    {
        "type": "image",
        "img_path": "images/0b92b81a7c670121596a9995aff39914c167ca625cc92dc802e14bf367d45534.jpg",
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    },
    {
        "type": "text",
        "text": "笔记 配方法一般算过渡矩阵不太方便，不过算标准型还行",
        "text_level": 2,
        "bbox": [
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    },
    {
        "type": "text",
        "text": "练习 7.3 配方法 (1) 用配方法计算 $f(x_{1}, x_{2}, x_{3}) = 2x_{1}x_{2} + 2x_{1}x_{3} - 6x_{2}x_{3}$ 的标准型解 (1) 无 $x_{1}^{2}$ 项，因此令 $x_{1} = y_{1} + y_{2}, x_{2} = y_{1} - y_{2}, x_{3} = y_{3}$ ，得到",
        "bbox": [
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        "page_idx": 80
    },
    {
        "type": "equation",
        "text": "$$\nf = 2 (y _ {1} - y _ {3}) ^ {2} - 2 y _ {2} ^ {2} - 2 y _ {3} ^ {2} + 8 y _ {2} y _ {3}\n$$",
        "text_format": "latex",
        "bbox": [
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    },
    {
        "type": "text",
        "text": "令 $x_{1} = y_{1} - y_{3}, x_{2} = y_{2}, x_{3} = y_{3}$ 得到 $f = 2x_{1}^{2} - 2(x_{2}^{2} - 4x_{2}x_{3} + 4x_{3}^{2}) + 6x_{3}^{2}$ , 再做换元 $y_{1} = x_{1}, y_{2} = x_{2} - 2x_{3}, y_{3} = x_{3}$ , 得到",
        "bbox": [
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    },
    {
        "type": "equation",
        "text": "$$\nf = 2 y _ {1} ^ {2} - 2 y _ {2} ^ {2} + 6 y _ {3} ^ {2}\n$$",
        "text_format": "latex",
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    },
    {
        "type": "header",
        "text": "7.2 二次型",
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    },
    {
        "type": "page_number",
        "text": "76",
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    },
    {
        "type": "image",
        "img_path": "images/1fbb47a9c475507d8ef1e789ee4ed7772a02bb0019876a7edda07571148f915d.jpg",
        "image_caption": [],
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    {
        "type": "text",
        "text": "7.2.3 二次型的标准型：初等变换法",
        "text_level": 2,
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    },
    {
        "type": "text",
        "text": "定理 7.7 (标准型的计算：初等变换法)",
        "text_level": 2,
        "bbox": [
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    },
    {
        "type": "text",
        "text": "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):",
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    },
    {
        "type": "equation",
        "text": "$$\n\\binom{{{{A}}}}{{{{E}}}} \\rightarrow \\binom{{{{C ^ {T} A C}}}}{{{{C}}}} \\quad \\text {对} \\mathbf {A} \\text {行变换和对应列变换，对} \\mathbf {I} \\text {只列变换}\n$$",
        "text_format": "latex",
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    },
    {
        "type": "text",
        "text": "具体的步骤就是证明标准型存在性中的过程：若 $A$ 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": "text",
        "text": "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": "text",
        "text": "练习7.4标准型计算：初等变换法(1)简单：计算 $f = x_{1}^{2} + 2x_{1}x_{2} + 2x_{2}^{2} + 4x_{2}x_{3} + 4x_{3}^{2}$ 的标准型",
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    },
    {
        "type": "text",
        "text": "(2) 对角均零：计算 $f(x_{1}, x_{2}, x_{3}) = 2x_{1}x_{2} + 2x_{1}x_{3} - 6x_{2}x_{3}$ ，计算其标准型，并求出过渡矩阵 $C$",
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    },
    {
        "type": "text",
        "text": "解(1)结果为 $f = y_{1}^{2} + y_{2}^{2}$ ，千万小心别行列变换错了",
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    },
    {
        "type": "text",
        "text": "(2) 首先显然 $A = \\left( \\begin{array}{ccc}0 & 1 & 1\\\\ 1 & 0 & -3\\\\ 1 & -3 & 0 \\end{array} \\right)$ ，列出 $\\left( \\begin{array}{c}A\\\\ E \\end{array} \\right)$ 并做行列初等变换，由于 $A$ 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",
        "text": "$$\n\\left(\\begin{array}{c c c}0&1&1\\\\1&0&- 3\\\\1&- 3&0\\\\1&0&0\\\\0&1&0\\\\0&0&1\\end{array}\\right)\\rightarrow \\left(\\begin{array}{c c c}2&1&- 2\\\\1&0&- 3\\\\- 2&- 3&0\\\\1&0&0\\\\1&1&0\\\\0&0&1\\end{array}\\right)\\rightarrow \\left(\\begin{array}{c c c}2&1&0\\\\1&0&- 2\\\\0&- 2&- 2\\\\1&0&1\\\\1&1&1\\\\0&0&1\\end{array}\\right)\\rightarrow \\dots\n$$",
        "text_format": "latex",
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    },
    {
        "type": "text",
        "text": "最终变为结果为:",
        "bbox": [
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    },
    {
        "type": "equation",
        "text": "$$\n\\left( \\begin{array}{c c c} 2 & 0 & 0 \\\\ 0 & - \\frac {1}{2} & 0 \\\\ 0 & 0 & 6 \\\\ 1 & - \\frac {1}{2} & 3 \\\\ 1 & \\frac {1}{2} & - 1 \\\\ 0 & 0 & 1 \\end{array} \\right) = \\binom{C ^ {T} A C}{C}\n$$",
        "text_format": "latex",
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        "page_idx": 81
    },
    {
        "type": "text",
        "text": "7.2.4 二次型的标准型：正交变换法",
        "text_level": 2,
        "bbox": [
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        "page_idx": 81
    },
    {
        "type": "text",
        "text": "定理 7.8 (正交对角化的可行性)",
        "text_level": 2,
        "bbox": [
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        "page_idx": 81
    },
    {
        "type": "text",
        "text": "任何实对称矩阵 A 一定正交相似于对角阵。",
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        "page_idx": 81
    },
    {
        "type": "text",
        "text": "证明 做数学归纳法，n=1 显然，假设 n-1 成立，任取特征值 $\\lambda$ ，对应单位特征向量 $\\alpha_{1}$ ，将 $\\alpha_{1}$ 扩充为 $R^{n}$ 的一组标准正交基 $\\alpha_{1},\\cdots,\\alpha_{n}$ ，令 $T=(\\alpha_{1},\\cdots,\\alpha_{n})$ ，则",
        "bbox": [
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        "page_idx": 81
    },
    {
        "type": "equation",
        "text": "$$\nA T = T \\left( \\begin{array}{c c} \\lambda & \\alpha^ {T} \\\\ 0 & A _ {n - 1} \\end{array} \\right) \\Rightarrow T ^ {T} A T = \\left( \\begin{array}{c c} \\lambda & \\alpha^ {T} \\\\ 0 & A _ {n - 1} \\end{array} \\right)\n$$",
        "text_format": "latex",
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        "page_idx": 81
    },
    {
        "type": "text",
        "text": "由于 $T^T A T$ 对称，因此 $\\alpha^T = 0$ ，再根据归纳假设，存在正交矩阵 $P_{1}$ 使得 $P_{1}^{T}A_{n - 1}P_{1}$ 为对角阵，因此令 $P =$",
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    },
    {
        "type": "header",
        "text": "7.2 二次型",
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    },
    {
        "type": "page_number",
        "text": "77",
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    },
    {
        "type": "text",
        "text": "$\\mathrm{diag}(1, P_1)$ ，此时",
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    },
    {
        "type": "equation",
        "text": "$$\nP ^ {T} T ^ {T} A T P = \\left( \\begin{array}{c c} \\lambda & 0 \\\\ 0 & P ^ {T} A _ {n - 1} P \\end{array} \\right)\n$$",
        "text_format": "latex",
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        "page_idx": 82
    },
    {
        "type": "text",
        "text": "显然 $TP = Q$ ，则 $Q$ 也是正交阵。",
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        "page_idx": 82
    },
    {
        "type": "text",
        "text": "定理 7.9 (标准型计算：正交变换)",
        "text_level": 2,
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        "page_idx": 82
    },
    {
        "type": "text",
        "text": "正交变换计算标准型过程如下：",
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        "page_idx": 82
    },
    {
        "type": "text",
        "text": "\\- 计算 $A$ 的特征值 $\\lambda_1, \\cdots, \\lambda_n$",
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        "page_idx": 82
    },
    {
        "type": "text",
        "text": "\\- 计算 $\\lambda_{i}$ 对应的特征向量 $\\mathbf{v}_{i}$ ，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",
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    },
    {
        "type": "text",
        "text": "\\- 将特征向量单位化，组成 $T = [\\mathbf{v}_1, \\dots, \\mathbf{v}_n]$",
        "bbox": [
            139,
            247,
            472,
            262
        ],
        "page_idx": 82
    },
    {
        "type": "text",
        "text": "练习 7.5 正交变换法 (1) 已知矩阵 A 如下，(a) 若 $tE + A$ 为正定矩阵，求 t 的取值范围 (b) 求正交矩阵 T 使得 $T^{T}AT$ 为对角矩阵",
        "bbox": [
            58,
            300,
            905,
            334
        ],
        "page_idx": 82
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{r r r} 2 & 2 & - 1 \\\\ 2 & 5 & - 4 \\\\ - 2 & - 4 & 5 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            408,
            332,
            584,
            390
        ],
        "page_idx": 82
    },
    {
        "type": "text",
        "text": "解 (1) $|\\lambda E - A| = (\\lambda - 1)^{2} (\\lambda - 10)$ ，特征值为 1, 10，正定要求 t > -1。",
        "bbox": [
            87,
            399,
            608,
            414
        ],
        "page_idx": 82
    },
    {
        "type": "text",
        "text": "先求 $(E - A)X = 0$ ，得到：",
        "bbox": [
            122,
            418,
            332,
            432
        ],
        "page_idx": 82
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c c} 1 & 2 & - 2 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right] X = 0 \\Rightarrow X = k _ {1} \\left( \\begin{array}{c} - 2 \\\\ 1 \\\\ 0 \\end{array} \\right) + k _ {2} \\left( \\begin{array}{c} 2 \\\\ 0 \\\\ 1 \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            297,
            436,
            700,
            495
        ],
        "page_idx": 82
    },
    {
        "type": "text",
        "text": "单位化得到 $\\frac{1}{\\sqrt{5}} (-2,1,0)^T,\\frac{1}{\\sqrt{5}} (2,0,1)^T$",
        "bbox": [
            90,
            498,
            384,
            527
        ],
        "page_idx": 82
    },
    {
        "type": "text",
        "text": "再求 $(10E - A)X = 0$",
        "bbox": [
            124,
            527,
            297,
            541
        ],
        "page_idx": 82
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c c c} 2 & 4 & 5 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 0 \\end{array} \\right] X = 0 \\Rightarrow X = k _ {1} \\left( \\begin{array}{c} 1 \\\\ 2 \\\\ - 2 \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            349,
            539,
            647,
            600
        ],
        "page_idx": 82
    },
    {
        "type": "text",
        "text": "单位化得到 $\\frac{1}{3} (1,2, - 2)$ 。",
        "bbox": [
            92,
            607,
            268,
            633
        ],
        "page_idx": 82
    },
    {
        "type": "text",
        "text": "最终结果为:",
        "bbox": [
            124,
            633,
            221,
            646
        ],
        "page_idx": 82
    },
    {
        "type": "equation",
        "text": "$$\nT = \\left[ \\begin{array}{c c c} - \\frac {2}{\\sqrt {5}} & \\frac {2}{3 \\sqrt {5}} & \\frac {1}{3} \\\\ - \\frac {1}{\\sqrt {5}} & \\frac {4}{3 \\sqrt {5}} & \\frac {2}{3} \\\\ 0 & \\frac {5}{3 \\sqrt {5}} & - \\frac {2}{3} \\end{array} \\right], T ^ {T} A T = \\left[ \\begin{array}{c c c} 1 & & \\\\ & 1 & \\\\ & & 1 0 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            297,
            643,
            695,
            727
        ],
        "page_idx": 82
    },
    {
        "type": "text",
        "text": "定理 7.10 (二次型与特征值)",
        "text_level": 2,
        "bbox": [
            122,
            747,
            329,
            763
        ],
        "page_idx": 82
    },
    {
        "type": "text",
        "text": "设 n 阶实对称矩阵 A 的全部特征值排序为： $\\lambda_{1} \\geq \\lambda_{2} \\geq \\cdots \\geq \\lambda_{n}$ ，则",
        "bbox": [
            115,
            769,
            618,
            783
        ],
        "page_idx": 82
    },
    {
        "type": "equation",
        "text": "$$\n\\max _ {X \\in \\mathbb {R} ^ {n}} \\frac {X ^ {T} A X}{X ^ {T} X} = \\lambda_ {1}, \\min _ {X \\in \\mathbb {R} ^ {n}} \\frac {X ^ {T} A X}{X ^ {T} X} = \\lambda_ {n}\n$$",
        "text_format": "latex",
        "bbox": [
            352,
            785,
            643,
            816
        ],
        "page_idx": 82
    },
    {
        "type": "text",
        "text": "证明 由于 $\\lambda_1E - A$ 的特征值均非负，故 $\\lambda_1E - A$ 半正定，因此 $\\forall X\\in \\mathbb{R}^n$ 有 $X^{T}(\\lambda E - A)X\\geq 0$ ，得到 $X^{T}AX\\leq \\lambda_{1}X^{T}X$ ，且 $\\lambda_{1}$ 是 $\\lambda_1E - A$ 的特征值，故 $\\exists X_0\\neq 0$ 使得 $X_0^T (\\lambda_1E - A)X_0 = 0$",
        "bbox": [
            87,
            835,
            905,
            870
        ],
        "page_idx": 82
    },
    {
        "type": "text",
        "text": "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": [
            58,
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            907,
            906
        ],
        "page_idx": 82
    },
    {
        "type": "page_number",
        "text": "78",
        "bbox": [
            489,
            953,
            507,
            965
        ],
        "page_idx": 82
    },
    {
        "type": "header",
        "text": "7.2 二次型",
        "bbox": [
            821,
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            905,
            68
        ],
        "page_idx": 82
    },
    {
        "type": "text",
        "text": "证明 $A + B$ 的特征值都大于 $a + b$",
        "bbox": [
            90,
            85,
            344,
            99
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "(2) B 是 n 阶实矩阵， $B^{T}B$ 的全部特征值按照大小顺序排为 $\\lambda_{1} \\geq \\lambda_{2} \\geq \\cdots \\geq \\lambda_{n}$ ，证明：B 的任意实特征值 $\\mu$ 满足 $\\sqrt{\\lambda_{n}} \\leq |\\mu| \\leq \\sqrt{\\lambda_{1}}$",
        "bbox": [
            89,
            103,
            905,
            137
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "证明 (1) 根据前面定理可知 $X^{T}AX > aX^{T}X, X^{T}BX > bX^{T}X$ ，从而 $X^{T}(A + B)X > (a + b)X^{T}X$ ，因此 $A + B$ 的特征值大于 $a + b$",
        "bbox": [
            89,
            139,
            905,
            173
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "(2) 由于 $B\\alpha = \\mu \\alpha$ ，因此 $\\alpha^T B^T = \\mu \\alpha^T$ ，得到",
        "bbox": [
            122,
            175,
            463,
            192
        ],
        "page_idx": 83
    },
    {
        "type": "equation",
        "text": "$$\n\\alpha^ {T} B ^ {T} B \\alpha = \\mu^ {2} \\alpha^ {T} \\alpha \\in [ \\lambda_ {n} \\alpha^ {T} \\alpha , \\lambda_ {1} \\alpha^ {T} \\alpha ] \\Rightarrow \\lambda_ {n} \\leq \\mu^ {2} \\leq \\lambda_ {1}\n$$",
        "text_format": "latex",
        "bbox": [
            292,
            199,
            702,
            216
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "7.2.5 R, C 上的规范型",
        "text_level": 2,
        "bbox": [
            94,
            242,
            295,
            258
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "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 $\\mathbb{R},\\mathbb{C}$ 上可以进一步划分，即每个规范型表示一个类。",
        "bbox": [
            90,
            269,
            905,
            303
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "定理 7.11 (C 上的规范型)",
        "text_level": 2,
        "bbox": [
            122,
            312,
            312,
            326
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "$\\mathbb{C}$ 上任一二次型 $f$ 都等价于",
        "bbox": [
            114,
            332,
            319,
            347
        ],
        "page_idx": 83
    },
    {
        "type": "equation",
        "text": "$$\ng = u _ {1} ^ {2} + \\dots + u _ {r} ^ {2}\n$$",
        "text_format": "latex",
        "bbox": [
            431,
            350,
            564,
            368
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "推论 7.1 (C 上合同充要条件)",
        "text_level": 2,
        "bbox": [
            121,
            392,
            337,
            407
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "C 上两个对称矩阵合同的充要条件为它们的秩相同。",
        "bbox": [
            114,
            412,
            495,
            426
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "定理 7.12 (R 上的规范型)",
        "text_level": 2,
        "bbox": [
            122,
            451,
            312,
            464
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "$\\mathbb{R}$ 上每个二次型 $f$ 都等价于一个如下二次型。将 $p$ 称为正惯性指数， $r - p$ 称为负惯性指数",
        "bbox": [
            114,
            470,
            783,
            486
        ],
        "page_idx": 83
    },
    {
        "type": "equation",
        "text": "$$\ng = u _ {1} ^ {2} + \\dots + u _ {p} ^ {2} - u _ {p + 1} ^ {2} - \\dots - u _ {r} ^ {2}\n$$",
        "text_format": "latex",
        "bbox": [
            364,
            492,
            631,
            511
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "推论 7.2 ( $\\mathbb{R}$ 上合同充要条件)",
        "text_level": 2,
        "bbox": [
            122,
            536,
            337,
            551
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "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": [
            114,
            557,
            546,
            570
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "练习7.7规范型的应用(1)设 $A$ 是 $n$ 阶实对称矩阵， $|A| < 0$ ，证明： $\\exists X \\in \\mathbb{R}^n$ 满足 $X^T A X < 0$",
        "bbox": [
            58,
            590,
            794,
            605
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "(2) $f(X) = X^{T}AX$ 为实二次型，若存在 $X_{1}, X_{2}$ 使得 $X_{1}^{T}AX_{1} > 0, X_{2}^{T}AX_{2} < 0$ ，证明： $\\exists X_0 \\neq 0$ 使得 $X_0^TAX_0 = 0$",
        "bbox": [
            89,
            608,
            904,
            643
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "证明 (1) 由于 $|A| < 0$ , 可知 $A$ 满秩且负惯性指数个数不为零, 做非退化线性替换 $X = CY$ 即可得到 $Y^{T}C^{T}ACX$ , 这里",
        "bbox": [
            87,
            645,
            905,
            677
        ],
        "page_idx": 83
    },
    {
        "type": "equation",
        "text": "$$\nC ^ {T} A C = \\operatorname{diag} \\{1, 1, \\dots , 1, - 1, - 1, \\dots , - 1 \\}\n$$",
        "text_format": "latex",
        "bbox": [
            337,
            682,
            655,
            697
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "取 $Y = e_{n}$ 时 $Y^{T}C^{T}ACY = -1$",
        "bbox": [
            92,
            709,
            324,
            725
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "(2) 正负惯性指数均非零，因此可转换为规范型 $f(X)=y_{1}^{2}+\\cdots+y_{p}^{2}-y_{p+1}^{2}-\\cdots-y_{n}^{2}$ ，取 $Y=e_{1}+e_{p+1}$ 即可",
        "bbox": [
            90,
            728,
            905,
            762
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "7.3 (半) 正定二次型的研究",
        "text_level": 2,
        "bbox": [
            94,
            789,
            383,
            809
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "7.3.1 正定二次型的概念与判别",
        "text_level": 2,
        "bbox": [
            94,
            828,
            374,
            845
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "正定二次型是 $\\mathbb{R}$ 中二次型中的特殊等价类，其代表为 $y^{T}Ey$",
        "bbox": [
            124,
            856,
            564,
            872
        ],
        "page_idx": 83
    },
    {
        "type": "page_number",
        "text": "79",
        "bbox": [
            489,
            953,
            507,
            964
        ],
        "page_idx": 83
    },
    {
        "type": "header",
        "text": "7.3 (半) 正定二次型的研究",
        "bbox": [
            705,
            55,
            905,
            68
        ],
        "page_idx": 83
    },
    {
        "type": "text",
        "text": "定义 7.9 (正定二次型)",
        "text_level": 2,
        "bbox": [
            122,
            86,
            287,
            102
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "给定实数域上的二次型 $f(x_{1},x_{2},\\cdots,x_{n})$ ，若对任意不全为 0 的 $x_{1},\\cdots,x_{n}$ 有 $f(x_{1},\\cdots,x_{n})>0$ ，则称 $f(x_{1},\\cdots,x_{n})$ 为正定矩阵。",
        "bbox": [
            112,
            105,
            884,
            143
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "定理 7.13 (正定二次型的规范型)",
        "text_level": 2,
        "bbox": [
            121,
            164,
            363,
            179
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "正定二次型的规范型是 $y^{T}Ey=y_{1}^{2}+y_{2}^{2}+\\cdots+y_{n}^{2}$",
        "bbox": [
            115,
            184,
            487,
            202
        ],
        "page_idx": 84
    },
    {
        "type": "image",
        "img_path": "images/c263772a56ac3460e77a1671c982bbce6d1734c60908b4b744d1e5be1797c031.jpg",
        "image_caption": [
            "图 7.1: 二次型分类示意图"
        ],
        "image_footnote": [],
        "content": "",
        "bbox": [
            334,
            224,
            660,
            343
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "定理 7.14 (正定充要条件)",
        "text_level": 2,
        "bbox": [
            121,
            388,
            312,
            403
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "$f = x^{T}Ax$ 是正定二次型当且仅当：",
        "bbox": [
            112,
            407,
            374,
            423
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "\\- $A$ 合同于矩阵 $E$",
        "bbox": [
            139,
            426,
            284,
            441
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "\\- 存在可逆矩阵 $C$ ，使得 $A = C^T C$ ，也可以分解为非方阵 $A = \\left( \\begin{array}{cc} C^T & O^T \\end{array} \\right) \\left( \\begin{array}{c} C \\\\ O \\end{array} \\right)$ .",
        "bbox": [
            137,
            441,
            771,
            478
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "所有特征值都大于0",
        "bbox": [
            139,
            476,
            312,
            492
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "\\- $A$ 的各阶顺序主子式/所有主子式行列式大于0",
        "bbox": [
            139,
            495,
            500,
            511
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "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": [
            90,
            529,
            536,
            544
        ],
        "page_idx": 84
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\alpha^ {T} & a _ {n n} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            426,
            548,
            566,
            583
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "正定推顺序主子式：若 $A$ 正定，则 $A_{1}$ 正定，因此 $A_{1}$ 特征值均为正， $|A_1| > 0, |A| > 0$ ，根据归纳法成立。顺序主子式推正定： $n = 1$ 时显然成立， $A$ 是 $n$ 阶矩阵时， $A_{1}$ 是 $n - 1$ 阶矩阵且顺序主子式均正，根据归纳假设 $A_{1}$ 正定，故存在 $P_{1}$ 使得 $P_{1}^{T} A_{1} P_{1} = E_{n - 1}$ ，因此令：",
        "bbox": [
            90,
            589,
            905,
            642
        ],
        "page_idx": 84
    },
    {
        "type": "equation",
        "text": "$$\nP _ {2} = \\left[ \\begin{array}{c c} P _ {1} & \\\\ & 1 \\end{array} \\right], P _ {3} = \\left[ \\begin{array}{c c} E _ {n - 1} & - P _ {1} ^ {T} \\alpha \\\\ & 1 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            337,
            644,
            653,
            684
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "得到：",
        "bbox": [
            90,
            686,
            137,
            701
        ],
        "page_idx": 84
    },
    {
        "type": "equation",
        "text": "$$\nP _ {3} ^ {T} P _ {2} ^ {T} A P _ {2} P _ {3} = \\left[ \\begin{array}{c c} E _ {n - 1} & \\\\ & a _ {n n} - \\alpha^ {T} P _ {1} P _ {1} ^ {T} \\alpha \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            327,
            699,
            663,
            737
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "而根据乘积矩阵的行列式性质得到",
        "bbox": [
            90,
            746,
            347,
            760
        ],
        "page_idx": 84
    },
    {
        "type": "equation",
        "text": "$$\na _ {n n} - \\alpha^ {T} P _ {1} P _ {1} ^ {T} \\alpha = | A | | P _ {2} | ^ {2} | P _ {3} | ^ {2} > 0\n$$",
        "text_format": "latex",
        "bbox": [
            361,
            768,
            636,
            785
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "令 $P = P_{2}P_{3}\\mathrm{diag}\\{E_{n - 1},\\frac{1}{\\sqrt{a_{nn} - \\alpha^{T}P_{1}P_{1}^{T}\\alpha}}\\}$ 得到",
        "bbox": [
            90,
            787,
            463,
            820
        ],
        "page_idx": 84
    },
    {
        "type": "equation",
        "text": "$$\nP ^ {T} A P = E\n$$",
        "text_format": "latex",
        "bbox": [
            452,
            822,
            544,
            837
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "因此正定",
        "bbox": [
            90,
            846,
            164,
            860
        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "练习 7.8 正定二次型的判定 (1)t 为何值时， $f = x_{1}^{2} + x_{2}^{2} + 5x_{3}^{2} + 2tx_{1}x_{2} - 2x_{1}x_{3} + 4x_{2}x_{3}$ 为正定的？",
        "bbox": [
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        "page_idx": 84
    },
    {
        "type": "text",
        "text": "(2) $A$ 为实对称矩阵，则 $\\exists M, \\forall t > M$ 有 $tE + A$ 为正定矩阵",
        "bbox": [
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        "page_idx": 84
    },
    {
        "type": "page_number",
        "text": "80",
        "bbox": [
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        ],
        "page_idx": 84
    },
    {
        "type": "header",
        "text": "7.3 (半) 正定二次型的研究",
        "bbox": [
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        ],
        "page_idx": 84
    },
    {
        "type": "text",
        "text": "解(1)算顺序主子式即可，得到 $1-t^{2}>0,-5t^{2}-4t>0$ ，因此 $-\\frac{4}{5}<t<0$",
        "bbox": [
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "(2) 看特征值即可",
        "bbox": [
            122,
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "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 $r(B^{T}AB) = r(B)$",
        "bbox": [
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "(2) $A$ 为 $n$ 阶正定矩阵， $B$ 为 $n \\times m$ 实矩阵，证明：若 $r(B) = m$ ，则 $C = B^T AB$ 为正定矩阵证明 (1) 由于 $A$ 正定，存在满秩矩阵 $C$ 使得 $A = C^T C$ ，因此 $B^T AB = (CB)^T (CB)$ ，而根据 $A^T A$ 的结论可知 $r((CB)^T (CB)) = r(CB) = r(B)$ ，因此 $r(B^T AB) = r(B)$",
        "bbox": [
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "(2) 由于 $A$ 正定，因此 $B^T AB$ 实对称， $\\forall X, X^T CX = (BX)^T ABX$ ，根据 $A$ 的正定性可知其大于0，因此为正定矩阵",
        "bbox": [
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "定理 7.15 (正定矩阵对角性质)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "$A$ 为 $n$ 阶正定矩阵，则",
        "bbox": [
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "1. $a_{ii} > 0, \\forall i = 1, 2, \\cdots, n$",
        "bbox": [
            136,
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            331
        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "2. $2|a_{ij}| < a_{ii} + a_{jj} (i \\neq j)$",
        "bbox": [
            134,
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            332,
            350
        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "3. $A$ 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": [
            134,
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "证明 (1) 不妨取 $X = e_{i}$ ，则 $X^{T}AX = a_{ii} > 0$",
        "bbox": [
            89,
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "(2) 取 $X = e_i + e_j$ ，则 $X^TAX = a_{ii} + a_{jj} + 2a_{ij} > 0$ ，取 $X = e_i - e_j$ ，则 $X^TAX = a_{ii} + a_{jj} - 2a_{ij} > 0$ ，综上得到结论成立。",
        "bbox": [
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "(3) 根据 (2) 可知 $\\forall i, j$ ，有 $|a_{ij}| < \\max\\{a_{ii}, a_{jj}\\}$ ，因此结论成立",
        "bbox": [
            122,
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "7.3.2 半正定二次型的概念与判别",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "定义 7.10 (半正定二次型)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "若对任意 $x_{1},\\cdots,x_{n}$ 均有 $f(x_{1},\\cdots,x_{n})\\geq0$ ，则称 $f(x_{1},\\cdots,x_{n})$ 为半正定二次型，对应矩阵 A 为半正定矩阵。",
        "bbox": [
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "定理 7.16 (半正定二次型的基础充要条件)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "$f(x_{1},\\cdots,x_{n})=X^{T}AX$ 为半正定二次型当且仅当：",
        "bbox": [
            115,
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            625
        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "\\- $A$ 和 $\\left( \\begin{array}{ll}E_r & O\\\\ O & O \\end{array} \\right)$ 合同， $A$ 的正惯性指数与 $A$ 的秩相同",
        "bbox": [
            137,
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "\\- $A$ 的特征值都大于等于0",
        "bbox": [
            139,
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            344,
            676
        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "\\- $A$ 的所有主子式 (不等价于顺序主子式) 大于等于 0",
        "bbox": [
            139,
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            529,
            695
        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "笔记 半正定不等价于顺序主子式大于等于 0，例如 $A = \\left( \\begin{array}{rr} 0 & 0 \\\\ 0 & -1 \\end{array} \\right)$ ，顺序主子式都为 0，但是半负定。",
        "bbox": [
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "定理 7.17 (半正定矩阵的 $C^{T}C$ 分解)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "$f(x_{1},\\cdots,x_{n})=X^{T}AX$ 为半正定二次型等价于下面两个命题",
        "bbox": [
            114,
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "\\- 存在不列满秩的 $n \\times n$ 矩阵 $C$ , 使得 $A = C^T C$",
        "bbox": [
            139,
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "\\- 存在行满秩 (列不满秩) 的 $r \\times n (r < n)$ 矩阵 $C$ ，使得 $A = C^T C$",
        "bbox": [
            139,
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        ],
        "page_idx": 85
    },
    {
        "type": "header",
        "text": "7.3 (半) 正定二次型的研究",
        "bbox": [
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        ],
        "page_idx": 85
    },
    {
        "type": "page_number",
        "text": "81",
        "bbox": [
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        ],
        "page_idx": 85
    },
    {
        "type": "text",
        "text": "证明 根据 $A = C^{T}\\left( \\begin{array}{cc}E_{r} & O\\\\ O & O \\end{array} \\right)C$ 以及",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "equation",
        "text": "$$\n\\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) = \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) = \\binom{E _ {r}}{O} \\left( \\begin{array}{c c} E _ {r} & O \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            257,
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "可以得到两种分解：",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} C _ {1} = \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) C \\Rightarrow A = C _ {1} ^ {T} C _ {1} \\\\ C _ {2} = \\left( \\begin{array}{c c} E _ {r} & O \\end{array} \\right) C \\Rightarrow A = C _ {2} ^ {T} C _ {2} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "无论哪种分解列均不满秩。",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "定理 7.18 (半正定矩阵的 $CC^{T}$ 分解)",
        "text_level": 2,
        "bbox": [
            122,
            300,
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            316
        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "$f(x_{1},\\cdots,x_{n})=X^{T}AX$ 为半正定二次型等价于下面两个命题",
        "bbox": [
            114,
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "\\- 存在不行满秩的 $n \\times n$ 矩阵 $C$ ，使得 $A = CC^T$",
        "bbox": [
            139,
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            354
        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "\\- 存在列满秩 (行不满秩) 的 $n \\times r (r < n)$ 矩阵 $C$ ，使得 $A = CC^T$",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "证明 右推左：直接 $X^{T}CC^{T}X = (C^{T}X)^{T}(C^{T}X)$ ，该多项式是 $C^{T}X$ 列向量中元素的平方和，因此大于等于0。左推右：合同的 $A = B^{T}\\mathrm{diag}\\{E_{r},O\\} B$ 也可以写为 $A = C\\mathrm{diag}\\{E_r,O\\} C^T$ ，因此把中间的分解了即可。笔记 特别的可以取 $A = \\alpha^{T}\\alpha$ ， $\\alpha$ 为列向量(列满秩)，此时 $A$ 是半正定的。",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "练习7.10半正定矩阵 $CC^T$ 分解 $(1)\\alpha \\in \\mathbb{R}^n$ ，证明： $\\alpha \\alpha^T$ 一定是半正定矩阵",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "(2) 重点: $A$ 的元素 $a_{ij} = (i \\times j)$ , $B$ 的元素 $b_{ij} = \\left(\\frac{1}{i \\times j}\\right)$ , 则 $A, B$ 都是半正定矩阵",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "证明 (1) 显然 $\\alpha \\alpha^T$ 是实对称矩阵，从而 $X^T \\alpha \\alpha^T X = (\\alpha^T X)^T (\\alpha^T X) = (\\alpha^T X)^2 \\geq 0$",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "(2) $A, B$ 可进行如下分解，根据 (1) 的结论可证是半正定矩阵",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c} 1 \\\\ 2 \\\\ \\vdots \\\\ n \\end{array} \\right] \\left[ \\begin{array}{l l l l} 1 & 2 & \\dots & n \\end{array} \\right], \\quad B = \\left[ \\begin{array}{c} 1 \\\\ \\frac {1}{2} \\\\ \\vdots \\\\ \\frac {1}{n} \\end{array} \\right] \\left[ \\begin{array}{l l l l} 1 & \\frac {1}{2} & \\dots & \\frac {1}{n} \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
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        "page_idx": 86
    },
    {
        "type": "text",
        "text": "定理 7.19 (半正定矩阵对角性质)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "半正定矩阵 $A$ 有以下性质：",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "1. 对角元素非负",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "2. $2|a_{ij}| \\leq a_{ik} + a_{ki}$",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "3. 若 $a_{ii} = 0$ ，则 $\\forall k, a_{ik} = a_{ki} = 0$",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "证明 (1) 只需要取 $x = e_i$ ， $x^T Ax \\geq 0$ 即可",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "(2) 取 $x = e_i + e_j$ ，得到 $a_{ii} + a_{jj} + 2a_{ij} \\geq 0$ ，同理取 $x = e_i - e_j$ ，得到 $a_{ii} + a_{jj} - 2a_{ij} \\geq 0$ ，因此 $2|a_{ij}| \\leq a_{ik} + a_{ki}$",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "(3) 根据 (2) 可以得到",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "7.3.3 $C^T C$ 的 (半) 正定性",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "$C^{T}C$ 对应着特殊的二次型 $f(X)=\\sum_{i=1}^{n}(a_{i1}x_{1}+\\cdots+a_{in}x_{n})^{2}$ ，且有着特殊的正定性，是正定二次型中特殊的一类。",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "header",
        "text": "7.3 (半) 正定二次型的研究",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "page_number",
        "text": "82",
        "bbox": [
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        ],
        "page_idx": 86
    },
    {
        "type": "text",
        "text": "引理7.1 $(r(A^T A)$ 与 $r(A))$",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "$A$ 为实数域上的 $m \\times n$ 的矩阵，则",
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "equation",
        "text": "$$\nr (A ^ {T} A) = r (A)\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "证明 在最小二乘法一节中已经给出了证明",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "定理 7.20 ( $A^{T}A$ 的正定性)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "考虑 $A$ 为实数域上的 $m \\times n$ 的矩阵，则",
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "1. $A^T A, AA^T$ 是半正定矩阵",
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "2. $A^T A$ 是正定矩阵当且仅当 $A$ 列满秩， $AA^T$ 正定当且仅当 $A$ 行满秩",
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "且 $A^{T}A$ 对应的二次型为 $f(X)=\\sum_{i=1}^{n}(a_{i1}x_{1}+a_{i2}x_{2}+\\cdots+a_{in}x_{n})^{2}$",
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "证明 (半) 正定性：显然 $A^T A$ 实对称，因此 $X^T A^T A X = (AX)^2 \\geq 0$ ，显然 $A^T A$ 半正定。而 $A^T A$ 正定当且仅当 $r(A^T A) = n$ ，根据 (1) 当且仅当 $r(A) = n$ ，这里 $n$ 等于 $A$ 的列数，因此等价于 $A$ 列满秩",
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "二次型表达式：在二次型矩阵一节已经作为练习给出。",
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "7.3.4 正定矩阵同时对角化",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "定理 7.21 (同时合同对角化)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "若 $A, B$ 是 $n$ 阶半正定矩阵，则存在一个 $n$ 阶可逆矩阵 $P$ ，使得 $P^T AP, P^T BP$ 同时为对角矩阵",
        "bbox": [
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        "page_idx": 87
    },
    {
        "type": "text",
        "text": "证明 由于 A, B 半正定，因此 $A + B$ 半正定，存在可逆矩阵 M 使得",
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "equation",
        "text": "$$\nM ^ {T} (A + B) M = \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) = M ^ {T} A M + M ^ {T} B M\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "由于 $M^T AM, M^T BM$ 半正定，因此根据半正定矩阵的对角性质，当 $r + 1 \\leq i \\leq n$ ，其第 $i$ 个对角元都非负，因此 $r + 1 \\sim n$ 的对角元都必须为0。再根据对角性质可知 $r + 1 \\sim n$ 行、列元素均为0，因此",
        "bbox": [
            90,
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        ],
        "page_idx": 87
    },
    {
        "type": "equation",
        "text": "$$\nM ^ {T} A M = \\left( \\begin{array}{c c} H _ {r} & O \\\\ O & O \\end{array} \\right), M ^ {T} B M = \\left( \\begin{array}{c c} E _ {r} - H _ {r} & O \\\\ O & O \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            292,
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            612
        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "存在正交矩阵 T 使得 $T^{T}H_{r}T = \\Lambda = \\operatorname{diag}\\{\\lambda_{1}, \\cdots, \\lambda_{r}\\}$ ，令 $P = M\\operatorname{diag}\\{T, E_{n-r}\\}$ ，此时有",
        "bbox": [
            90,
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        ],
        "page_idx": 87
    },
    {
        "type": "equation",
        "text": "$$\nP ^ {T} A P = \\left( \\begin{array}{c c} \\Lambda & O \\\\ O & O \\end{array} \\right), P ^ {T} B P = \\left( \\begin{array}{c c} E _ {r} - \\Lambda & O \\\\ O & O \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            310,
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "同时为对角矩阵",
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "定理 7.22 (AB = BA 同时相似对角化)",
        "text_level": 2,
        "bbox": [
            121,
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "若 $A, B$ 可相似对角化，且 $AB = BA$ ，则存在可逆矩阵 $P$ ，使得",
        "bbox": [
            114,
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        ],
        "page_idx": 87
    },
    {
        "type": "equation",
        "text": "$$\nP ^ {- 1} A P, P ^ {- 1} B P\n$$",
        "text_format": "latex",
        "bbox": [
            433,
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "同时为对角矩阵",
        "bbox": [
            115,
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        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "证明 由于 A 可相似对角化，因此存在 M，使得 $M^{-1}AM = \\text{diag}\\{\\lambda_{1}E, \\lambda_{2}E, \\cdots, \\lambda_{r}E\\}$ ，而",
        "bbox": [
            90,
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            764,
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        ],
        "page_idx": 87
    },
    {
        "type": "equation",
        "text": "$$\nM ^ {- 1} A B M = (M ^ {- 1} A M) (M ^ {- 1} B M) = M ^ {- 1} B A M = (M ^ {- 1} B M) (M ^ {- 1} A M)\n$$",
        "text_format": "latex",
        "bbox": [
            220,
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            841
        ],
        "page_idx": 87
    },
    {
        "type": "text",
        "text": "将 $M^{-1}BM$ 以同样的形式进行分块，得到 $M^{-1}BM = (B_{ij})_{r\\times r}$ ，对于每个 $B_{ii}$ ， $\\exists Q_i$ 使得 $Q_i^{-1}B_{ii}Q_i = \\Lambda_i$ ，因此令",
        "bbox": [
            90,
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        ],
        "page_idx": 87
    },
    {
        "type": "equation",
        "text": "$$\nT = M \\operatorname{diag} \\left\\{Q _ {1}, \\dots , Q _ {r} \\right\\}\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "page_number",
        "text": "83",
        "bbox": [
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        ],
        "page_idx": 87
    },
    {
        "type": "header",
        "text": "7.3 (半) 正定二次型的研究",
        "bbox": [
            705,
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            68
        ],
        "page_idx": 87
    },
    {
        "type": "image",
        "img_path": "images/18f1321b9a28cd1e7a912d0030429bba3abaea7c68f82d6ab7aab02fb1299de2.jpg",
        "image_caption": [],
        "image_footnote": [],
        "content": "",
        "bbox": [
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "可以得到",
        "bbox": [
            90,
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            99
        ],
        "page_idx": 88
    },
    {
        "type": "equation",
        "text": "$$\nT ^ {- 1} A T = \\operatorname{diag} \\left\\{\\lambda_ {1} E, \\dots , \\lambda_ {r} E \\right\\}, T ^ {- 1} B T = \\operatorname{diag} \\left\\{\\Lambda_ {1}, \\dots , \\Lambda_ {r} \\right\\}\n$$",
        "text_format": "latex",
        "bbox": [
            275,
            102,
            719,
            118
        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "同时为对角矩阵。",
        "bbox": [
            90,
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            221,
            146
        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "7.3.5 正定矩阵的四则运算性质",
        "text_level": 2,
        "bbox": [
            92,
            168,
            373,
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "定理 7.23 (正定的四则运算)",
        "text_level": 2,
        "bbox": [
            122,
            200,
            329,
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "若 $A, B$ 是正定矩阵，则 $A^{-1}, A^{k}, A + B, A^{*}$ 都是正定矩阵，但 $AB$ 不一定正定 (原因在于不一定对称)，若 $AB$ 实对称，则 $AB$ 是正定矩阵",
        "bbox": [
            114,
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "证明 (1) $A^{-1}, A^{k}, A^{*}$ : 由于 A 正定, 因此 A 的特征值 $\\lambda_{i}$ 均大于 0, $A^{-1}$ 的特征值 $\\lambda_{i}^{-1} > 0$ , $A^{*}$ 的特征值 $\\prod_{j \\neq i} \\lambda_{j} > 0$ , $A^{k}$ 的特征值 $\\lambda_{i}^{k} > 0$ , 因此均正定。",
        "bbox": [
            90,
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "(2) $A + B$ 不能用特征值，只能用合同，显然 $A + B$ 实对称，而任意非零 $x \\in \\mathbb{R}^n$ ，有 $X^T (A + B)X = X^T AX + X^T BX > 0$ ，因此 $A + B$ 实对称。",
        "bbox": [
            90,
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "(3) $AB$ 正定：由于 $AB$ 实对称，此时 $(AB)^T = B^TA^T = BA$ 。由于 $A, B$ 正定，故存在可逆 $P, Q$ 使得 $A = P^TP, B = Q^TQ, AB = P^TPQ^TQ$ ，而",
        "bbox": [
            89,
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        ],
        "page_idx": 88
    },
    {
        "type": "equation",
        "text": "$$\nQ A B Q ^ {- 1} = Q P ^ {T} P Q ^ {T} (Q Q ^ {- 1}) = (P Q ^ {T}) ^ {T} (P Q ^ {T})\n$$",
        "text_format": "latex",
        "bbox": [
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "因此 AB 与 $(PQ^{T})^{T}(PQ^{T})$ 相似，而后者正定（因为列满秩），因此 AB 为正定矩阵。",
        "bbox": [
            90,
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "7.3.6 正定矩阵分解进阶",
        "text_level": 2,
        "bbox": [
            94,
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            312,
            476
        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "定理 7.24 (正定矩阵平方分解)",
        "text_level": 2,
        "bbox": [
            121,
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "$A$ 是 $n$ 阶正定矩阵，则存在实对称矩阵 $C_1$ ，使得 $A = C_1^2$",
        "bbox": [
            114,
            512,
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            527
        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "证明 由于 $\\exists P$ 使得 $P^{T}AP = \\Lambda = \\text{diag}\\{\\lambda_{1}, \\cdots, \\lambda_{n}\\}$ ，根据正定，所有特征值都为正，因此取",
        "bbox": [
            90,
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            769,
            564
        ],
        "page_idx": 88
    },
    {
        "type": "equation",
        "text": "$$\nC _ {1} = P ^ {T} \\sqrt {\\Lambda} P\n$$",
        "text_format": "latex",
        "bbox": [
            443,
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "即可得到 $A = C_1^2$",
        "bbox": [
            92,
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "笔记 上述 $A = C_{1}^{2}$ 也是正定矩阵分解！且是唯一的。",
        "bbox": [
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "定理 7.25 (正定矩阵三角分解)",
        "text_level": 2,
        "bbox": [
            122,
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "$A$ 是 $n$ 阶正定矩阵，则存在可逆上三角阵 $R$ ，使得 $A = R^{T}R$",
        "bbox": [
            114,
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "证明 首先 $A = C^T C$ ， $C$ 为可逆矩阵，再对 $C$ 做 $QR$ 分解得到 $C = QR$ ，因此 $A = R^T Q^T QR = R^TR$ ，即所求。",
        "bbox": [
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "定理 7.26 (极分解定理)",
        "text_level": 2,
        "bbox": [
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "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 $S_{1}, S_{2}$ 使得",
        "bbox": [
            114,
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            771
        ],
        "page_idx": 88
    },
    {
        "type": "equation",
        "text": "$$\nA = T S _ {1} = S _ {2} T\n$$",
        "text_format": "latex",
        "bbox": [
            436,
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "且这种分解是唯一的。",
        "bbox": [
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "证明 (1) 存在性: 由于 A 可逆, 因此 $A^{T}A$ 正定, 根据前面结论存在正定矩阵 $S_{1}$ , 满足 $A^{T}A = S_{1}^{2}$ , 设 $T = (A^{T})^{-1}S_{1}$ , 则",
        "bbox": [
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            869
        ],
        "page_idx": 88
    },
    {
        "type": "equation",
        "text": "$$\nA = (A ^ {T}) ^ {- 1} S _ {1} ^ {2} = T S _ {1}\n$$",
        "text_format": "latex",
        "bbox": [
            415,
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "经过验证可发现 $TT^{T} = (A^{T})^{-1}S_{1}S_{1}^{T}A^{-1} = (A^{T})^{-1}S_{1}^{2}A^{-1} = (A^{T})^{-1}A^{T}AA^{-1} = E$ ，因此为正交矩阵。",
        "bbox": [
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            917
        ],
        "page_idx": 88
    },
    {
        "type": "header",
        "text": "7.3 (半) 正定二次型的研究",
        "bbox": [
            705,
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        ],
        "page_idx": 88
    },
    {
        "type": "page_number",
        "text": "84",
        "bbox": [
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        ],
        "page_idx": 88
    },
    {
        "type": "text",
        "text": "另一方面， $A = TS_{1}T^{-1}T$ ，记 $S_{2} = TS_{1}T^{-1}$ ，则 $A = S_{2}T$ ， $S_{2}$ 正定。",
        "bbox": [
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        ],
        "page_idx": 89
    },
    {
        "type": "text",
        "text": "(2) 唯一性：设 $A = TS_{1} = \\tilde{T}\\tilde{S}_{1}$ ，则",
        "bbox": [
            124,
            103,
            391,
            117
        ],
        "page_idx": 89
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} A ^ {T} A = S _ {1} ^ {2} \\\\ A ^ {T} A = \\tilde {S} _ {1} ^ {2} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
            442,
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            162
        ],
        "page_idx": 89
    },
    {
        "type": "text",
        "text": "根据前面正定矩阵分解唯一性可知唯一。",
        "bbox": [
            90,
            170,
            389,
            185
        ],
        "page_idx": 89
    },
    {
        "type": "text",
        "text": "7.3.7 负定二次型",
        "text_level": 2,
        "bbox": [
            95,
            225,
            253,
            243
        ],
        "page_idx": 89
    },
    {
        "type": "text",
        "text": "定义 7.11 (负定二次型)",
        "text_level": 2,
        "bbox": [
            122,
            258,
            295,
            273
        ],
        "page_idx": 89
    },
    {
        "type": "text",
        "text": "若对于任意不全为零的 $x_{1},\\dots ,x_{n}$ ，满足 $f(x_{1},\\dots ,x_{n}) < 0$ ，则称 $f(x_{1},\\dots ,x_{n})$ 为负定二次型",
        "bbox": [
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        ],
        "page_idx": 89
    },
    {
        "type": "text",
        "text": "定理 7.27 (负定二次型的充要条件)",
        "text_level": 2,
        "bbox": [
            122,
            319,
            379,
            334
        ],
        "page_idx": 89
    },
    {
        "type": "text",
        "text": "$A$ 负定当且仅当 $-A$ 正定，因此充要条件为： $A$ 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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            373
        ],
        "page_idx": 89
    },
    {
        "type": "text",
        "text": "证明 用特征值判断即可。",
        "bbox": [
            90,
            391,
            280,
            405
        ],
        "page_idx": 89
    },
    {
        "type": "header",
        "text": "7.3 (半) 正定二次型的研究",
        "bbox": [
            705,
            55,
            905,
            68
        ],
        "page_idx": 89
    },
    {
        "type": "page_number",
        "text": "85",
        "bbox": [
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            964
        ],
        "page_idx": 89
    },
    {
        "type": "text",
        "text": "第8章 内积空间",
        "text_level": 2,
        "bbox": [
            396,
            87,
            618,
            111
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "8.1 Euclid 空间",
        "text_level": 2,
        "bbox": [
            92,
            146,
            265,
            164
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "8.1.1 Euclid 空间与正交",
        "text_level": 2,
        "bbox": [
            92,
            185,
            312,
            200
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "定义 8.1 (欧式空间)",
        "text_level": 2,
        "bbox": [
            122,
            217,
            270,
            231
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "$V$ 是 $\\mathbb{R}$ 上线性空间，定义内积 $(\\alpha, \\beta)$ 满足：",
        "bbox": [
            114,
            237,
            433,
            250
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "\\- 线性性： $(k_{1}\\alpha_{1} + k_{2}\\alpha_{2}, \\beta) = k_{1}(\\alpha_{1}, \\beta) + k_{2}(\\alpha_{2}, \\beta)$",
        "bbox": [
            139,
            256,
            522,
            269
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "\\- 对称性： $(\\alpha ,\\beta) = (\\alpha ,\\beta)$",
        "bbox": [
            139,
            274,
            336,
            288
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "\\- 正定性： $(\\alpha, \\alpha) = 0$ 当且仅当 $\\alpha = 0$",
        "bbox": [
            139,
            293,
            420,
            307
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "定义 8.2 (度量矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            332,
            268,
            347
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "给定 Euclid 空间 V 的一组基 $\\epsilon_{1},\\cdots,\\epsilon_{n}$ ，则定义这组基对应的度量矩阵为",
        "bbox": [
            114,
            353,
            653,
            368
        ],
        "page_idx": 90
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{c c c c} (\\epsilon_ {1}, \\epsilon_ {1}) & (\\epsilon_ {1}, \\epsilon_ {2}) & \\dots & (\\epsilon_ {1}, \\epsilon_ {n}) \\\\ (\\epsilon_ {2}, \\epsilon_ {1}) & (\\epsilon_ {2}, \\epsilon_ {2}) & \\dots & (\\epsilon_ {2}, \\epsilon_ {n}) \\\\ \\vdots & \\vdots & & \\vdots \\\\ (\\epsilon_ {n}, \\epsilon_ {1}) & (\\epsilon_ {n}, \\epsilon_ {2}) & \\dots & (\\epsilon_ {n}, \\epsilon_ {n}) \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            346,
            370,
            647,
            453
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "定理 8.1 (内积的度量矩阵表示)",
        "text_level": 2,
        "bbox": [
            122,
            476,
            354,
            492
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "若 Euclid 空间中的向量 $\\alpha, \\beta$ 在基 $(\\epsilon_{1}, \\cdots, \\epsilon_{n})$ 中的坐标分别为 X, Y, $(\\epsilon_{1}, \\cdots, \\epsilon_{n})$ 对应的度量矩阵为 A，则",
        "bbox": [
            115,
            497,
            880,
            513
        ],
        "page_idx": 90
    },
    {
        "type": "equation",
        "text": "$$\n(\\alpha , \\beta) = X ^ {T} A Y\n$$",
        "text_format": "latex",
        "bbox": [
            438,
            519,
            557,
            536
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "练习 8.1 度量矩阵 已知 Euclid 空间 V 的一组基为:",
        "bbox": [
            58,
            574,
            465,
            589
        ],
        "page_idx": 90
    },
    {
        "type": "equation",
        "text": "$$\n\\epsilon_ {1} = e _ {1} + e _ {2}, \\epsilon_ {2} = e _ {1} + e _ {3}, \\epsilon_ {3} = e _ {4} - e _ {1}, \\epsilon_ {4} = e _ {1} - e _ {2} - e _ {3} + e _ {4}\n$$",
        "text_format": "latex",
        "bbox": [
            270,
            601,
            724,
            613
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "向量 $\\alpha,\\beta$ 在 $(\\epsilon_{1},\\cdots,\\epsilon_{4})$ 下的坐标分别为 $(1,2,3,4)^{T},(2,0,1,0)^{T}$ ，计算 $(\\epsilon_{1},\\cdots,\\epsilon_{4})$ 的度量矩阵 A，以及 $(\\alpha,\\beta)$ 解 把 $\\epsilon_{1},\\cdots,\\epsilon_{4}$ 看成 $R^{n}$ 中的坐标：",
        "bbox": [
            87,
            620,
            897,
            653
        ],
        "page_idx": 90
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} \\epsilon_ {1} = (1, 1, 0, 0) \\\\ \\epsilon_ {2} = (1, 0, 1, 0) \\\\ \\epsilon_ {3} = (- 1, 0, 0, 1) \\\\ \\epsilon_ {4} = (1, - 1, - 1, 1) \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
            415,
            655,
            564,
            741
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "因此根据向量的坐标内积公式可以得到度量矩阵为：",
        "bbox": [
            90,
            751,
            472,
            765
        ],
        "page_idx": 90
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{r r r r} 2 & 1 & - 1 & 0 \\\\ 1 & 2 & - 1 & 0 \\\\ - 1 & - 1 & 2 & 0 \\\\ 0 & 0 & 0 & 4 \\end{array} \\right], (\\alpha , \\beta) = 5\n$$",
        "text_format": "latex",
        "bbox": [
            354,
            769,
            642,
            843
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "定理 8.2 (正交向量线性无关)",
        "text_level": 2,
        "bbox": [
            122,
            860,
            336,
            876
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "若 Eulid 空间 V 中的向量 $\\alpha_{1},\\cdots,\\alpha_{s}$ 彼此正交，则它们线性无关。",
        "bbox": [
            115,
            881,
            598,
            896
        ],
        "page_idx": 90
    },
    {
        "type": "text",
        "text": "证明 不妨设 $c_{1}\\alpha_{1}+\\cdots+c_{s}\\alpha_{s}=0$ ，则两侧与 $\\alpha_{i}$ 做内积得到：",
        "bbox": [
            87,
            85,
            549,
            99
        ],
        "page_idx": 91
    },
    {
        "type": "equation",
        "text": "$$\n\\left(c _ {1} \\alpha_ {1} + \\dots + c _ {s} \\alpha_ {s}, \\alpha_ {i}\\right) = (0, \\alpha_ {i}) = 0\n$$",
        "text_format": "latex",
        "bbox": [
            364,
            108,
            633,
            123
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "根据正交定义可知左侧等于 $c_{i}(\\alpha_{i},\\alpha_{i})$ ，而 $(\\alpha_{i},\\alpha_{i})\\neq 0$ ，因此 $c_{i} = 0$ ，根据 $i$ 的任意性可知线性无关。",
        "bbox": [
            90,
            131,
            826,
            146
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "8.1.2 标准正交基",
        "text_level": 2,
        "bbox": [
            92,
            168,
            253,
            185
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "一、标准正交基的概念与存在性",
        "text_level": 2,
        "bbox": [
            90,
            195,
            331,
            211
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "定义 8.3 (标准正交基)",
        "text_level": 2,
        "bbox": [
            122,
            219,
            287,
            235
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "将 $n$ 维Eulid空间中 $n$ 个两两正交的单位向量称为标准正交基",
        "bbox": [
            112,
            240,
            569,
            256
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "二、标准正交基之间的过渡矩阵",
        "text_level": 2,
        "bbox": [
            90,
            274,
            331,
            290
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "定理 8.3 (标准正交基的过渡矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            300,
            371,
            315
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "$\\epsilon_{1},\\cdots,\\epsilon_{n}$ 是 n 维 Euclid 空间 V 的一组基，若 $\\eta_{1},\\cdots,\\eta_{n}$ 下述等式，则 $\\eta_{1},\\cdots,\\eta_{n}$ 是标准正交基当且仅当 T 是正交矩阵。",
        "bbox": [
            112,
            319,
            880,
            351
        ],
        "page_idx": 91
    },
    {
        "type": "equation",
        "text": "$$\n(\\eta_ {1}, \\dots , \\eta_ {n}) = (\\epsilon_ {1}, \\dots , \\epsilon_ {n}) T\n$$",
        "text_format": "latex",
        "bbox": [
            393,
            357,
            605,
            373
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "三、标准正交基的计算：Schmidt 正交化",
        "text_level": 2,
        "bbox": [
            90,
            391,
            391,
            407
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "定理 8.4 (Schmidt 正交化)",
        "text_level": 2,
        "bbox": [
            121,
            417,
            317,
            432
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "给定一组线性无关的向量 $\\alpha_{1},\\cdots,\\alpha_{s}$ ，要求做出一组相互正交的向量 (单位化单独进行) $\\epsilon_{1},\\cdots,\\epsilon_{s}$ ，可以按照以下步骤：",
        "bbox": [
            114,
            437,
            882,
            470
        ],
        "page_idx": 91
    },
    {
        "type": "equation",
        "text": "$$\n\\epsilon_ {1} = \\alpha_ {1}\n$$",
        "text_format": "latex",
        "bbox": [
            410,
            474,
            470,
            486
        ],
        "page_idx": 91
    },
    {
        "type": "equation",
        "text": "$$\n\\epsilon_ {2} = \\alpha_ {2} - \\frac {\\left(\\alpha_ {2} , \\epsilon_ {1}\\right)}{\\left(\\epsilon_ {1} , \\epsilon_ {1}\\right)} \\epsilon_ {1}\n$$",
        "text_format": "latex",
        "bbox": [
            408,
            483,
            563,
            518
        ],
        "page_idx": 91
    },
    {
        "type": "equation",
        "text": "$$\n\\epsilon_ {s} = \\alpha_ {s} - \\sum_ {i = 1} ^ {s - 1} \\frac {(\\alpha_ {s} , \\epsilon_ {i})}{(\\epsilon_ {i} , \\epsilon_ {i})} \\epsilon_ {i}\n$$",
        "text_format": "latex",
        "bbox": [
            408,
            538,
            586,
            577
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "练习 8.2Schmidt 正交化 已知 $\\epsilon_{1}=(1,2,-1)^{T},\\epsilon_{2}=(-1,3,1)^{T},\\epsilon_{3}=(4,-1,0)^{T}$ ，用 Schmidt 正交化将其化为标准正交基",
        "bbox": [
            58,
            614,
            905,
            646
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "解 最终结果为 $\\eta_{1} = \\frac{1}{\\sqrt{6}} (1,2, - 1)^{T},\\eta_{2} = \\frac{1}{\\sqrt{3}} (-1,1,1)^{T},\\eta_{3} = \\frac{1}{\\sqrt{2}} (1,0,1)^{T}$",
        "bbox": [
            87,
            645,
            631,
            677
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "定理 8.5 (QR 分解)",
        "text_level": 2,
        "bbox": [
            122,
            694,
            267,
            709
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "证明 假设原本的矩阵 A 的列向量为 $\\alpha_{1},\\cdots,\\alpha_{n}$ ，经过 Schmidt 正交化后得到 $\\beta_{1},\\cdots,\\beta_{n}$ ，则满足：",
        "bbox": [
            89,
            747,
            811,
            764
        ],
        "page_idx": 91
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} \\alpha_ {1} = \\beta_ {1} \\\\ \\alpha_ {2} = a _ {1 2} \\beta_ {1} + \\beta_ {2} \\\\ \\quad \\vdots \\\\ \\alpha_ {n} = a _ {1 n} \\beta_ {1} + a _ {2 n} \\beta_ {2} + \\dots + \\beta_ {n} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
            368,
            770,
            611,
            858
        ],
        "page_idx": 91
    },
    {
        "type": "page_number",
        "text": "87",
        "bbox": [
            489,
            953,
            507,
            964
        ],
        "page_idx": 91
    },
    {
        "type": "header",
        "text": "8.1 Euclid 空间",
        "bbox": [
            791,
            55,
            905,
            68
        ],
        "page_idx": 91
    },
    {
        "type": "text",
        "text": "再将 $\\beta$ 进行单位化得到 $\\eta_{i} = \\frac{\\beta_{i}}{|\\beta_{i}|}$ ，此时写为",
        "bbox": [
            90,
            80,
            416,
            112
        ],
        "page_idx": 92
    },
    {
        "type": "equation",
        "text": "$$\n(\\alpha_ {1}, \\dots , \\alpha_ {n}) = (\\eta_ {1}, \\dots , \\eta_ {n}) \\left( \\begin{array}{c c c c} b _ {1 1} & b _ {1 2} & \\dots & b _ {1 n} \\\\ & b _ {2 2} & \\dots & b _ {2 n} \\\\ & & \\ddots & \\vdots \\\\ & & & b _ {n n} \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            302,
            116,
            690,
            196
        ],
        "page_idx": 92
    },
    {
        "type": "text",
        "text": "此时对角 $b_{11}=|\\beta_{1}|\\cdots,b_{nn}=|\\beta_{n}|$ ，而 $(\\eta_{1},\\cdots,\\eta_{n})$ 是标准正交基。因此是正交矩阵乘上三角阵",
        "bbox": [
            90,
            200,
            798,
            216
        ],
        "page_idx": 92
    },
    {
        "type": "text",
        "text": "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": [
            58,
            237,
            774,
            252
        ],
        "page_idx": 92
    },
    {
        "type": "equation",
        "text": "$$\nA = \\left[ \\begin{array}{l l l} 1 & 1 & 1 \\\\ 1 & 0 & 1 \\\\ 1 & 2 & 0 \\end{array} \\right]\n$$",
        "text_format": "latex",
        "bbox": [
            428,
            255,
            564,
            315
        ],
        "page_idx": 92
    },
    {
        "type": "text",
        "text": "解 先进行 $\\alpha_{1}, \\alpha_{2}, \\alpha_{3}$ 正交化到 $\\beta_{1}, \\beta_{2}, \\beta_{3}$ :",
        "bbox": [
            89,
            318,
            389,
            332
        ],
        "page_idx": 92
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} \\beta_ {1} = \\alpha_ {1} = (1, 1, 1) ^ {T}, | \\beta_ {1} | = \\sqrt {3} \\\\ \\beta_ {2} = \\alpha_ {2} - \\frac {3}{3} (1, 1, 1) ^ {T} = (0, - 1, 1) ^ {T}, | \\beta_ {2} | = \\sqrt {2} \\\\ \\beta_ {3} = \\alpha_ {3} + \\frac {1}{2} \\beta_ {2} - \\frac {2}{3} \\beta_ {1} = (\\frac {1}{3}, - \\frac {1}{6}, - \\frac {1}{6}), | \\beta_ {3} | = \\sqrt {\\frac {1}{6}} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
            302,
            338,
            678,
            414
        ],
        "page_idx": 92
    },
    {
        "type": "text",
        "text": "进行单位化得到",
        "bbox": [
            90,
            416,
            213,
            430
        ],
        "page_idx": 92
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} \\eta_ {1} = (\\frac {\\sqrt {3}}{3}, \\frac {\\sqrt {3}}{3}, \\frac {\\sqrt {3}}{3}) \\\\ \\eta_ {2} = (0, - \\frac {\\sqrt {2}}{2}, \\frac {\\sqrt {2}}{2}) \\\\ \\eta_ {3} = (\\frac {\\sqrt {6}}{3}, - \\frac {\\sqrt {6}}{6}, - \\frac {\\sqrt {6}}{6}) \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
            396,
            426,
            584,
            513
        ],
        "page_idx": 92
    },
    {
        "type": "text",
        "text": "反向得到",
        "bbox": [
            90,
            519,
            163,
            533
        ],
        "page_idx": 92
    },
    {
        "type": "equation",
        "text": "$$\n\\left\\{ \\begin{array}{l} \\alpha_ {1} = \\beta_ {1} \\\\ \\alpha_ {2} = \\beta_ {1} + \\beta_ {2} \\\\ \\alpha_ {3} = \\frac {2}{3} \\beta_ {1} - \\frac {1}{2} \\beta_ {2} + \\beta_ {3} \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
            401,
            533,
            578,
            604
        ],
        "page_idx": 92
    },
    {
        "type": "text",
        "text": "因此",
        "bbox": [
            92,
            609,
            131,
            623
        ],
        "page_idx": 92
    },
    {
        "type": "equation",
        "text": "$$\n(\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}) = (\\beta_ {1}, \\beta_ {2}, \\beta_ {3}) \\left( \\begin{array}{c c c} 1 & 1 & \\frac {2}{3} \\\\ 0 & 1 & - \\frac {1}{2} \\\\ 0 & 0 & 1 \\end{array} \\right) = (\\eta_ {1}, \\eta_ {2}, \\eta_ {3}) \\left( \\begin{array}{c c c} | \\beta_ {1} | & & \\\\ & | \\beta_ {2} | & \\\\ & & | \\beta_ {3} | \\end{array} \\right) \\left( \\begin{array}{c c c} 1 & 1 & \\frac {2}{3} \\\\ 0 & 1 & - \\frac {1}{2} \\\\ 0 & 0 & 1 \\end{array} \\right)\n$$",
        "text_format": "latex",
        "bbox": [
            149,
            624,
            843,
            693
        ],
        "page_idx": 92
    },
    {
        "type": "text",
        "text": "8.1.3 正交和与正交补",
        "text_level": 2,
        "bbox": [
            94,
            713,
            292,
            730
        ],
        "page_idx": 92
    },
    {
        "type": "text",
        "text": "定义 8.4 (正交和、正交补)",
        "text_level": 2,
        "bbox": [
            122,
            746,
            321,
            760
        ],
        "page_idx": 92
    },
    {
        "type": "text",
        "text": "$V_{1},\\cdots,V_{s}$ 为一列子空间，若 $\\forall i,j$ 有 $V_{i}\\perp V_{j}$ ，则称 $V_{1}+\\cdots+V_{s}$ 为正交和。若 $V=V_{1}+V_{2}$ 且为正交和，则 $V_{1},V_{2}$ 互为正交补。",
        "bbox": [
            114,
            765,
            882,
            799
        ],
        "page_idx": 92
    },
    {
        "type": "text",
        "text": "定理 8.6 (正交和必为直和)",
        "text_level": 2,
        "bbox": [
            122,
            825,
            321,
            839
        ],
        "page_idx": 92
    },
    {
        "type": "text",
        "text": "若 $V_{1} + \\cdots + V_{s}$ 为正交和，则其必为直和。",
        "bbox": [
            115,
            844,
            431,
            859
        ],
        "page_idx": 92
    },
    {
        "type": "text",
        "text": "证明 设 $0 = \\alpha_{1} + \\cdots + \\alpha_{s}$ ，两侧与 $\\alpha_{i}$ 做内积可知",
        "bbox": [
            90,
            879,
            462,
            892
        ],
        "page_idx": 92
    },
    {
        "type": "equation",
        "text": "$$\n0 = (\\alpha_ {i}, \\alpha_ {i})\n$$",
        "text_format": "latex",
        "bbox": [
            452,
            902,
            542,
            917
        ],
        "page_idx": 92
    },
    {
        "type": "header",
        "text": "8.1 Euclid 空间",
        "bbox": [
            791,
            55,
            905,
            68
        ],
        "page_idx": 92
    },
    {
        "type": "page_number",
        "text": "88",
        "bbox": [
            489,
            954,
            507,
            965
        ],
        "page_idx": 92
    },
    {
        "type": "text",
        "text": "因此 $\\alpha_{i}=0$ ，故零元素分解唯一，为直和。",
        "bbox": [
            90,
            85,
            406,
            99
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "定理 8.7 (正交补的唯一性)",
        "text_level": 2,
        "bbox": [
            122,
            110,
            321,
            124
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "Euclid 空间 V 的任一子空间 $V_{1}$ 有唯一的正交补 $W^{\\perp}$",
        "bbox": [
            114,
            130,
            500,
            145
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "证明 $V_{1}=\\{0\\}$ 时，即 $V_{1}^{\\perp}=V$ 。否则设 $V_{1}=L(\\epsilon_{1},\\cdots,\\epsilon_{s})$ 为标准正交基，将其扩充为 V 的标准正交基：",
        "bbox": [
            90,
            164,
            860,
            180
        ],
        "page_idx": 93
    },
    {
        "type": "equation",
        "text": "$$\nV = L \\left(\\epsilon_ {1}, \\dots , \\epsilon_ {n}\\right)\n$$",
        "text_format": "latex",
        "bbox": [
            428,
            187,
            566,
            203
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "则 $V_{1}^{\\perp}=L(\\epsilon_{s+1},\\cdots,\\epsilon_{n})$ 为正交补",
        "bbox": [
            90,
            211,
            347,
            225
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "唯一性: 若 $V_{1}$ 有正交补 $V_{2}, V_{3}$ ，则 $\\forall \\alpha \\in V_{2} \\subset V_{1} \\oplus V_{3}, \\exists \\alpha_{1} \\in V_{1}, \\alpha_{3} \\in V_{3}$ 使得 $\\alpha = \\alpha_{1} + \\alpha_{3}$ ，而 $\\alpha \\perp \\alpha_{1}, \\alpha_{1} \\perp \\alpha_{3}$ ，因此",
        "bbox": [
            90,
            229,
            905,
            263
        ],
        "page_idx": 93
    },
    {
        "type": "equation",
        "text": "$$\n0 = (\\alpha , \\alpha_ {1}) = (\\alpha_ {1}, \\alpha_ {1}) + (\\alpha_ {3}, \\alpha_ {1}) = (\\alpha_ {1}, \\alpha_ {1})\n$$",
        "text_format": "latex",
        "bbox": [
            336,
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        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "即 $\\alpha_{1} = 0, \\alpha = \\alpha_{3} \\in V_{3}$ ，得到 $V_{2} \\subseteq V_{3}, V_{3} \\subseteq V_{2}$ ，因此 $V_{2} = V_{3}$",
        "bbox": [
            90,
            294,
            549,
            309
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "定理 8.8 (正交补的具体表达式)",
        "text_level": 2,
        "bbox": [
            122,
            319,
            354,
            335
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "$V_{1}$ 是线性空间 $V$ 的子空间，则 $V_{1}^{\\perp}$ 恰由所有与 $V_{1}$ 正交的向量组成，即",
        "bbox": [
            115,
            341,
            642,
            356
        ],
        "page_idx": 93
    },
    {
        "type": "equation",
        "text": "$$\nV _ {1} ^ {\\perp} = \\{\\alpha \\in V, \\alpha \\perp V _ {1} \\}\n$$",
        "text_format": "latex",
        "bbox": [
            413,
            363,
            581,
            380
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "证明 令 $W = \\{\\alpha \\in V : \\alpha \\perp V_1\\}$ ，下面证明 $V_1^\\perp = W$ 。首先 $V_1 \\perp V_1^\\perp$ ，因此 $V_1^\\perp \\subseteq W$ 。",
        "bbox": [
            89,
            398,
            721,
            414
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "反之， $\\forall \\alpha \\in W$ ，根据 $V = V_{1} \\oplus V_{1}^{\\perp}$ ，因此可分解 $\\alpha = \\alpha_{1} + \\alpha_{2}$ ，得到",
        "bbox": [
            122,
            418,
            636,
            432
        ],
        "page_idx": 93
    },
    {
        "type": "equation",
        "text": "$$\n\\left(\\alpha_ {1}, \\alpha_ {1}\\right) = \\left(\\alpha - \\alpha_ {2}, \\alpha_ {1}\\right) = \\left(\\alpha , \\alpha_ {1}\\right) - \\left(\\alpha_ {2}, \\alpha_ {1}\\right) = 0\n$$",
        "text_format": "latex",
        "bbox": [
            321,
            441,
            675,
            457
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "因此 $\\alpha_{1} = 0$ ，即 $\\alpha \\in V_1^\\perp$",
        "bbox": [
            92,
            464,
            275,
            479
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "定理 8.9 (正交补的性质)",
        "text_level": 2,
        "bbox": [
            122,
            491,
            302,
            505
        ],
        "page_idx": 93
    },
    {
        "type": "equation",
        "text": "$$\n(1) (U ^ {\\perp}) ^ {\\perp} = U (2) (U _ {1} + U _ {2}) ^ {\\perp} = U _ {1} ^ {\\perp} \\cap U _ {2} ^ {\\perp} (3) (U _ {1} \\cap U _ {2}) ^ {\\perp} = U _ {1} ^ {\\perp} + U _ {2} ^ {\\perp}\n$$",
        "text_format": "latex",
        "bbox": [
            114,
            510,
            616,
            527
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "证明 (1) 根据正交补的具体表达式一下得出",
        "bbox": [
            90,
            546,
            413,
            562
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "(2) 显然 $(U_{1} + U_{2})^{\\perp}\\subseteq U_{1}^{\\perp}$ ， $(U_{1} + U_{2})^{\\perp}\\subseteq U_{2}^{\\perp}$ ，因此 $(U_{1} + U_{2})^{\\perp}\\subseteq U_{1}^{\\perp}\\cap U_{2}^{\\perp}$ 。",
        "bbox": [
            122,
            564,
            697,
            581
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "另一方面， $\\forall \\alpha \\in U_1^\\perp \\cap U_2^\\perp, \\beta \\in U_1 + U_2$ ，有 $\\beta = \\beta_1 + \\beta_2$ ，因此",
        "bbox": [
            124,
            583,
            594,
            599
        ],
        "page_idx": 93
    },
    {
        "type": "equation",
        "text": "$$\n(\\alpha , \\beta) = (\\alpha , \\beta_ {1} + \\beta_ {2}) = (\\alpha , \\beta_ {1}) + (\\alpha , \\beta_ {2}) = 0\n$$",
        "text_format": "latex",
        "bbox": [
            334,
            607,
            662,
            623
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "因此 $U_{1}^{\\perp}\\cap U_{2}^{\\perp}\\subseteq (U_{1} + U_{2})^{\\perp}$",
        "bbox": [
            92,
            629,
            305,
            645
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "(3) 可转换为 (2)",
        "bbox": [
            124,
            649,
            243,
            663
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "8.1.4 正交变换",
        "text_level": 2,
        "bbox": [
            94,
            684,
            231,
            701
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "定义 8.5 (正交变换)",
        "text_level": 2,
        "bbox": [
            122,
            718,
            268,
            733
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "$V$ 是 $n$ 维Eulid空间， $\\mathcal{A}$ 是 $V$ 上线性变换，若 $\\forall \\alpha ,\\beta \\in V$ 有 $(\\mathcal{A}\\alpha ,\\mathcal{A}\\beta) = (\\alpha ,\\beta)$ ，则称 $\\mathcal{A}$ 为正交变换",
        "bbox": [
            114,
            738,
            847,
            753
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "定理 8.10 (正交变换充要条件)",
        "text_level": 2,
        "bbox": [
            122,
            778,
            344,
            793
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "$\\mathcal{A}$ 是正交变换与下面几个命题等价：",
        "bbox": [
            114,
            797,
            379,
            813
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "\\- $\\mathcal{A}$ 在标准正交基下矩阵为正交矩阵",
        "bbox": [
            139,
            816,
            415,
            831
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "\\- 标准正交基在 $\\mathcal{A}$ 的作用下仍为标准正交基",
        "bbox": [
            139,
            835,
            468,
            850
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "\\- $\\mathcal{A}$ 保持向量长度不变",
        "bbox": [
            139,
            854,
            312,
            868
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "若 $\\epsilon_1, \\dots, \\epsilon_n$ 是 $V$ 的标准正交基，则 $\\mathcal{A}(\\epsilon_1), \\dots, \\mathcal{A}(\\epsilon_n)$ 也是 $V$ 的标准正交基",
        "bbox": [
            139,
            872,
            717,
            888
        ],
        "page_idx": 93
    },
    {
        "type": "header",
        "text": "8.1 Euclid 空间",
        "bbox": [
            791,
            55,
            905,
            68
        ],
        "page_idx": 93
    },
    {
        "type": "page_number",
        "text": "89",
        "bbox": [
            489,
            953,
            507,
            964
        ],
        "page_idx": 93
    },
    {
        "type": "text",
        "text": "8.1.5 镜面反射",
        "text_level": 2,
        "bbox": [
            94,
            85,
            233,
            102
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "定义 8.6 (镜面反射)",
        "text_level": 2,
        "bbox": [
            122,
            118,
            270,
            133
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "若对 $\\forall \\alpha \\in V$ ， $\\mathcal{A}:V\\to V$ 满足",
        "bbox": [
            117,
            137,
            346,
            153
        ],
        "page_idx": 94
    },
    {
        "type": "equation",
        "text": "$$\n\\mathcal {A} \\alpha = \\alpha - 2 (\\eta , \\alpha) \\eta\n$$",
        "text_format": "latex",
        "bbox": [
            428,
            158,
            569,
            173
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "其中 $\\eta$ 是单位向量",
        "bbox": [
            115,
            185,
            255,
            200
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "引理 8.1 (镜面反射是正交变换)",
        "text_level": 2,
        "bbox": [
            122,
            224,
            352,
            239
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "镜面反射是正交变换",
        "bbox": [
            115,
            244,
            272,
            259
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "证明",
        "bbox": [
            92,
            278,
            129,
            292
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "定理 8.11 (镜面反射的特征)",
        "bbox": [
            124,
            303,
            329,
            318
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "8.2 Unitary 空间",
        "text_level": 2,
        "bbox": [
            94,
            370,
            278,
            388
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "8.2.1 Unitary 空间概念",
        "text_level": 2,
        "bbox": [
            94,
            407,
            304,
            426
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "定义 8.7 (Unitary 空间)",
        "text_level": 2,
        "bbox": [
            122,
            441,
            295,
            456
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "$V$ 是 $\\mathbb{C}$ 上内积空间，其上定义内积 $(\\alpha, \\beta)$ 满足 $\\forall k_1, k_2 \\in \\mathbb{C}$",
        "bbox": [
            115,
            462,
            547,
            476
        ],
        "page_idx": 94
    },
    {
        "type": "equation",
        "text": "$$\n\\bullet \\left(k _ {1} \\alpha_ {1} + k _ {2} \\alpha_ {2}, \\beta\\right) = k _ {1} (\\alpha_ {1}, \\beta) + k _ {2} (\\alpha_ {2}, \\beta)\n$$",
        "text_format": "latex",
        "bbox": [
            139,
            480,
            455,
            495
        ],
        "page_idx": 94
    },
    {
        "type": "equation",
        "text": "$$\n\\bullet (\\alpha , \\beta) = \\overline {{(\\beta , \\alpha)}}\n$$",
        "text_format": "latex",
        "bbox": [
            142,
            499,
            268,
            514
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "$(\\alpha, \\alpha) \\geq 0$ 且 $(\\alpha, \\alpha) = 0$ 当且仅当 $\\alpha = 0$",
        "bbox": [
            139,
            517,
            450,
            532
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "定义 8.8 (酉矩阵)",
        "text_level": 2,
        "bbox": [
            122,
            557,
            253,
            571
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "$\\mathbb{C}$ 上矩阵 $U^{-1} = \\overline{U}^T$ ，则称 $U$ 为酉矩阵。",
        "bbox": [
            114,
            579,
            413,
            595
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "定义 8.9 (酉变换)",
        "text_level": 2,
        "bbox": [
            124,
            620,
            252,
            634
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "$\\mathcal{U}$ 满足 $(\\mathcal{U}\\alpha, \\mathcal{U}\\beta) = (\\alpha, \\beta)$ ，则称 $\\mathcal{U}$ 为酉变换",
        "bbox": [
            112,
            640,
            443,
            655
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "定义 8.10 (共轭变换)",
        "text_level": 2,
        "bbox": [
            124,
            680,
            278,
            695
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "$\\mathcal{A}$ 是酉空间 $V$ 的线性变换，若 $\\mathcal{A}^*$ 满足 $(\\mathcal{A}\\alpha, \\beta) = (\\alpha, \\mathcal{A}^*\\beta)$ ，则称 $\\mathcal{A}^*$ 是 $\\mathcal{A}$ 的共轭变换",
        "bbox": [
            114,
            700,
            754,
            715
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "定义 8.11 (正规变换)",
        "text_level": 2,
        "bbox": [
            122,
            740,
            278,
            755
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "若酉空间 $V$ 的线性变换 $\\mathcal{A}$ 与其共轭变换 $\\mathcal{A}^*$ 可交换，则称 $\\mathcal{A}$ 为正规变换。",
        "bbox": [
            115,
            760,
            658,
            775
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "定义 8.12 (Hermite 变换)",
        "text_level": 2,
        "bbox": [
            122,
            800,
            309,
            815
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "若酉空间 $V$ 的线性变换 $\\mathcal{A}$ 满足 $\\mathcal{A} = \\mathcal{A}^*$ ，则称 $\\mathcal{A}$ 为Hermite变换",
        "bbox": [
            115,
            819,
            596,
            835
        ],
        "page_idx": 94
    },
    {
        "type": "header",
        "text": "8.2 Unitary 空间",
        "bbox": [
            784,
            55,
            905,
            68
        ],
        "page_idx": 94
    },
    {
        "type": "page_number",
        "text": "90",
        "bbox": [
            489,
            953,
            509,
            965
        ],
        "page_idx": 94
    },
    {
        "type": "text",
        "text": "第9章 对偶空间",
        "text_level": 2,
        "bbox": [
            396,
            87,
            618,
            111
        ],
        "page_idx": 95
    },
    {
        "type": "text",
        "text": "9.1 对偶空间与对偶基",
        "text_level": 2,
        "bbox": [
            92,
            145,
            337,
            165
        ],
        "page_idx": 95
    },
    {
        "type": "text",
        "text": "定义 9.1 (对偶空间)",
        "text_level": 2,
        "bbox": [
            121,
            186,
            270,
            200
        ],
        "page_idx": 95
    },
    {
        "type": "text",
        "text": "$V$ 是 $K$ 上的线性空间，将定义于 $V$ 上的所有有界线性泛函组成的空间称为对偶空间 $V^{*}$ 。",
        "bbox": [
            114,
            205,
            769,
            221
        ],
        "page_idx": 95
    },
    {
        "type": "text",
        "text": "定义 9.2 (对偶基)",
        "text_level": 2,
        "bbox": [
            122,
            244,
            253,
            261
        ],
        "page_idx": 95
    },
    {
        "type": "text",
        "text": "给定 V 的一组基 $\\epsilon_{1},\\cdots,\\epsilon_{n}$ ，定义线性泛函 $f_{i}$ 满足：",
        "bbox": [
            114,
            266,
            499,
            280
        ],
        "page_idx": 95
    },
    {
        "type": "equation",
        "text": "$$\nf _ {i} (\\epsilon_ {j}) = \\left\\{ \\begin{array}{l l} 1, & i = j \\\\ 0, & i \\neq j \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
            421,
            285,
            573,
            329
        ],
        "page_idx": 95
    },
    {
        "type": "text",
        "text": "则 $\\{f_{1},\\cdots,f_{n}\\}$ 是 $V^{*}$ 的一组基，称其为 $\\epsilon_{1},\\cdots,\\epsilon_{n}$ 的对偶基。",
        "bbox": [
            115,
            332,
            569,
            349
        ],
        "page_idx": 95
    },
    {
        "type": "text",
        "text": "笔记 计算对偶基时，将 $f_{i}$ 视为一个向量，即 $f_{i}(\\alpha)=f_{i1}x_{1}+\\cdots+f_{in}x_{n}=f_{i}^{T}\\alpha$ 。计算 $\\epsilon_{1},\\cdots,\\epsilon_{n}$ 的对偶基只需要满足：",
        "bbox": [
            87,
            368,
            905,
            401
        ],
        "page_idx": 95
    },
    {
        "type": "equation",
        "text": "$$\n\\left[ \\begin{array}{c} f _ {1} ^ {T} \\\\ f _ {2} ^ {T} \\\\ \\vdots \\\\ f _ {n} ^ {T} \\end{array} \\right] \\left[ \\begin{array}{c c c c} \\epsilon_ {1} & \\epsilon_ {2} & \\dots & \\epsilon_ {n} \\end{array} \\right] = E\n$$",
        "text_format": "latex",
        "bbox": [
            376,
            399,
            625,
            481
        ],
        "page_idx": 95
    },
    {
        "type": "text",
        "text": "练习9.1给定 $\\epsilon_{1} = [1,1,0]^{T},\\epsilon_{2} = [0,1,1]^{T},\\epsilon_{3} = [0,1, - 1]^{T}$ ，求其对偶基",
        "bbox": [
            60,
            497,
            613,
            514
        ],
        "page_idx": 95
    },
    {
        "type": "text",
        "text": "解显然 $f_{1}, f_{2}, f_{3}$ 都是三个系数组成的线性函数，不妨记 $A = [f_{1}, f_{2}, f_{3}]^{T}, P = [\\epsilon_{1}, \\epsilon_{2}, \\epsilon_{3}]$ ，则",
        "bbox": [
            87,
            516,
            769,
            532
        ],
        "page_idx": 95
    },
    {
        "type": "equation",
        "text": "$$\nA P = \\left[ \\begin{array}{l} f _ {1} ^ {T} \\\\ f _ {2} ^ {T} \\\\ f _ {3} ^ {T} \\end{array} \\right] [ \\epsilon_ {1}, \\epsilon_ {2}, \\epsilon_ {3} ] = E\n$$",
        "text_format": "latex",
        "bbox": [
            388,
            533,
            610,
            595
        ],
        "page_idx": 95
    },
    {
        "type": "text",
        "text": "因此 $A = P^{-1}$ ，只需求出 $P^{-1}$ 即可。最终求出：",
        "bbox": [
            90,
            596,
            448,
            612
        ],
        "page_idx": 95
    },
    {
        "type": "equation",
        "text": "$$\nP ^ {- 1} = \\left[ \\begin{array}{c c c} 1 & 0 & 0 \\\\ - \\frac {1}{2} & \\frac {1}{2} & \\frac {1}{2} \\\\ - \\frac {1}{2} & \\frac {1}{2} & - \\frac {1}{2} \\end{array} \\right] \\Rightarrow \\left\\{ \\begin{array}{l} f _ {1} ^ {T} = (1, 0, 0) \\\\ f _ {2} ^ {T} = (- \\frac {1}{2}, \\frac {1}{2}, \\frac {1}{2}) \\\\ f _ {3} ^ {T} = (- \\frac {1}{2}, \\frac {1}{2}, - \\frac {1}{2}) \\end{array} \\right.\n$$",
        "text_format": "latex",
        "bbox": [
            302,
            618,
            673,
            689
        ],
        "page_idx": 95
    }
]