{"pdfData":[[{"bbox":[0,0,595,335],"type":"image","angle":0,"index":0,"img_path":"/6aea91c268d2eff96016e37721df9752a8666330531a40b4f741b7b4de061d23.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":1,"page_idx":0,"id":"d3ffc6db-ed9a-4b49-8a41-73aa5145a42b","page_size":[595,841],"block_position":"0-0"},{"bbox":[20,456,172,484],"type":"title","angle":0,"index":1,"level":2,"text":"## 高等代数笔记","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"75ce4726-4194-4740-91cf-d39e0830b05c","page_size":[595,841],"block_position":"0-1"},{"bbox":[31,551,104,565],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"作者：Reichtum ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"a604abfd-6cdf-4244-aa01-0a0f03a7c53d","page_size":[595,841],"block_position":"0-2"},{"bbox":[30,570,145,585],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"组织：ZheJiang University ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"da4c79b8-6c80-4904-8623-599d3cf27189","page_size":[595,841],"block_position":"0-3"},{"bbox":[30,588,145,602],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"时间：September 20, 2022 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"77d83cf7-536a-4c60-b2a7-530705e3c84e","page_size":[595,841],"block_position":"0-4"},{"bbox":[31,608,77,620],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"版本：0.1 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"aa95de93-f57f-469b-a604-b3d5672f1191","page_size":[595,841],"block_position":"0-5"},{"bbox":[196,726,398,739],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"There was no royal road to geometry. ——Euclid ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"4c75c5e6-4a5a-4929-9d5d-029f17bd8ab0","page_size":[595,841],"block_position":"0-6"},{"bbox":[420,617,537,733],"type":"image","angle":0,"index":7,"img_path":"/d840dcbb2187119fefcbbad478a371deb4b6159c9ef6f1e6f1baa4fef154e71b.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":0,"id":"27461bc6-f0e2-4374-aca8-ac0f55ce9b4b","page_size":[595,841],"block_position":"0-7"}],[{"bbox":[279,74,315,95],"type":"title","angle":0,"index":0,"level":2,"text":"## 目录","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":1,"id":"55438bb3-ad21-4b83-ad80-977ff9284ae3","page_size":[595,841],"block_position":"1-0"},{"bbox":[55,126,542,759],"type":"text","angle":0,"index":1,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":1,"id":"0614bae8-1f66-49a7-982a-cffad5f89019","page_size":[595,841],"block_position":"1-1"}],[{"bbox":[518,47,539,58],"type":"header","angle":0,"index":-99999,"text":"目录 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":2,"is_discarded":true,"id":"f06c25b6-2629-469b-baa2-c631529e4071","page_size":[595,841],"block_position":"2-0"},{"bbox":[55,69,542,770],"type":"text","angle":0,"index":1,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":2,"id":"9228aa67-2103-4777-9d7b-c03658affb5f","page_size":[595,841],"block_position":"2-1"},{"bbox":[293,803,301,812],"type":"page_number","angle":0,"index":1000001,"text":"ii ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":2,"is_discarded":true,"id":"97ea268c-6679-40ba-8086-ea0cb255e0f3","page_size":[595,841],"block_position":"2-2"}],[{"bbox":[518,47,539,58],"type":"header","angle":0,"index":-99999,"text":"目录 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":3,"is_discarded":true,"id":"3603192f-d3f4-4654-ae53-a4dc23f56d34","page_size":[595,841],"block_position":"3-0"},{"bbox":[65,69,542,735],"type":"text","angle":0,"index":1,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":3,"id":"b8996838-5d73-4e7e-9dfd-5472b3cdd30b","page_size":[595,841],"block_position":"3-1"},{"bbox":[54,743,185,756],"type":"footer","angle":0,"index":1000001,"text":"第7章 双线性函数与二次型 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":3,"is_discarded":true,"id":"be3c276f-ede7-45fc-b01b-502dffa8d395","page_size":[595,841],"block_position":"3-2"},{"bbox":[527,745,539,755],"type":"page_number","angle":0,"index":1000002,"text":"73 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":3,"is_discarded":true,"id":"9412ba46-331b-41be-aefa-a9902d3b5e97","page_size":[595,841],"block_position":"3-3"},{"bbox":[292,803,302,812],"type":"page_number","angle":0,"index":1000003,"text":"iii ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":3,"is_discarded":true,"id":"61e55950-6c84-4ea6-9c6c-aa06003e3ba4","page_size":[595,841],"block_position":"3-4"}],[{"bbox":[518,47,539,58],"type":"header","angle":0,"index":-99999,"text":"目录 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":4,"is_discarded":true,"id":"c0d987d2-6cbf-4fb0-a7e7-36f38d52d823","page_size":[595,841],"block_position":"4-0"},{"bbox":[64,70,541,542],"type":"text","angle":0,"index":1,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":4,"id":"1615805b-09fd-4fa6-b6f0-e0ef19cf6989","page_size":[595,841],"block_position":"4-1"},{"bbox":[292,803,301,811],"type":"page_number","angle":0,"index":1000001,"text":"iv ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":4,"is_discarded":true,"id":"333c0719-03e3-4a5d-997c-c715ac6a1655","page_size":[595,841],"block_position":"4-2"}],[{"bbox":[245,74,359,94],"type":"title","angle":0,"index":0,"level":2,"text":"## 第1章 多项式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"9d28a9bf-abf0-4486-a3a7-1985214cd75e","page_size":[595,841],"block_position":"5-0"},{"bbox":[57,123,158,138],"type":"title","angle":0,"index":1,"level":2,"text":"## 1.1 一元多项式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"8a0e52af-8f21-4142-8b39-8de0ab401428","page_size":[595,841],"block_position":"5-1"},{"bbox":[57,155,139,169],"type":"title","angle":0,"index":2,"level":2,"text":"## 1.1.1 整除理论","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"0213a740-b7cf-4dbd-83d6-e30dbed79a8f","page_size":[595,841],"block_position":"5-2"},{"bbox":[72,183,142,195],"type":"title","angle":0,"index":3,"level":2,"text":"## 定义 1.1 (整除)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"47084f69-64b7-45bc-8391-378da69cc2ec","page_size":[595,841],"block_position":"5-3"},{"bbox":[67,198,528,230],"type":"text","angle":0,"index":4,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"ca248d88-efd6-4683-8848-05adc3c30869","page_size":[595,841],"block_position":"5-4"},{"bbox":[72,249,201,262],"type":"title","angle":0,"index":5,"level":2,"text":"## 定理 1.1 (整除的数域不变性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"e7655ff3-246a-428d-88ff-fd197d0c9319","page_size":[595,841],"block_position":"5-5"},{"bbox":[67,265,506,280],"type":"text","angle":0,"index":6,"merge_prev":false,"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 $g(x)|f(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"19f04940-69ae-43b5-8fd5-c78a0be5b631","page_size":[595,841],"block_position":"5-6"},{"bbox":[35,295,474,309],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"f27a283c-27b4-4ed5-b4a4-aabe2608a068","page_size":[595,841],"block_position":"5-7"},{"bbox":[72,317,161,330],"type":"title","angle":0,"index":8,"level":2,"text":"## 定理 1.2 (带余除法)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"3414335c-b7f3-4ae4-b42f-600bae9d4936","page_size":[595,841],"block_position":"5-8"},{"bbox":[67,333,536,369],"type":"text","angle":0,"index":9,"merge_prev":false,"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 $\\partial(r(x)) < \\partial(g(x))$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"a1d15403-6cac-4921-9b06-3e2bd193f3ae","page_size":[595,841],"block_position":"5-9"},{"bbox":[35,395,515,423],"type":"text","angle":0,"index":10,"merge_prev":false,"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 $32x - 7$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"0ac514fc-56bd-4a57-a7c3-2bb738593db3","page_size":[595,841],"block_position":"5-10"},{"bbox":[35,442,484,455],"type":"text","angle":0,"index":11,"merge_prev":false,"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 $g(x)|f(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"87b442d2-a20b-448e-bda9-90c1066e8ff6","page_size":[595,841],"block_position":"5-11"},{"bbox":[72,457,539,470],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"f69ddd67-a8ea-4456-a201-942939b2a35c","page_size":[595,841],"block_position":"5-12"},{"bbox":[87,474,130,486],"type":"interline_equation","angle":0,"index":13,"text":"$$\nf (x), g (x)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":5,"id":"67948ff3-6ca2-4e2f-b68a-c69c9e64ee13","page_size":[595,841],"block_position":"5-13"},{"bbox":[210,473,268,486],"type":"interline_equation","angle":0,"index":14,"text":"$$\ng ^ {m} (x) | f ^ {m} (x)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":5,"id":"6a3d9870-d656-46d4-869f-d52410c5f849","page_size":[595,841],"block_position":"5-14"},{"bbox":[311,474,353,486],"type":"interline_equation","angle":0,"index":15,"text":"$$\ng (x) | f (x)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":5,"id":"e8597184-ecb2-4351-a3c0-996abc3c763b","page_size":[595,841],"block_position":"5-15"},{"bbox":[52,487,538,518],"type":"text","angle":0,"index":16,"merge_prev":false,"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 $r(x) = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"df771e89-8e1d-4f38-b58b-8c7d7a3bb25c","page_size":[595,841],"block_position":"5-16"},{"bbox":[73,519,375,533],"type":"text","angle":0,"index":17,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"56f5b7c2-fed5-4eb2-8248-a7f7aa6e7b31","page_size":[595,841],"block_position":"5-17"},{"bbox":[188,539,406,552],"type":"interline_equation","angle":0,"index":18,"text":"$$\nf (x) - g (x) = h (x) \\left[ q _ {1} (x) - q _ {2} (x) \\right] + r _ {1} (x) - r _ {2} (x)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":5,"id":"4fb1de96-66ed-4d29-97e0-49baf0d0d80f","page_size":[595,841],"block_position":"5-18"},{"bbox":[55,558,435,571],"type":"text","angle":0,"index":19,"merge_prev":false,"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 $r_1(x) = r_2(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"75de7f57-997f-4c1d-bea1-935f3cf4ccf2","page_size":[595,841],"block_position":"5-19"},{"bbox":[73,575,311,587],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"(3) 右推左：用 $f(x)=h(x)g(x)$ ，两侧取 m 次方即可。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"76b6a460-54a7-4a42-8b74-5b78a947046e","page_size":[595,841],"block_position":"5-20"},{"bbox":[54,589,539,618],"type":"text","angle":0,"index":21,"merge_prev":false,"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 $g(x)|f(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"6badeba4-5913-4c17-9bb0-9c6939ffc31d","page_size":[595,841],"block_position":"5-21"},{"bbox":[73,620,381,633],"type":"text","angle":0,"index":22,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"7d909651-8e88-4529-b786-550a15f691af","page_size":[595,841],"block_position":"5-22"}],[{"bbox":[470,47,539,58],"type":"header","angle":0,"index":-99999,"text":"1.1 一元多项式 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":6,"is_discarded":true,"id":"89d02e65-bd79-406c-8653-b327d468c13d","page_size":[595,841],"block_position":"6-0"},{"bbox":[57,72,151,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 1.1.2 最大公因式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"afc38ef8-bb63-448a-9afd-a3271c4fb341","page_size":[595,841],"block_position":"6-1"},{"bbox":[73,100,171,112],"type":"title","angle":0,"index":2,"level":2,"text":"## 定义 1.2 (最大公因式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"6571ab4e-a3fa-4ae9-acbe-522126b50ab3","page_size":[595,841],"block_position":"6-2"},{"bbox":[67,116,523,144],"type":"text","angle":0,"index":3,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"10096aa0-6c48-4f32-ad36-674d087c257c","page_size":[595,841],"block_position":"6-3"},{"bbox":[253,148,339,162],"type":"interline_equation","angle":0,"index":4,"text":"$$\nd (x) = (f (x), g (x))\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":6,"id":"667294c9-b478-4cb2-bd8d-d7e350267c42","page_size":[595,841],"block_position":"6-4"},{"bbox":[73,181,211,195],"type":"title","angle":0,"index":5,"level":2,"text":"## 定理 1.3 (最大公因式表示定理)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"b2a296bf-2916-44ad-b40f-306784c4a759","page_size":[595,841],"block_position":"6-5"},{"bbox":[67,198,534,232],"type":"text","angle":0,"index":6,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"82768116-1a38-4b4b-b89b-059efa4bc89b","page_size":[595,841],"block_position":"6-6"},{"bbox":[54,244,196,256],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"证明 假设通过辗转相除法得到: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"20784c7b-c6ba-4e11-afe7-b181f3d22eba","page_size":[595,841],"block_position":"6-7"},{"bbox":[209,259,374,391],"type":"interline_equation","angle":0,"index":8,"text":"$$\n\\left\\{ \\begin{array}{l} f (x) = q _ {1} (x) g (x) + r _ {1} (x) \\\\ g (x) = q _ {2} (x) r _ {1} (x) + r _ {2} (x) \\\\ r _ {1} (x) = q _ {3} (x) r _ {2} (x) + r _ {3} (x) \\\\ \\vdots \\\\ r _ {s - 3} (x) = q _ {s - 1} (x) r _ {s - 2} (x) + r _ {s - 1} (x) \\\\ r _ {s - 2} (x) = q _ {s} (x) r _ {s - 1} (x) + r _ {s} (x) \\\\ r _ {s - 1} (x) = q _ {s + 1} (x) r _ {s} (x) + 0 \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":6,"id":"b4d50dfc-ae7f-4105-8bb1-dadf559fa7ad","page_size":[595,841],"block_position":"6-8"},{"bbox":[54,396,461,408],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"812ab3c9-59b8-49a6-abfe-042727cb1f97","page_size":[595,841],"block_position":"6-9"},{"bbox":[185,416,408,484],"type":"interline_equation","angle":0,"index":11,"text":"$$\n\\begin{array}{r l} r _ {s} (x) & = r _ {s - 2} (x) - q _ {s} (x) r _ {s - 1} (x) \\\\ & = r _ {s - 2} (x) - q _ {s} (x) (r _ {s - 3} (x) - q _ {s - 1} (x) r _ {s - 2} (x)) \\\\ & = \\dots \\dots \\\\ & = u (x) f (x) + v (x) g (x) \\end{array}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":6,"id":"115bd584-97e0-44c2-8f9f-27785ef7e5c5","page_size":[595,841],"block_position":"6-10"},{"bbox":[54,491,449,503],"type":"text","angle":0,"index":12,"merge_prev":false,"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 $g(x), f(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"7ea3e5ce-3547-4c4b-9798-81d4a95fe0d7","page_size":[595,841],"block_position":"6-11"},{"bbox":[52,502,539,534],"type":"text","angle":0,"index":13,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"817d85bf-5783-41e2-8e65-a7b804386a0a","page_size":[595,841],"block_position":"6-12"},{"bbox":[35,551,539,581],"type":"text","angle":0,"index":14,"merge_prev":false,"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 $u(x)f(x) + v(x)g(x) = (f(x), g(x))$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"fe6b18fb-7d0b-44fe-a101-e694e90deb61","page_size":[595,841],"block_position":"6-13"},{"bbox":[53,582,539,613],"type":"text","angle":0,"index":15,"merge_prev":false,"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 $u(x), v(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"69816672-327e-4ff0-bde8-b3e4c99f1ed7","page_size":[595,841],"block_position":"6-14"},{"bbox":[53,615,337,627],"type":"text","angle":0,"index":16,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"102fa670-2194-4560-8b83-50640bc89c69","page_size":[595,841],"block_position":"6-15"},{"bbox":[73,650,181,662],"type":"title","angle":0,"index":17,"level":2,"text":"## 定理 1.4 (保最大公因式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"5dd3980b-db9b-451c-a061-23271cea12e9","page_size":[595,841],"block_position":"6-16"},{"bbox":[68,666,314,679],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"$f(x), g(x) \\in P[x]$ ，若 $a, b, c, d \\in P$ 满足 $ad - bc \\neq 0$ ，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"5bac7231-3fda-4acf-bf94-8d4de477217a","page_size":[595,841],"block_position":"6-17"},{"bbox":[197,686,396,699],"type":"interline_equation","angle":0,"index":19,"text":"$$\n(f (x), g (x)) = (a f (x) + b g (x), c f (x) + d g (x))\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":6,"id":"f0f2f473-f625-4950-b16c-5f53896a2dcc","page_size":[595,841],"block_position":"6-18"},{"bbox":[54,715,476,729],"type":"text","angle":0,"index":20,"merge_prev":false,"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 $d(x)|d_{1}(x), d_{1}(x)|d(x)$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"6a539a0c-e590-415d-9b4a-a709c6f62c86","page_size":[595,841],"block_position":"6-19"},{"bbox":[72,731,471,744],"type":"text","angle":0,"index":21,"merge_prev":false,"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 $d(x)|d_{1}(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"1971edc6-0eb4-4e51-bc55-0c72a46d4626","page_size":[595,841],"block_position":"6-20"},{"bbox":[73,746,540,760],"type":"text","angle":0,"index":22,"merge_prev":false,"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， ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":6,"id":"5fb9dcad-ce2d-4f54-803d-6145752a1c84","page_size":[595,841],"block_position":"6-21"},{"bbox":[293,802,301,811],"type":"page_number","angle":0,"index":1000008,"text":"2 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":6,"is_discarded":true,"id":"629e7236-8329-46be-9b87-c3b2a5528ca3","page_size":[595,841],"block_position":"6-22"}],[{"bbox":[470,47,539,58],"type":"header","angle":0,"index":-99999,"text":"1.1 一元多项式 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":7,"is_discarded":true,"id":"eced1101-879d-40cc-9cc6-f55412401be7","page_size":[595,841],"block_position":"7-0"},{"bbox":[127,78,435,252],"type":"interline_equation","angle":0,"index":1,"text":"$$\n\\begin{array}{c} \\text {   ①   } \\\\ q _ {1} (x) = x - \\frac {1}{3} \\left| \\begin{array}{l l} f (x) & \\frac {1}{3} g (x) \\\\ x ^ {4} + 3 x ^ {3} - x ^ {2} - 4 x - 3 & x ^ {3} + \\frac {1 0}{3} x ^ {2} + \\frac {2}{3} x - 1 \\\\ x ^ {4} + \\frac {1 0}{3} x ^ {3} + \\frac {2}{3} x ^ {2} - x & x ^ {3} + 5 x ^ {2} + 6 x \\\\ \\hline - \\frac {1}{3} x ^ {3} - \\frac {5}{3} x ^ {2} - 3 x - 3 & - \\frac {5}{3} x ^ {2} - \\frac {1 6}{3} x - 1 \\\\ - \\frac {1}{3} x ^ {3} - \\frac {1 0}{9} x ^ {2} - \\frac {9}{9} x + \\frac {1}{3} & - \\frac {5}{3} x ^ {2} - \\frac {2 5}{3} x - 1 0 \\\\ \\hline r _ {1} (x) = - \\frac {5}{9} x ^ {2} - \\frac {2 5}{9} x - \\frac {1 0}{3} & r _ {2} (x) = 3 x + 9 \\\\ - \\frac {9}{5} r _ {1} (x) = x ^ {2} + 5 x + 6 & \\frac {1}{3} r _ {2} (x) = x + 3. \\end{array} \\right| \\quad \\text {   ②   } \\\\ q _ {2} (x) = x - \\frac {5}{3} \\end{array}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":7,"id":"e445d221-0df9-40fa-b5ac-f2a8aee706d4","page_size":[595,841],"block_position":"7-1"},{"bbox":[131,256,464,347],"type":"interline_equation","angle":0,"index":2,"text":"$$\n\\begin{array}{r l} & {\\mathrm{因此}: ① f (x) = \\frac {1}{3} g _ {1} (x) g (x) + r _ {1} (x) \\quad ② g (x) = g _ {2} (x) (- \\frac {9}{5} r _ {1} (x)) + r _ {2} (x)} \\\\ & {③ - \\frac {9}{5} r _ {1} (x) = (x + 2) \\frac {1}{3} r _ {2} (x) \\qquad r _ {2} (x) \\mathrm{即大公因式}} \\\\ & {\\Rightarrow ① ^ {\\prime}: r _ {1} (x) = f (x) - \\frac {1}{3} g _ {1} (x) g (x), ② ^ {\\prime} r _ {2} (x) = g (x) + \\frac {9}{5} g _ {2} (x) r _ {1} (x)} \\\\ & {\\mathrm{将} r _ {1} (x) \\mathrm{代入} ② ^ {\\prime} \\mathrm{中即可}} \\end{array}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":7,"id":"f2aab264-f786-4d4c-bc64-44c9740b6f0b","page_size":[595,841],"block_position":"7-2"},{"bbox":[54,364,484,391],"type":"text","angle":0,"index":3,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":7,"id":"95248bba-cb9a-4db7-8aa6-0a3bf898e112","page_size":[595,841],"block_position":"7-3"},{"bbox":[73,399,201,411],"type":"title","angle":0,"index":4,"level":2,"text":"## 推论 1.1 (加减保最大公因式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":7,"id":"fbe35639-44be-474b-bca3-fd3b8737639d","page_size":[595,841],"block_position":"7-4"},{"bbox":[69,415,312,428],"type":"interline_equation","angle":0,"index":5,"text":"$$\n(f (x), f (x) - g (x)) = (f (x), f (x) + g (x)) = (f (x), g (x))\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":7,"id":"fa3edd2e-e6de-4936-9f80-e8210c0ddfa5","page_size":[595,841],"block_position":"7-5"},{"bbox":[73,464,200,476],"type":"title","angle":0,"index":6,"level":2,"text":"## 定理 1.5 (最小公倍式的表示)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":7,"id":"0f4826ad-5ced-4cdf-969e-78cb3ef1e951","page_size":[595,841],"block_position":"7-6"},{"bbox":[68,481,311,507],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"$f(x), g(x)$ 的首 1 最小公倍式 $[f(x), g(x)] = \\frac{f(x)g(x)}{(f(x), g(x))}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":7,"id":"f19c516a-06e0-4329-92de-2264f84639c7","page_size":[595,841],"block_position":"7-7"},{"bbox":[52,519,539,587],"type":"text","angle":0,"index":8,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":7,"id":"c79dde93-eb4a-4acf-9667-1d50fc74b4b7","page_size":[595,841],"block_position":"7-8"},{"bbox":[57,603,114,618],"type":"title","angle":0,"index":9,"level":2,"text":"## 1.1.3 互素","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":7,"id":"8b609be1-4295-46f5-9cb7-7db2ee93ac5d","page_size":[595,841],"block_position":"7-9"},{"bbox":[73,630,142,643],"type":"title","angle":0,"index":10,"level":2,"text":"## 定义 1.3 (互素)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":7,"id":"58dc0f74-6ef3-4837-be79-4880d84f4d9d","page_size":[595,841],"block_position":"7-10"},{"bbox":[69,647,310,661],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"若 $f(x), g(x) \\in \\mathbb{P}[x]$ 的最大公因式为1，则称它们互素。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":7,"id":"b5126970-089c-4896-9efa-64039e415d5b","page_size":[595,841],"block_position":"7-11"},{"bbox":[73,682,181,693],"type":"title","angle":0,"index":12,"level":2,"text":"## 定理 1.6 (互素充要条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":7,"id":"58ceb80b-e642-4e92-b1db-f7f3cfeee46c","page_size":[595,841],"block_position":"7-12"},{"bbox":[68,698,417,712],"type":"text","angle":0,"index":13,"merge_prev":false,"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 $u(x)f(x) + v(x)g(x) = 1$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":7,"id":"08103d17-be74-4c72-9137-e5331ec12c53","page_size":[595,841],"block_position":"7-13"},{"bbox":[53,727,538,756],"type":"text","angle":0,"index":14,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":7,"id":"e409f95b-1858-4257-8c69-52a056e3200d","page_size":[595,841],"block_position":"7-14"},{"bbox":[224,759,370,772],"type":"interline_equation","angle":0,"index":15,"text":"$$\nd (x) | [ u (x) f _ {1} (x) + v (x) g _ {1} (x) ] = 1\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":7,"id":"5a585159-2f3c-466c-9b4a-1e36f53b03a9","page_size":[595,841],"block_position":"7-15"},{"bbox":[293,803,300,812],"type":"page_number","angle":0,"index":1000015,"text":"3 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":7,"is_discarded":true,"id":"46336464-b0d4-4906-9752-41c81c3d8eee","page_size":[595,841],"block_position":"7-16"}],[{"bbox":[470,47,539,58],"type":"header","angle":0,"index":-99998,"text":"1.1 一元多项式 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":8,"is_discarded":true,"id":"3e094164-3339-499d-9b41-9b3b5a5a6874","page_size":[595,841],"block_position":"8-0"},{"bbox":[53,71,327,100],"type":"text","angle":0,"index":2,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"855d7ea0-8913-4837-afd2-dee9f0eca6ae","page_size":[595,841],"block_position":"8-1"},{"bbox":[35,117,371,131],"type":"text","angle":0,"index":3,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"dc02bdbe-9434-4134-b8b7-9bb22f3c2bab","page_size":[595,841],"block_position":"8-2"},{"bbox":[73,133,361,146],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"(2) 重点：若 $(f(x), g(x)) = d(x)$ ，证明： $(f(x^m), g(x^m)) = d(x^m)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"04c36464-558a-496d-b5a7-dcb7ea84e185","page_size":[595,841],"block_position":"8-3"},{"bbox":[73,149,373,162],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"(3) 设 $(f(x), g(x)) = 1, (f(x), h(x)) = 1$ ，证明： $(f(x), g(x)h(x)) = 1$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"1268cce3-8b0e-411d-ab43-84c811aa64d7","page_size":[595,841],"block_position":"8-4"},{"bbox":[54,163,538,193],"type":"text","angle":0,"index":6,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"6503b1e9-1a1d-464d-8de8-281c33849c53","page_size":[595,841],"block_position":"8-5"},{"bbox":[54,195,538,224],"type":"text","angle":0,"index":7,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"9918dfa3-bc98-41bd-9627-cee64438a812","page_size":[595,841],"block_position":"8-6"},{"bbox":[230,226,363,240],"type":"interline_equation","angle":0,"index":8,"text":"$$\nu (x ^ {k}) m (x ^ {k}) + v (x ^ {k}) n (x ^ {k}) = 1\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":8,"id":"0faef624-8b87-4ed5-bee8-c67feb6e6c52","page_size":[595,841],"block_position":"8-7"},{"bbox":[54,248,486,263],"type":"text","angle":0,"index":9,"merge_prev":false,"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 $(f(x^{k}), g(x^{k})) = d(x^{k})$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"7e26b804-62ad-4ab2-a662-88044b5bcf99","page_size":[595,841],"block_position":"8-8"},{"bbox":[73,265,458,278],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"3e47185b-0cc5-4a53-8baa-b9d02d5a2b4e","page_size":[595,841],"block_position":"8-9"},{"bbox":[137,285,456,298],"type":"interline_equation","angle":0,"index":11,"text":"$$\n\\left[ u _ {1} (x) u _ {2} (x) f (x) + v _ {1} (x) g (x) + v _ {2} (x) h (x) \\right] f (x) + v _ {1} (x) v _ {2} (x) g (x) h (x) = 1\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":8,"id":"243b2481-a91f-4e60-b3f8-4c448b195b59","page_size":[595,841],"block_position":"8-10"},{"bbox":[54,305,246,317],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"因此根据充要条件得到 $(f(x), g(x)h(x)) = 1$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"2d07e969-0ef4-4adf-a2ed-8a38f9c42d4c","page_size":[595,841],"block_position":"8-11"},{"bbox":[36,335,539,364],"type":"text","angle":0,"index":13,"merge_prev":false,"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 $\\partial(u(x)) < \\partial(g(x))$ , $\\partial(v(x)) < \\partial(f(x))$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"9090785c-836b-444a-9d19-ea78e62b8455","page_size":[595,841],"block_position":"8-12"},{"bbox":[52,366,539,395],"type":"text","angle":0,"index":14,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"4cf83f0e-7993-4b4b-ac4e-63570874dff5","page_size":[595,841],"block_position":"8-13"},{"bbox":[124,401,468,414],"type":"interline_equation","angle":0,"index":15,"text":"$$\n(q (x) g (x) + u (x)) f (x) + b (x) g (x) = 1 \\Rightarrow u (x) f (x) + (b (x) + q (x) f (x)) g (x) = 1\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":8,"id":"eab4e039-8fd0-4431-8f09-df8243145d0a","page_size":[595,841],"block_position":"8-14"},{"bbox":[54,420,537,449],"type":"text","angle":0,"index":16,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"be6e5259-b898-4d88-b5ab-f52b4ca4040d","page_size":[595,841],"block_position":"8-15"},{"bbox":[72,452,537,465],"type":"text","angle":0,"index":17,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"4a9086da-2ae4-4d46-8ea0-fd76cbbb38ef","page_size":[595,841],"block_position":"8-16"},{"bbox":[207,471,386,485],"type":"interline_equation","angle":0,"index":18,"text":"$$\n(u (x) - u _ {1} (x)) f (x) = (v _ {1} (x) - v (x)) g (x)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":8,"id":"e3eea77b-7ca3-4a84-802e-a61548e12e45","page_size":[595,841],"block_position":"8-17"},{"bbox":[55,491,448,504],"type":"text","angle":0,"index":19,"merge_prev":false,"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 $v_{1}(x)=v(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"2aa29c29-8d51-409f-8bc5-a6e6ed4fcc46","page_size":[595,841],"block_position":"8-18"},{"bbox":[57,521,223,535],"type":"title","angle":0,"index":20,"level":2,"text":"## 1.1.4 不可约多项式与因式分解","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"1a09ebbf-dbad-4c93-ab9f-c58f6759ec0b","page_size":[595,841],"block_position":"8-19"},{"bbox":[73,549,181,561],"type":"title","angle":0,"index":21,"level":2,"text":"## 定义 1.4 (不可约多项式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"fa35deeb-6fef-4e9c-bf96-886436768386","page_size":[595,841],"block_position":"8-20"},{"bbox":[67,565,526,595],"type":"text","angle":0,"index":22,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"82c3101a-5e4e-45c2-afcb-c0ea36871236","page_size":[595,841],"block_position":"8-21"},{"bbox":[73,615,210,628],"type":"title","angle":0,"index":23,"level":2,"text":"## 定理 1.7 (不可约多项式的性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"d513dd39-1897-4dbf-a99b-d75d1fdd8cb8","page_size":[595,841],"block_position":"8-22"},{"bbox":[69,632,224,645],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"若 $p(x)\\in \\mathbb{P}[x]$ 是不可约多项式，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"5fed7dfe-bc78-4104-8347-951df46d0596","page_size":[595,841],"block_position":"8-23"},{"bbox":[83,648,301,661],"type":"text","angle":0,"index":25,"merge_prev":false,"text":"- $p(x)$ 与 $\\forall f(x) \\in \\mathbb{P}[x]$ 要么互素，要么 $p(x) | f(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"5561b54c-fa97-4a7d-b264-da0494a4208d","page_size":[595,841],"block_position":"8-24"},{"bbox":[83,664,315,677],"type":"text","angle":0,"index":26,"merge_prev":false,"text":"若 $p(x)|f(x)g(x)$ ，则要么 $p(x)|f(x)$ 要么 $p(x)|g(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"c43eb007-2c40-4f36-b861-ca610282719b","page_size":[595,841],"block_position":"8-25"},{"bbox":[35,689,47,704],"type":"image","angle":0,"index":27,"img_path":"/94c21327ea1ce13f97d4196886e4a3c8357fc7b23d1e4155e67153e99c609ca1.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":8,"id":"b8cef8a6-abd1-4001-8eac-2fcf16504c4c","page_size":[595,841],"block_position":"8-26"},{"bbox":[53,693,539,722],"type":"text","angle":0,"index":28,"merge_prev":false,"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 $g(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"37d38ebf-26e3-4b6f-a217-f76ba0301390","page_size":[595,841],"block_position":"8-27"},{"bbox":[35,740,524,753],"type":"text","angle":0,"index":29,"merge_prev":false,"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 $(f(x)g(x), f(x) + g(x)) = 1$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":8,"id":"be688e70-3813-4cfa-94af-18866654e688","page_size":[595,841],"block_position":"8-28"},{"bbox":[293,803,301,811],"type":"page_number","angle":0,"index":999999,"text":"4 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":8,"is_discarded":true,"id":"62fbad9e-387c-4c7e-a138-3f489f975f42","page_size":[595,841],"block_position":"8-29"}],[{"bbox":[470,47,539,58],"type":"header","angle":0,"index":-99998,"text":"1.1 一元多项式 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":9,"is_discarded":true,"id":"e0e6047d-b7af-41c0-8422-da53a1efff40","page_size":[595,841],"block_position":"9-0"},{"bbox":[54,72,567,99],"type":"text","angle":0,"index":2,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"5c919b23-8905-40f7-8a29-7a41baabef12","page_size":[595,841],"block_position":"9-1"},{"bbox":[72,102,446,116],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"(3) 稍加变化: 复数域上的多项式 $f(x)$ 没有重因式, 证明 $(f(x) + 2f'(x), 3f^2(x)) = 1$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"df8ffc0d-2574-4d38-aa32-61a1d98bfd55","page_size":[595,841],"block_position":"9-2"},{"bbox":[54,116,540,161],"type":"text","angle":0,"index":4,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"2e4bdee7-2db4-4f40-864c-3c10ec386d82","page_size":[595,841],"block_position":"9-3"},{"bbox":[73,164,277,177],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"(2) 同种方法，随便挑 $F_{i}(x), F_{j}(x)$ 说一说就行 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"b2139047-5158-43fc-9a15-aa7c8a68ce33","page_size":[595,841],"block_position":"9-4"},{"bbox":[53,177,553,214],"type":"text","angle":0,"index":6,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"21e64616-df09-4215-a24f-e9126ce023a5","page_size":[595,841],"block_position":"9-5"},{"bbox":[35,232,538,261],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"87e20078-720d-4c9e-8c32-3f1aa897b2ec","page_size":[595,841],"block_position":"9-6"},{"bbox":[52,263,539,292],"type":"text","angle":0,"index":8,"merge_prev":false,"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 $p(x) \\nmid f(x), p(x) \\nmid g(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"160bcb2a-51ac-4a98-9f6b-9bb5da69b8ba","page_size":[595,841],"block_position":"9-7"},{"bbox":[72,313,191,327],"type":"title","angle":0,"index":9,"level":2,"text":"## 定理 1.8 (因式分解唯一性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"eed890d3-611d-4dea-8efe-2ae1bf03a93a","page_size":[595,841],"block_position":"9-8"},{"bbox":[68,330,474,343],"type":"text","angle":0,"index":10,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"fa3ed71b-d8fe-4907-93ea-18357d88edc1","page_size":[595,841],"block_position":"9-9"},{"bbox":[229,350,364,364],"type":"interline_equation","angle":0,"index":11,"text":"$$\nf (x) = c p _ {1} ^ {r _ {1}} (x) p _ {2} ^ {r _ {2}} (x) \\dots p _ {s} ^ {r _ {s}} (x)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":9,"id":"d0f02191-b36b-4a0f-97a3-930028cc5a9a","page_size":[595,841],"block_position":"9-10"},{"bbox":[57,396,186,410],"type":"title","angle":0,"index":12,"level":2,"text":"## 1.1.5 余数定理及其应用","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"cb4542e8-acaf-4ea9-bdcd-47352cf964a4","page_size":[595,841],"block_position":"9-11"},{"bbox":[73,423,161,436],"type":"title","angle":0,"index":13,"level":2,"text":"## 定理 1.9 (余数定理)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"5512fd6f-7c3c-41d9-accc-85ede59bf900","page_size":[595,841],"block_position":"9-12"},{"bbox":[68,440,203,453],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"$f(x)\\in \\mathbb{P}[x]$ ，对 $\\forall \\alpha \\in \\mathbb{P}$ ，总有 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"3889031f-d39c-4380-bd67-333eb5103fc2","page_size":[595,841],"block_position":"9-13"},{"bbox":[236,456,356,469],"type":"interline_equation","angle":0,"index":15,"text":"$$\nf (x) = (x - \\alpha) q (x) + f (\\alpha)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":9,"id":"ff62ae0f-124f-4ae1-8d88-88804db7779b","page_size":[595,841],"block_position":"9-14"},{"bbox":[52,486,491,498],"type":"text","angle":0,"index":16,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"51a53441-199e-4f71-a5b4-efbe0c3657d2","page_size":[595,841],"block_position":"9-15"},{"bbox":[236,506,356,518],"type":"interline_equation","angle":0,"index":17,"text":"$$\nf (x) = (x - \\alpha) q (x) + f (\\alpha)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":9,"id":"9fba0289-e839-42e7-abc9-e0c22a7acd57","page_size":[595,841],"block_position":"9-16"},{"bbox":[72,531,113,543],"type":"title","angle":0,"index":18,"level":2,"text":"## 推论 1.2","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"8bbc718f-9a23-4c88-8e03-2a7ce27e44a6","page_size":[595,841],"block_position":"9-17"},{"bbox":[67,547,285,560],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"$\\alpha \\in \\mathbb{P}$ 是多项式 $f(x)$ 的根当且仅当 $(x - \\alpha) \\mid f(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"40b9331c-3c0e-484f-b3ab-1cb83e736ac9","page_size":[595,841],"block_position":"9-18"},{"bbox":[52,577,539,605],"type":"text","angle":0,"index":20,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"d918d4b1-7ef4-4af8-93dd-1d6cc484f809","page_size":[595,841],"block_position":"9-19"},{"bbox":[35,624,367,637],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"练习 1.8 余数定理的应用 (1) 若 $(x-1)\\mid f(x^{m})$ ，证明 $(x^{m}-1)\\mid f(x^{m})$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"8e0c23ed-0776-4e03-be63-1a727d051770","page_size":[595,841],"block_position":"9-20"},{"bbox":[73,639,259,651],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"(2) 证明 $x^{2} + x + 1 \\mid x^{3m} + x^{3n + 1} + x^{3p + 2}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"32510aae-c530-45b5-b0d4-859cdb527fbe","page_size":[595,841],"block_position":"9-21"},{"bbox":[73,655,410,667],"type":"text","angle":0,"index":23,"merge_prev":false,"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 $(x - 1) \\mid f_{2}(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"e96868bf-5662-41c3-b667-33c144f63070","page_size":[595,841],"block_position":"9-22"},{"bbox":[73,670,371,683],"type":"text","angle":0,"index":24,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"1c6c8bf3-be55-4133-bc08-25c7c0c6ddf8","page_size":[595,841],"block_position":"9-23"},{"bbox":[53,685,539,714],"type":"text","angle":0,"index":25,"merge_prev":false,"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 $(x^m - 1) \\mid f(x^m)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"d27b6be8-0b08-4626-b923-9ff8e5fb60d8","page_size":[595,841],"block_position":"9-24"},{"bbox":[53,715,537,745],"type":"text","angle":0,"index":26,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"33f98ff8-e00e-4f3b-ace6-d1f45646fcb2","page_size":[595,841],"block_position":"9-25"},{"bbox":[73,747,508,761],"type":"text","angle":0,"index":27,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":9,"id":"322f88a0-7563-4e71-b277-2016b8da4e99","page_size":[595,841],"block_position":"9-26"},{"bbox":[293,803,301,812],"type":"page_number","angle":0,"index":999999,"text":"5 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":9,"is_discarded":true,"id":"301dac60-43e2-4ea7-adb4-7e1b58f1c9a0","page_size":[595,841],"block_position":"9-27"}],[{"bbox":[470,47,539,58],"type":"header","angle":0,"index":-99999,"text":"1.1 一元多项式 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":10,"is_discarded":true,"id":"4f824005-d404-4e03-92ff-0b65abbb1835","page_size":[595,841],"block_position":"10-0"},{"bbox":[57,72,163,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 1.1.6 重因式与重根","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"c87b4bc9-b06e-46ea-b623-cc40ffcb1fc0","page_size":[595,841],"block_position":"10-1"},{"bbox":[73,100,196,112],"type":"title","angle":0,"index":2,"level":2,"text":"## 定理 1.10 (重因式性质汇总)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"803e011b-f777-4907-adef-c590e50528b3","page_size":[595,841],"block_position":"10-2"},{"bbox":[68,116,314,129],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"$f(x)$ 是次数大于0的多项式， $p(x)$ 是不可约多项式，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"088a5822-0aaa-43f9-b149-eab9ab8473a6","page_size":[595,841],"block_position":"10-3"},{"bbox":[83,132,384,145],"type":"text","angle":0,"index":4,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"1f54b05a-fc53-478f-bfd4-ce9785060b02","page_size":[595,841],"block_position":"10-4"},{"bbox":[83,148,354,160],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"- 无重因式充要： $f(x)$ 没有重因式当且仅当 $f(x), f'(x)$ 互素。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"23bc715e-8165-45a8-88c4-54102aafce57","page_size":[595,841],"block_position":"10-5"},{"bbox":[83,163,437,187],"type":"text","angle":0,"index":6,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"e1736561-ab34-496e-84b1-b7e6e9a9dc9a","page_size":[595,841],"block_position":"10-6"},{"bbox":[52,201,539,231],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"fd1b0e2c-00a4-45d2-935f-50279c4ff74a","page_size":[595,841],"block_position":"10-7"},{"bbox":[35,248,539,263],"type":"text","angle":0,"index":8,"merge_prev":false,"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 $x^{2} + 1|f(x), x^{2} + 1|g(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"396aa86b-4b38-4696-a622-95b5bd7b3657","page_size":[595,841],"block_position":"10-8"},{"bbox":[188,266,396,303],"type":"interline_equation","angle":0,"index":9,"text":"$$\n\\left\\{ \\begin{array}{l} (x ^ {2} + 1) h (x) + (x + 1) f (x) + (x - 2) g (x) = 0 \\\\ (x ^ {2} + 1) k (x) + (x - 1) f (x) + (x + 2) g (x) = 0 \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":10,"id":"863bde3f-484d-42e2-a98e-5ec2c4e76cdf","page_size":[595,841],"block_position":"10-9"},{"bbox":[52,306,539,334],"type":"text","angle":0,"index":10,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"1f0119c0-3360-4cfb-ab67-324230bb9520","page_size":[595,841],"block_position":"10-10"},{"bbox":[224,333,359,371],"type":"interline_equation","angle":0,"index":11,"text":"$$\n\\left\\{ \\begin{array}{l} (i + 1) f (i) + (i - 2) g (i) = 0 \\\\ (i - 1) f (i) + (i + 2) g (i) = 0 \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":10,"id":"a11ee46c-d520-4b87-851b-8fd03f8f41bc","page_size":[595,841],"block_position":"10-11"},{"bbox":[54,379,279,392],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"解得 $f(i) = g(i) = 0$ 。同理可证 $f(-i) = g(-i) = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"76c2ed59-91e3-4536-bbc2-5a03691bd4d5","page_size":[595,841],"block_position":"10-12"},{"bbox":[73,414,186,427],"type":"title","angle":0,"index":13,"level":2,"text":"## 定理 1.11 (重根性质汇总)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"94518ba0-6ebb-4229-b290-a1d5711a542c","page_size":[595,841],"block_position":"10-13"},{"bbox":[68,431,246,444],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"$f(x)$ 是 $\\mathbb{P}[x]$ 上次数大于0的多项式，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"e54235db-f836-4196-ba94-389ce368a54d","page_size":[595,841],"block_position":"10-14"},{"bbox":[83,445,498,460],"type":"text","angle":0,"index":15,"merge_prev":false,"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 $f^{(k)}(x_0) \\neq 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"cd31aa3e-19f8-4bf9-a106-9d57bae1d519","page_size":[595,841],"block_position":"10-15"},{"bbox":[83,461,398,475],"type":"text","angle":0,"index":16,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"c24d00ae-5eb1-4281-8f88-62d3b9c2fd76","page_size":[595,841],"block_position":"10-16"},{"bbox":[52,491,539,520],"type":"text","angle":0,"index":17,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"d109fe22-ba4c-490c-8288-8d8beee104d2","page_size":[595,841],"block_position":"10-17"},{"bbox":[35,532,383,555],"type":"text","angle":0,"index":18,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"53bbbd5b-67d9-473d-9dc1-48f8f0675a6d","page_size":[595,841],"block_position":"10-18"},{"bbox":[73,554,345,567],"type":"text","angle":0,"index":20,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"88385798-7043-478f-8871-fe60f41d2cda","page_size":[595,841],"block_position":"10-19"},{"bbox":[54,565,537,615],"type":"text","angle":0,"index":21,"merge_prev":false,"lines_deleted":true,"text":"","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"88385798-7043-478f-8871-fe60f41d2cda","page_size":[595,841],"block_position":"10-20"},{"bbox":[161,613,432,637],"type":"interline_equation","angle":0,"index":22,"text":"$$\n(f (x), f ^ {\\prime} (x)) = (f (x), f (x) - f ^ {\\prime} (x)) = (f (x), \\frac {x ^ {n}}{n !}) = (f (x), x ^ {n})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":10,"id":"6cc7f9ec-c025-4a33-b0a7-7a6dc08ee5ce","page_size":[595,841],"block_position":"10-21"},{"bbox":[54,643,539,671],"type":"text","angle":0,"index":23,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"cd7e839b-913e-4e10-bf04-d9bb459eb56a","page_size":[595,841],"block_position":"10-22"},{"bbox":[73,674,171,687],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"(2) 也用无重因式证明 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"48a99ac3-bb25-4003-94c5-a258183b7e15","page_size":[595,841],"block_position":"10-23"},{"bbox":[35,705,430,718],"type":"text","angle":0,"index":25,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"8aa5956b-8d19-4562-b841-1d94272f77c6","page_size":[595,841],"block_position":"10-24"},{"bbox":[73,720,282,733],"type":"text","angle":0,"index":26,"merge_prev":false,"text":"(2) 重点: 求 $A, B$ 使得 $(x - 1)^{2} | Ax^{4} + Bx^{2} + 1$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"46583ba3-ea25-4f71-9165-9bc93ed9dcdc","page_size":[595,841],"block_position":"10-25"},{"bbox":[73,736,305,749],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"(3) 重点：求 $t$ 使得 $f(x) = x^{3} - 3x^{2} + tx - 1$ 有重根 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"fa068222-aea7-48b0-8155-80dd32781e8a","page_size":[595,841],"block_position":"10-26"},{"bbox":[73,751,390,764],"type":"text","angle":0,"index":28,"merge_prev":false,"text":"(4) 重点 ZJU2018.2: 证明 $x^{3} + px + q$ 有重根的充要条件 $4p^{3} + 27q^{2} = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":10,"id":"bd053800-61fd-479c-9ffd-60b697f764af","page_size":[595,841],"block_position":"10-27"},{"bbox":[293,803,301,812],"type":"page_number","angle":0,"index":1000018,"text":"6 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":10,"is_discarded":true,"id":"e2988d79-390a-43d8-b359-606a064043e7","page_size":[595,841],"block_position":"10-28"}],[{"bbox":[390,47,539,58],"type":"header","angle":0,"index":-99998,"text":"1.2 特殊数域上的一元多项式理论 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":11,"is_discarded":true,"id":"eb320b17-0202-4569-9795-0bd52d547d27","page_size":[595,841],"block_position":"11-0"},{"bbox":[52,71,471,100],"type":"text","angle":0,"index":2,"merge_prev":false,"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 $(x + 1)^4(x - 3)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"61d2de46-96d3-43a2-9cbd-a768fefb15b2","page_size":[595,841],"block_position":"11-1"},{"bbox":[54,103,539,130],"type":"text","angle":0,"index":3,"merge_prev":false,"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 $A = -2, B = 1$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"165a4612-203e-4948-94eb-cec34249e39b","page_size":[595,841],"block_position":"11-2"},{"bbox":[54,130,540,195],"type":"text","angle":0,"index":4,"merge_prev":false,"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 $t = -\\frac{15}{4}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"03b31ecd-f9c6-4e69-baab-69c22048aece","page_size":[595,841],"block_position":"11-3"},{"bbox":[54,197,539,242],"type":"text","angle":0,"index":5,"merge_prev":false,"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 $4p^3 + 27q^2 = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"9ecc79c4-fa2d-4ddc-962c-0e5e789e1b15","page_size":[595,841],"block_position":"11-4"},{"bbox":[73,242,335,253],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"(5)t=12 时有 3 重 x=-2 根，t=-15 时有 2 重 x=1 根。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"e23fda24-d749-41c4-ba3a-09c4e3f04246","page_size":[595,841],"block_position":"11-5"},{"bbox":[35,272,538,300],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"5bd1c104-95e4-44c8-be6d-e97d892aafeb","page_size":[595,841],"block_position":"11-6"},{"bbox":[54,302,202,315],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"解例如 $f(x) = x^{k + 1} + 1$ ， $x_0 = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"6c1c726b-cbd4-4b79-ae53-02781c4cf3a6","page_size":[595,841],"block_position":"11-7"},{"bbox":[52,318,447,331],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"5aee8a39-0f56-4913-95ad-d2a99df113b6","page_size":[595,841],"block_position":"11-8"},{"bbox":[57,349,163,364],"type":"title","angle":0,"index":10,"level":2,"text":"## 1.1.7 整数与多项式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"11ff2c06-0666-4b9c-8bd5-1169b7875cc3","page_size":[595,841],"block_position":"11-9"},{"bbox":[73,376,186,389],"type":"title","angle":0,"index":11,"level":2,"text":"## 定理 1.12 (整数与多项式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"bc734d6d-ff17-4dec-8f33-948e8fa48844","page_size":[595,841],"block_position":"11-10"},{"bbox":[64,383,538,414],"type":"text","angle":0,"index":12,"merge_prev":false,"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 $(x^{m} - 1, x^{n} - 1) = x^{d} - 1$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"4b2afd06-1a37-48a9-acbb-f8af18f4bace","page_size":[595,841],"block_position":"11-11"},{"bbox":[52,423,538,452],"type":"text","angle":0,"index":13,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"ff49b6bb-b8b8-46ad-8a28-36c2536dd99a","page_size":[595,841],"block_position":"11-12"},{"bbox":[54,455,538,500],"type":"text","angle":0,"index":14,"merge_prev":false,"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 $p(x)|x^d - 1$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"d10ac2a2-437e-423d-a93e-cd3e16559f21","page_size":[595,841],"block_position":"11-13"},{"bbox":[73,502,503,514],"type":"text","angle":0,"index":15,"merge_prev":false,"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 $d_{1} = d$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"6ae78d77-af38-4fea-970b-a2e49541d6f9","page_size":[595,841],"block_position":"11-14"},{"bbox":[35,533,387,545],"type":"title","angle":0,"index":16,"level":2,"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 $d|n$","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"74769945-1c61-4169-8a07-583356cd9a60","page_size":[595,841],"block_position":"11-15"},{"bbox":[54,548,537,576],"type":"text","angle":0,"index":17,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"a2997544-bf0b-4a5d-a748-3fa94b89623a","page_size":[595,841],"block_position":"11-16"},{"bbox":[72,580,423,592],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"(3) $g(x)=1+x+x^{2}+\\cdots+x^{n},f(x)=g(x^{k}),(k,n+1)=1$ ，证明： $g(x)|f(x)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"bc7fc695-4038-4941-ae87-823333d3f4dd","page_size":[595,841],"block_position":"11-17"},{"bbox":[54,596,200,608],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"证明 (1) 本题即前面题目的特例。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"53644139-e8de-4a51-bd2a-2aad631ce710","page_size":[595,841],"block_position":"11-18"},{"bbox":[52,610,539,638],"type":"text","angle":0,"index":20,"merge_prev":false,"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 $x^{d}-1|x^{n}-1$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"9d2aa968-e996-455f-8bbf-1ebf60f0baab","page_size":[595,841],"block_position":"11-19"},{"bbox":[52,641,537,671],"type":"text","angle":0,"index":21,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"aa04423d-bee6-4c28-8666-4f30073ee6fd","page_size":[595,841],"block_position":"11-20"},{"bbox":[54,674,539,739],"type":"text","angle":0,"index":22,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":11,"id":"489fedc8-dd8e-4a20-8ca6-62188e82f7e4","page_size":[595,841],"block_position":"11-21"},{"bbox":[293,802,300,811],"type":"page_number","angle":0,"index":999999,"text":"7 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":11,"is_discarded":true,"id":"40724640-a2fc-4f14-a013-0b9948108aa2","page_size":[595,841],"block_position":"11-22"}],[{"bbox":[390,46,539,58],"type":"header","angle":0,"index":-99999,"text":"1.2 特殊数域上的一元多项式理论 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":12,"is_discarded":true,"id":"3542ed2c-5e0d-44fa-8140-6ccf0fd0528d","page_size":[595,841],"block_position":"12-0"},{"bbox":[57,74,273,90],"type":"title","angle":0,"index":1,"level":2,"text":"## 1.2 特殊数域上的一元多项式理论","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"7a2af948-7303-4d18-bb89-e829c728d438","page_size":[595,841],"block_position":"12-1"},{"bbox":[57,106,210,121],"type":"title","angle":0,"index":2,"level":2,"text":"## 1.2.1 复数域上的一元多项式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"c2e433c8-772d-44a7-80ea-0397f6047882","page_size":[595,841],"block_position":"12-2"},{"bbox":[73,135,196,147],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"定理 1.13 (代数学基本定理) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"d75f235c-f3e5-486f-978e-92e370d4b8a8","page_size":[595,841],"block_position":"12-3"},{"bbox":[69,151,298,163],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"任意次数大于等于1的多项式在 $\\mathbb{C}$ 上至少有一个根。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"b680c3f6-f7b8-4e19-b241-8963774235a2","page_size":[595,841],"block_position":"12-4"},{"bbox":[73,184,196,197],"type":"title","angle":0,"index":5,"level":2,"text":"## 定理 1.14 (复数域因式分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"9f7f460b-bacc-4dcb-9fd5-4fd6303da569","page_size":[595,841],"block_position":"12-5"},{"bbox":[67,200,526,229],"type":"text","angle":0,"index":6,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"c114fb9d-4531-4fef-84aa-67258a030dd9","page_size":[595,841],"block_position":"12-6"},{"bbox":[57,259,210,274],"type":"title","angle":0,"index":7,"level":2,"text":"## 1.2.2 实数域上的一元多项式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"df9955eb-9079-42d5-8951-99f4b18de77c","page_size":[595,841],"block_position":"12-7"},{"bbox":[73,287,196,300],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"定理 1.15 (实数域因式分解) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"6973aeee-bf7c-415b-9b79-aa95cf59991e","page_size":[595,841],"block_position":"12-8"},{"bbox":[68,304,365,317],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"33056059-4b05-4da5-9cea-490595e2579b","page_size":[595,841],"block_position":"12-9"},{"bbox":[57,349,258,363],"type":"title","angle":0,"index":10,"level":2,"text":"## 1.2.3 有理数域上一元多项式因式分解","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"c3db4cda-3287-4f4c-b4d9-ba22eacca38b","page_size":[595,841],"block_position":"12-10"},{"bbox":[54,371,539,400],"type":"text","angle":0,"index":11,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"4fc13f06-0fb3-43c3-afcb-0baea295e928","page_size":[595,841],"block_position":"12-11"},{"bbox":[73,408,170,420],"type":"title","angle":0,"index":12,"level":2,"text":"## 定义 1.5 (本原多项式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"1057243d-bda4-4d61-84b6-3db46ef107d8","page_size":[595,841],"block_position":"12-12"},{"bbox":[69,425,506,438],"type":"text","angle":0,"index":13,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"29916113-2e00-41df-97f7-ad39b81c99c5","page_size":[595,841],"block_position":"12-13"},{"bbox":[261,444,333,458],"type":"interline_equation","angle":0,"index":14,"text":"$$\n(b _ {1}, \\dots , b _ {n}) = 1\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":12,"id":"2145cf4f-04d8-4a14-997c-ebd9c926a204","page_size":[595,841],"block_position":"12-14"},{"bbox":[69,464,195,476],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"则称 $g(x)$ 是一个本原多项式 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"254f1634-7ccf-42fe-b74b-0a993a9bdf69","page_size":[595,841],"block_position":"12-15"},{"bbox":[73,498,174,510],"type":"title","angle":0,"index":16,"level":2,"text":"## 定理 1.16 (Gauss 引理)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"02021571-5293-471d-863b-7913af71ee56","page_size":[595,841],"block_position":"12-16"},{"bbox":[69,514,241,526],"type":"title","angle":0,"index":17,"level":2,"text":"## 两个本原多项式的乘积仍为本原多项式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"479ef6f3-b637-43d9-b649-7f08a8a35188","page_size":[595,841],"block_position":"12-17"},{"bbox":[52,538,538,582],"type":"text","angle":0,"index":18,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"3cdd554f-5cbb-4b5f-8a3f-7c1c2983f896","page_size":[595,841],"block_position":"12-18"},{"bbox":[233,585,350,623],"type":"interline_equation","angle":0,"index":19,"text":"$$\n\\left\\{ \\begin{array}{l} p \\mid a _ {0}, \\dots , p \\mid a _ {i - 1}, p \\nmid a _ {i} \\\\ p \\mid b _ {0}, \\dots , p \\mid b _ {j - 1}, p \\nmid b _ {j} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":12,"id":"d741d433-ac40-4487-97a9-71cf8623ca7a","page_size":[595,841],"block_position":"12-19"},{"bbox":[54,628,539,671],"type":"text","angle":0,"index":20,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"ab866c7a-a6e0-4046-9615-5aa4de6e0450","page_size":[595,841],"block_position":"12-20"},{"bbox":[73,681,236,693],"type":"title","angle":0,"index":21,"level":2,"text":"## 定理 1.17 (有理多项式变本原多项式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"cb6407e5-6636-4482-b027-df5101bd1ac2","page_size":[595,841],"block_position":"12-21"},{"bbox":[69,698,494,711],"type":"text","angle":0,"index":22,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"c3035a43-4e47-4f9b-9622-fe219b66e64c","page_size":[595,841],"block_position":"12-22"},{"bbox":[53,728,539,756],"type":"text","angle":0,"index":23,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":12,"id":"81a23fa9-6c6c-443e-b0ce-8171b1f82a13","page_size":[595,841],"block_position":"12-23"},{"bbox":[293,803,300,812],"type":"page_number","angle":0,"index":1000023,"text":"8 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":12,"is_discarded":true,"id":"ee4cfce9-c48c-425e-8a66-0d2f8bf14615","page_size":[595,841],"block_position":"12-24"}],[{"bbox":[390,47,539,58],"type":"header","angle":0,"index":-99998,"text":"1.2 特殊数域上的一元多项式理论 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":13,"is_discarded":true,"id":"ed79e052-f36b-4664-94ed-d634a4765ecd","page_size":[595,841],"block_position":"13-0"},{"bbox":[35,70,461,99],"type":"text","angle":0,"index":2,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"b7dce30e-11ab-46cf-99db-2c89301795bd","page_size":[595,841],"block_position":"13-1"},{"bbox":[72,94,324,106],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"(2) $f(x)$ 是本原多项式，证明： $f(x+1)$ 也是本原多项式。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"98c235b2-ee58-4e8e-a718-c0f8394448a8","page_size":[595,841],"block_position":"13-2"},{"bbox":[52,110,537,138],"type":"text","angle":0,"index":4,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"a9f67fbf-011b-4e9d-9417-7da83560ef55","page_size":[595,841],"block_position":"13-3"},{"bbox":[255,141,337,154],"type":"interline_equation","angle":0,"index":5,"text":"$$\nf (x) = r g (x) q _ {1} (x)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":13,"id":"40599b7e-0856-4ae8-b517-82ca4c46e255","page_size":[595,841],"block_position":"13-4"},{"bbox":[54,163,486,177],"type":"text","angle":0,"index":6,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"cc2217d8-8947-4fcb-97e6-d9493d1fa2e4","page_size":[595,841],"block_position":"13-5"},{"bbox":[52,180,539,218],"type":"text","angle":0,"index":7,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"39139129-0dae-4207-a2d1-a658926c87fa","page_size":[595,841],"block_position":"13-6"},{"bbox":[196,216,386,261],"type":"interline_equation","angle":0,"index":8,"text":"$$\n\\left\\{ \\begin{array}{l} d \\mid b _ {n} \\Rightarrow d \\mid a _ {n} \\\\ d \\mid b _ {n - 1} \\Rightarrow d \\mid \\binom {n} {1} a _ {n} + a _ {n - 1} \\Rightarrow d \\mid a _ {n - 1} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":13,"id":"a0a8ee6d-b942-4385-92c7-495a7d1a2668","page_size":[595,841],"block_position":"13-7"},{"bbox":[54,268,367,280],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"b48a82ec-73b5-4280-8e6b-1f648acd3d0d","page_size":[595,841],"block_position":"13-8"},{"bbox":[73,302,226,315],"type":"title","angle":0,"index":10,"level":2,"text":"## 定理 1.18 (整系数多项式因式分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"3ef136ae-c7d2-4a1d-8d7a-86df282ea1bf","page_size":[595,841],"block_position":"13-9"},{"bbox":[68,319,533,333],"type":"text","angle":0,"index":11,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"c0489848-dbff-460d-adcb-32ebd03a3b12","page_size":[595,841],"block_position":"13-10"},{"bbox":[54,349,464,362],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"5c3f241b-c3f2-49f9-a18b-14ca9daddf82","page_size":[595,841],"block_position":"13-11"},{"bbox":[167,368,427,381],"type":"interline_equation","angle":0,"index":13,"text":"$$\nf (x) = a f _ {1} (x), g (x) = r g _ {1} (x), h (x) = s h _ {1} (x), a \\in \\mathbb {Z}, r, s \\in \\mathbb {Q}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":13,"id":"2e7360a8-1645-4f59-bbd8-7131da7535ae","page_size":[595,841],"block_position":"13-12"},{"bbox":[52,387,538,417],"type":"text","angle":0,"index":14,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"2f40bd23-6385-4c67-841a-b7f1b5467aed","page_size":[595,841],"block_position":"13-13"},{"bbox":[246,419,346,432],"type":"interline_equation","angle":0,"index":15,"text":"$$\nf (x) = (r s g _ {1} (x)) h _ {1} (x)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":13,"id":"755da10c-3326-4ccc-bccb-d6ed6dc71f41","page_size":[595,841],"block_position":"13-14"},{"bbox":[53,441,227,454],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"其中 $rsg_{1}(x), h_{1}(x)$ 都是整系数多项式。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"6990b27b-63b9-4c98-bcc9-308143f88e65","page_size":[595,841],"block_position":"13-15"},{"bbox":[35,467,531,502],"type":"text","angle":0,"index":17,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"868a2b7c-164a-4767-82eb-0f75caeb759b","page_size":[595,841],"block_position":"13-16"},{"bbox":[52,506,538,534],"type":"text","angle":0,"index":18,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"8383786f-2942-449b-b207-3b5511d5c20c","page_size":[595,841],"block_position":"13-17"},{"bbox":[52,535,540,563],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"3560a392-41ce-42c5-abbc-d483e49de4d0","page_size":[595,841],"block_position":"13-18"},{"bbox":[143,561,451,587],"type":"interline_equation","angle":0,"index":20,"text":"$$\nf (1) = (1 - \\alpha) (b _ {n - 1} + \\dots + b _ {1} + b _ {0}) \\Rightarrow \\frac {f (1)}{\\alpha - 1} = - b _ {0} - \\dots - b _ {n - 1} \\in \\mathbb {Z}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":13,"id":"78133e4e-0efc-4aac-ac37-4bd31e22aa7c","page_size":[595,841],"block_position":"13-19"},{"bbox":[53,592,539,620],"type":"text","angle":0,"index":21,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"588a6eaa-1d03-4522-91ca-322ff0e7130e","page_size":[595,841],"block_position":"13-20"},{"bbox":[52,622,540,668],"type":"text","angle":0,"index":22,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":13,"id":"37dac394-ccc2-48ec-80bc-f93dd8f41557","page_size":[595,841],"block_position":"13-21"},{"bbox":[293,802,300,811],"type":"page_number","angle":0,"index":999999,"text":"9 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":13,"is_discarded":true,"id":"b132331a-b95b-487d-9b79-75da03f597d9","page_size":[595,841],"block_position":"13-22"}],[{"bbox":[470,46,539,58],"type":"header","angle":0,"index":-99999,"text":"1.3 多元多项式 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":14,"is_discarded":true,"id":"0c5a9252-7d57-4969-adc4-f829b0d4628a","page_size":[595,841],"block_position":"14-0"},{"bbox":[57,72,198,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 1.2.4 一元有理多项式的根","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"c019b7cc-873a-49bc-8c1b-c1e047ea57a1","page_size":[595,841],"block_position":"14-1"},{"bbox":[73,100,196,113],"type":"title","angle":0,"index":2,"level":2,"text":"## 定理 1.19 (有理多项式的根)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"a32d538e-ed1e-416b-8a87-c53b4b098023","page_size":[595,841],"block_position":"14-2"},{"bbox":[64,115,533,147],"type":"text","angle":0,"index":3,"merge_prev":false,"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 $q\\mid a_{0},p\\mid a_{n}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"26b08788-faa6-44e9-a641-06f7fae7572b","page_size":[595,841],"block_position":"14-3"},{"bbox":[54,153,229,175],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"证明 假设 $\\frac{q}{p}$ 是解，则代入多项式得到： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"361731be-735c-4463-8a68-ffd90f43b1b2","page_size":[595,841],"block_position":"14-4"},{"bbox":[235,178,358,203],"type":"interline_equation","angle":0,"index":5,"text":"$$\na _ {n} \\frac {q ^ {n}}{p ^ {n}} + \\dots + a _ {1} \\frac {q}{p} + a _ {0} = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":14,"id":"e827a390-8507-4609-8e4f-6f6c1f6f2f59","page_size":[595,841],"block_position":"14-5"},{"bbox":[54,205,167,217],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"两边乘上 $p^n$ 并移项得到： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"7a1115b2-daef-4300-b6f2-92e80b1f0b9f","page_size":[595,841],"block_position":"14-6"},{"bbox":[175,221,408,258],"type":"interline_equation","angle":0,"index":7,"text":"$$\n\\left\\{ \\begin{array}{l} a _ {n} q ^ {n} = p (- a _ {n - 1} q ^ {n - 1} - a _ {n - 2} p q ^ {n - 2} - \\dots - a _ {0} p ^ {n - 1}) \\\\ a _ {0} p ^ {n} = q (- a _ {n} q ^ {n - 1} - a _ {n - 1} p q ^ {n - 2} - \\dots - a _ {n - 1} p ^ {n - 1}) \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":14,"id":"5a4f0376-1eb0-44d7-90c9-bd8ee193006b","page_size":[595,841],"block_position":"14-7"},{"bbox":[54,262,379,275],"type":"text","angle":0,"index":8,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"da47cbed-5825-428d-9f7b-31e05f62e2a0","page_size":[595,841],"block_position":"14-8"},{"bbox":[268,282,324,295],"type":"interline_equation","angle":0,"index":9,"text":"$$\np \\mid a _ {0}, q \\mid a _ {n}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":14,"id":"edd7b35a-d550-4329-9462-5a52abdfb0fb","page_size":[595,841],"block_position":"14-9"},{"bbox":[36,316,395,329],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"de5dd78b-a033-4109-9415-a9d01960480d","page_size":[595,841],"block_position":"14-10"},{"bbox":[73,331,285,344],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"(2) 求 $f(x) = 3x^{4} + 5x^{3} + x^{2} + 5x - 2$ 的有理根 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"71edb145-1fd2-4e4f-a276-98d0a42e5482","page_size":[595,841],"block_position":"14-11"},{"bbox":[54,347,439,359],"type":"text","angle":0,"index":12,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"75f20215-c752-4d61-a5a0-d33a6958a8ff","page_size":[595,841],"block_position":"14-12"},{"bbox":[73,361,229,381],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"(2) 全部列出来验证，结果为 $\\frac{1}{3}, -2$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"e714e2cf-e573-4fbc-a649-f83a2b5f8558","page_size":[595,841],"block_position":"14-13"},{"bbox":[73,399,202,412],"type":"title","angle":0,"index":14,"level":2,"text":"## 定理 1.20 (Eisenstein 判别法)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"32770141-c230-4753-b13e-319597164327","page_size":[595,841],"block_position":"14-14"},{"bbox":[67,416,527,445],"type":"text","angle":0,"index":15,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"90fd5f4b-c23e-4b24-98cb-8e6ee34bc704","page_size":[595,841],"block_position":"14-15"},{"bbox":[36,477,275,490],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"练习 1.17 使用 Eisenstein 判别法 证明在 Q 不可约 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"7ccda847-c19a-4c07-9157-58cfdd2ec498","page_size":[595,841],"block_position":"14-16"},{"bbox":[73,492,232,506],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"(1) $f(x)=2x^{4}+3x^{3}-9x^{2}-3x+6$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"9febd434-5d39-4208-a260-35b68dfe76bb","page_size":[595,841],"block_position":"14-17"},{"bbox":[73,508,173,521],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"(2) $f(x)=x^{n}+2x+2$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"8bdc26ee-59af-4dbf-800a-6a8cbaaf62e4","page_size":[595,841],"block_position":"14-18"},{"bbox":[54,523,154,536],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"证明 (1) 取 $p = 3$ 即可 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"50c9fc8b-3dbf-4f05-9182-6f7cd9a30bbf","page_size":[595,841],"block_position":"14-19"},{"bbox":[73,540,128,553],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"(2) 取 $p = 2$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"7fefddf4-d431-4cc7-9bd6-104fb006c37a","page_size":[595,841],"block_position":"14-20"},{"bbox":[57,575,158,591],"type":"title","angle":0,"index":22,"level":2,"text":"## 1.3 多元多项式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"c4e71eed-71d8-43c1-acd4-2ce7ca193a7c","page_size":[595,841],"block_position":"14-21"},{"bbox":[57,608,138,622],"type":"title","angle":0,"index":23,"level":2,"text":"## 1.3.1 基本概念","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"ad2caa25-d1f5-486f-879c-97c2427d0644","page_size":[595,841],"block_position":"14-22"},{"bbox":[73,634,161,647],"type":"title","angle":0,"index":24,"level":2,"text":"## 定义 1.6 (字典排序)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"8e01c1f2-31ad-4771-b8b7-4c614da2721d","page_size":[595,841],"block_position":"14-23"},{"bbox":[67,652,525,680],"type":"text","angle":0,"index":25,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"042ede7e-e725-47ae-9bc6-5006f3fa0b10","page_size":[595,841],"block_position":"14-24"},{"bbox":[73,701,176,714],"type":"title","angle":0,"index":26,"level":2,"text":"## 定理 1.21 (乘积保首项)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"ed49c2df-83d9-4416-983e-1f149103dce6","page_size":[595,841],"block_position":"14-25"},{"bbox":[68,717,420,730],"type":"text","angle":0,"index":27,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":14,"id":"395101a9-05cc-46a6-a871-bc305b16370c","page_size":[595,841],"block_position":"14-26"},{"bbox":[292,803,303,812],"type":"page_number","angle":0,"index":1000020,"text":"10 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":14,"is_discarded":true,"id":"58eceacd-9a80-4a2d-8679-1946f70815fa","page_size":[595,841],"block_position":"14-27"}],[{"bbox":[470,46,539,58],"type":"header","angle":0,"index":-99999,"text":"1.3 多元多项式 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":15,"is_discarded":true,"id":"4efcbb77-c9e2-4d42-a1a5-abfc1c703486","page_size":[595,841],"block_position":"15-0"},{"bbox":[57,72,151,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 1.3.2 对称多项式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"9f1e0b78-1d35-4e53-857e-3e11e887378a","page_size":[595,841],"block_position":"15-1"},{"bbox":[73,100,170,112],"type":"title","angle":0,"index":2,"level":2,"text":"## 定义 1.7 (对称多项式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"ab9724e2-50f4-4e30-9870-638bc0aea1eb","page_size":[595,841],"block_position":"15-2"},{"bbox":[69,116,443,130],"type":"text","angle":0,"index":3,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"d7a93b2b-a9c5-4a28-94a1-9378998ed355","page_size":[595,841],"block_position":"15-3"},{"bbox":[73,151,206,163],"type":"title","angle":0,"index":4,"level":2,"text":"## 定理 1.22 (对称多项式的性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"ba91c266-ef97-4a3b-98e4-f794062dcc8a","page_size":[595,841],"block_position":"15-4"},{"bbox":[69,168,326,180],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"对称多项式的和、差、乘积、多项式复合仍为对称多项式。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"45454349-083d-41c3-8200-ad9364d2b7ac","page_size":[595,841],"block_position":"15-5"},{"bbox":[73,201,190,213],"type":"title","angle":0,"index":6,"level":2,"text":"## 定义 1.8 (初等对称多项式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"bf73eba5-4522-4456-9db4-5137abf807b5","page_size":[595,841],"block_position":"15-6"},{"bbox":[68,219,391,244],"type":"text","angle":0,"index":7,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"cf904eaa-61b1-45f4-a794-1ed0cc08614b","page_size":[595,841],"block_position":"15-7"},{"bbox":[214,250,370,324],"type":"interline_equation","angle":0,"index":8,"text":"$$\n\\left\\{ \\begin{array}{l} \\sigma_ {1} = x _ {1} + x _ {2} + \\dots + x _ {n} \\\\ \\sigma_ {2} = x _ {1} x _ {2} + x _ {1} x _ {3} + \\dots + x _ {n - 1} x _ {n} \\\\ \\quad \\vdots \\\\ \\sigma_ {n} = x _ {1} x _ {2} \\dots x _ {n} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":15,"id":"da954d38-347a-4721-8005-ce11ac5e7320","page_size":[595,841],"block_position":"15-8"},{"bbox":[73,348,171,359],"type":"title","angle":0,"index":9,"level":2,"text":"## 定理 1.23 (Vieta 定理)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"f6633334-d07f-49f9-a7f9-3d9b7cb45025","page_size":[595,841],"block_position":"15-9"},{"bbox":[68,364,365,377],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"db7cbd2f-b23f-40ac-87b5-8088f1b43b90","page_size":[595,841],"block_position":"15-10"},{"bbox":[254,381,329,497],"type":"interline_equation","angle":0,"index":11,"text":"$$\n\\left\\{ \\begin{array}{l} a _ {n - 1} = - \\sigma_ {1} \\\\ a _ {n - 2} = \\sigma_ {2} \\\\ \\vdots \\\\ a _ {n - i} = (- 1) ^ {i} \\sigma_ {i} \\\\ \\vdots \\\\ a _ {0} = (- 1) ^ {n} \\sigma_ {n} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":15,"id":"52903216-d268-4f58-89ed-c5dc089edb8f","page_size":[595,841],"block_position":"15-11"},{"bbox":[73,519,166,531],"type":"title","angle":0,"index":12,"level":2,"text":"## 定理 1.24 (复合定理)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"b02e2670-de7f-4690-9a79-c414c3b4f51c","page_size":[595,841],"block_position":"15-12"},{"bbox":[69,536,417,550],"type":"text","angle":0,"index":13,"merge_prev":false,"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 $\\varphi(\\sigma_{1},\\cdots,\\sigma_{n})$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"22a0752c-254a-4602-941b-ad3af99e5265","page_size":[595,841],"block_position":"15-13"},{"bbox":[54,565,539,610],"type":"text","angle":0,"index":14,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"0b6065de-5117-418a-9a6f-d792e75491fa","page_size":[595,841],"block_position":"15-14"},{"bbox":[35,628,364,641],"type":"text","angle":0,"index":15,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"93f67219-3519-436d-ac82-45338e13b966","page_size":[595,841],"block_position":"15-15"},{"bbox":[52,643,406,656],"type":"text","angle":0,"index":16,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"fa97d682-cfe6-41f1-af0e-d143bfc25e3b","page_size":[595,841],"block_position":"15-16"},{"bbox":[175,663,418,677],"type":"interline_equation","angle":0,"index":17,"text":"$$\nf _ {1} (x) = f (x) - \\varphi_ {1} = - 3 (x _ {1} ^ {2} x _ {2} + x _ {1} ^ {2} x _ {3} + \\dots) - 6 x _ {1} x _ {2} x _ {3}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":15,"id":"55acae51-0a90-4139-a8ae-fda0ee188edb","page_size":[595,841],"block_position":"15-17"},{"bbox":[53,682,539,711],"type":"text","angle":0,"index":19,"merge_prev":false,"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 $f(x_{1},x_{2},x_{3})=3\\sigma_{3}-3\\sigma_{1}\\sigma_{2}+\\sigma_{1}^{3}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":15,"id":"1abae148-d533-4aba-a10d-859d4594dcc5","page_size":[595,841],"block_position":"15-18"},{"bbox":[292,803,302,811],"type":"page_number","angle":0,"index":1000017,"text":"11 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":15,"is_discarded":true,"id":"e0d69791-25f7-4067-ab19-8c640cc03562","page_size":[595,841],"block_position":"15-19"}],[{"bbox":[470,47,539,58],"type":"header","angle":0,"index":-99998,"text":"1.3 多元多项式 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":16,"is_discarded":true,"id":"fe4a75c1-455c-4f7f-94e4-f8314181ff0f","page_size":[595,841],"block_position":"16-0"},{"bbox":[57,72,127,86],"type":"title","angle":0,"index":2,"level":2,"text":"## 1.3.3 判别式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"735fe7ff-cf4d-46ec-ad99-e675a9cc9c0e","page_size":[595,841],"block_position":"16-1"},{"bbox":[73,96,460,109],"type":"text","angle":0,"index":3,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"086a534d-9f61-4290-bae3-38cd31eeb614","page_size":[595,841],"block_position":"16-2"},{"bbox":[73,116,151,129],"type":"title","angle":0,"index":4,"level":2,"text":"## 定义 1.9 (判别式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"af6e7b85-e9d4-4ae2-8298-d7eb0365cf42","page_size":[595,841],"block_position":"16-3"},{"bbox":[67,133,502,148],"type":"text","angle":0,"index":5,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"071ec2d5-d5ec-4641-ac04-ff5dc2305f0b","page_size":[595,841],"block_position":"16-4"},{"bbox":[237,152,356,178],"type":"interline_equation","angle":0,"index":6,"text":"$$\nD (f) := \\prod_ {1 \\leq i <   j \\leq n} (x _ {i} - x _ {j}) ^ {n}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":16,"id":"cde9f09c-c9ab-4973-8b04-bb7adb5f6dda","page_size":[595,841],"block_position":"16-5"},{"bbox":[69,180,230,194],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"显然 $f(x)$ 有重根当且仅当 $D(f) = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"9f5c6685-2fd0-4d99-8863-6986a2067555","page_size":[595,841],"block_position":"16-6"},{"bbox":[73,215,186,227],"type":"title","angle":0,"index":8,"level":2,"text":"## 定理 1.25 (判别式的计算)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"5399fc0a-91ee-4045-92d8-deeaf128f3ae","page_size":[595,841],"block_position":"16-7"},{"bbox":[68,232,524,260],"type":"text","angle":0,"index":9,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"ec9aabe0-b947-4fb1-8498-3ce8fc42147c","page_size":[595,841],"block_position":"16-8"},{"bbox":[35,292,326,305],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"练习 1.19 判别式的计算 (1) 计算 $f(x)=x^{2}+px+q$ 的判别式 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"9f26610c-0e01-4707-9fb2-b843191022e7","page_size":[595,841],"block_position":"16-9"},{"bbox":[52,308,418,322],"type":"text","angle":0,"index":11,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"ae1a43ea-ce81-4690-b3d5-684b48901520","page_size":[595,841],"block_position":"16-10"},{"bbox":[249,327,343,342],"type":"interline_equation","angle":0,"index":12,"text":"$$\ng (x _ {1}, x _ {2}) = \\sigma_ {1} ^ {2} - 4 \\sigma_ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":16,"id":"d93573eb-307b-4dd8-9a48-2bc415544712","page_size":[595,841],"block_position":"16-11"},{"bbox":[54,348,313,360],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"而根据 Vieta 定理， $p = -\\sigma_{1}, q = \\sigma_{2}$ ，因此 $D(f) = p^{2} - 4q$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"d5701913-7513-4c0a-8177-8ecf95b209fe","page_size":[595,841],"block_position":"16-12"},{"bbox":[56,378,116,391],"type":"title","angle":0,"index":14,"level":2,"text":"## 1.3.4 结式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"f3081d28-5013-4bcb-8880-82c76389ed85","page_size":[595,841],"block_position":"16-13"},{"bbox":[54,401,539,430],"type":"text","angle":0,"index":15,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"fcf798c6-81cc-403d-8c19-89b94ade44b8","page_size":[595,841],"block_position":"16-14"},{"bbox":[73,438,146,450],"type":"title","angle":0,"index":16,"level":2,"text":"## 定义 1.10 (结式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"871b5188-286a-4221-afbf-4f3945f08c5f","page_size":[595,841],"block_position":"16-15"},{"bbox":[68,454,524,482],"type":"text","angle":0,"index":17,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"d58b23ad-9c43-4834-bd8a-310e4b0551ea","page_size":[595,841],"block_position":"16-16"},{"bbox":[150,483,442,552],"type":"interline_equation","angle":0,"index":18,"text":"$$\nA = \\left[ \\begin{array}{c c c c c c c c} a _ {n} & a _ {n - 1} & a _ {n - 2} & \\dots & a _ {0} & 0 & \\dots & 0 \\\\ 0 & a _ {n} & a _ {n - 1} & a _ {n - 2} & \\dots & a _ {0} & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\ddots & \\ddots & & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & a _ {n} & a _ {n - 1} & a _ {n - 2} & \\dots & a _ {0} \\end{array} \\right] _ {m \\times (n + m)}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":16,"id":"27e61a5e-c36d-4061-8716-e205d175e237","page_size":[595,841],"block_position":"16-17"},{"bbox":[148,559,445,629],"type":"interline_equation","angle":0,"index":19,"text":"$$\nB = \\left[ \\begin{array}{c c c c c c c c} b _ {m} & b _ {m - 1} & b _ {m - 2} & \\dots & b _ {0} & 0 & \\dots & 0 \\\\ 0 & b _ {m} & b _ {m - 1} & b _ {m - 2} & \\dots & b _ {0} & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\ddots & \\ddots & & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & b _ {m} & b _ {m - 1} & b _ {m - 2} & \\dots & b _ {0} \\end{array} \\right] _ {n \\times (n + m)}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":16,"id":"e3d67c65-cd1b-4920-a09c-7d28e9bea5a2","page_size":[595,841],"block_position":"16-18"},{"bbox":[68,636,222,668],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"则定义它们的结式 $R(f,g) = \\left| \\begin{array}{l}A\\\\ B \\end{array} \\right|$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"9fe30b41-2ca5-456e-9372-febeff71c5b8","page_size":[595,841],"block_position":"16-19"},{"bbox":[73,690,186,702],"type":"title","angle":0,"index":21,"level":2,"text":"## 定理 1.26 (结式判定定理)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"df5f61bb-2d6c-48d9-af18-f2ad9807b4b3","page_size":[595,841],"block_position":"16-20"},{"bbox":[67,705,537,724],"type":"text","angle":0,"index":22,"merge_prev":false,"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 $R(f,g)=0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"9c8b48d3-d4ab-450b-8fdd-c28017755cdd","page_size":[595,841],"block_position":"16-21"},{"bbox":[54,736,537,750],"type":"text","angle":0,"index":23,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":16,"id":"f6cfbbcc-8c6d-41e1-be03-02ea7793adf1","page_size":[595,841],"block_position":"16-22"},{"bbox":[292,803,303,811],"type":"page_number","angle":0,"index":999999,"text":"12 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":16,"is_discarded":true,"id":"721b0380-eef7-4ebc-bea2-ff38d096115b","page_size":[595,841],"block_position":"16-23"}],[{"bbox":[470,47,539,58],"type":"header","angle":0,"index":-99998,"text":"1.3 多元多项式 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":17,"is_discarded":true,"id":"aadad483-ff09-4067-b5bf-a5f17112e81a","page_size":[595,841],"block_position":"17-0"},{"bbox":[526,360,535,370],"type":"text","angle":0,"index":0,"text":"# ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":17,"id":"60778f5a-375a-40b1-803c-6d3dcb7e4827","page_size":[595,841],"block_position":"17-1"},{"bbox":[54,72,186,84],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"$u(x)f(x) = v(x)g(x)$ ，不妨设 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":17,"id":"b5cf494d-101f-4dbd-b3e1-77ffc632f8ff","page_size":[595,841],"block_position":"17-2"},{"bbox":[223,84,360,121],"type":"interline_equation","angle":0,"index":3,"text":"$$\n\\left\\{ \\begin{array}{l} u (x) = u _ {m - 1} x ^ {m - 1} + \\dots + u _ {0} \\\\ v (x) = v _ {n - 1} x ^ {n - 1} + \\dots + v _ {0} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":17,"id":"347f6bd2-3a12-4958-a816-8d29903c33e1","page_size":[595,841],"block_position":"17-3"},{"bbox":[54,129,240,142],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"将 $u(x)f(x) = v(x)g(x)$ 展开对比系数得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":17,"id":"6373bc9d-c224-451b-8fe1-fd5854f3e81f","page_size":[595,841],"block_position":"17-4"},{"bbox":[191,148,392,221],"type":"interline_equation","angle":0,"index":5,"text":"$$\n\\left\\{ \\begin{array}{l} a _ {n} u _ {m - 1} = b _ {m} v _ {n - 1} \\\\ a _ {n - 1} u _ {m - 1} + a _ {n} u _ {m - 2} = b _ {m - 1} v _ {n - 1} + b _ {m} v _ {n - 2} \\\\ \\vdots \\\\ a _ {0} u _ {0} = b _ {0} v _ {0} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":17,"id":"1af9d63c-7671-40d7-a36c-933b9b3451f6","page_size":[595,841],"block_position":"17-5"},{"bbox":[54,227,539,239],"type":"text","angle":0,"index":6,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":17,"id":"cb9067b9-2d62-4eac-bc7d-fb27d982fd07","page_size":[595,841],"block_position":"17-6"},{"bbox":[260,243,331,274],"type":"interline_equation","angle":0,"index":7,"text":"$$\nC ^ {T} = \\binom{A}{- B}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":17,"id":"5cb17cf9-3097-4c7f-96d0-ef3f0fd85ad4","page_size":[595,841],"block_position":"17-7"},{"bbox":[54,278,279,291],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"因此有解当且仅当 $|C^T| = 0$ ，也等价于 $R(f,g) = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":17,"id":"369dca6b-fa55-4234-8448-efcb27ada876","page_size":[595,841],"block_position":"17-8"},{"bbox":[73,301,186,313],"type":"title","angle":0,"index":9,"level":2,"text":"## 定理 1.27 (结式计算公式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":17,"id":"9a5ec6d8-90f2-42ef-9088-a82de8873a2a","page_size":[595,841],"block_position":"17-9"},{"bbox":[68,317,414,330],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":17,"id":"b1fcdee6-aef4-4dca-974e-949b8e140676","page_size":[595,841],"block_position":"17-10"},{"bbox":[226,333,367,365],"type":"interline_equation","angle":0,"index":11,"text":"$$\nR (f, g) = a _ {n} ^ {m} b _ {m} ^ {n} \\prod_ {i = 1} ^ {n} \\prod_ {j = 1} ^ {m} (\\alpha_ {i} - \\beta_ {i})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":17,"id":"7a55a959-caa9-44e8-802c-a148bbe4e17d","page_size":[595,841],"block_position":"17-11"},{"bbox":[74,387,195,399],"type":"title","angle":0,"index":12,"level":2,"text":"## 定理 1.28 (判别式计算公式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":17,"id":"a1444b53-8346-4bb1-96c1-a343cc82d38e","page_size":[595,841],"block_position":"17-12"},{"bbox":[68,404,413,417],"type":"text","angle":0,"index":13,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":17,"id":"07d42e9c-c6f2-4521-aa4e-efdec5158bd0","page_size":[595,841],"block_position":"17-13"},{"bbox":[218,421,375,438],"type":"interline_equation","angle":0,"index":14,"text":"$$\nD (f) = (- 1) ^ {\\frac {n (n - 1)}{2}} a _ {0} ^ {- (2 n - 1)} R (f, f ^ {\\prime})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":17,"id":"4b3114a0-ecbf-470d-8a35-c4172c8517f2","page_size":[595,841],"block_position":"17-14"},{"bbox":[292,803,302,811],"type":"page_number","angle":0,"index":1000014,"text":"13 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":17,"is_discarded":true,"id":"40a4c4be-b4a9-4e17-a6cd-7c1851312cb5","page_size":[595,841],"block_position":"17-15"}],[{"bbox":[35,513,47,530],"type":"image","angle":0,"index":0,"img_path":"/a3107efcc333af95e29aadb82eccf9ae18205fa0aa42d2c811430f586e035cee.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":18,"id":"d136ebc5-f544-456d-9249-2e8814731e9e","page_size":[595,841],"block_position":"18-0"},{"bbox":[253,74,351,94],"type":"title","angle":0,"index":1,"level":2,"text":"## 第2章 矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"dbfad95d-6757-4932-9cef-9dc1c6b9d1f1","page_size":[595,841],"block_position":"18-1"},{"bbox":[56,122,143,139],"type":"title","angle":0,"index":2,"level":2,"text":"## 2.1 矩阵基础","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"f014ee10-add4-4884-a2ca-0fa3f9ef2604","page_size":[595,841],"block_position":"18-2"},{"bbox":[56,155,174,169],"type":"title","angle":0,"index":3,"level":2,"text":"## 2.1.1 矩阵概念与运算","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"5a7ac59b-4312-44e2-ab40-cbcc87f93a80","page_size":[595,841],"block_position":"18-3"},{"bbox":[74,179,88,191],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"略 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"66afb0b1-f0c1-49b6-8791-c74839aa7e97","page_size":[595,841],"block_position":"18-4"},{"bbox":[56,210,139,224],"type":"title","angle":0,"index":5,"level":2,"text":"## 2.1.2 矩阵的逆","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"e4e0a4a2-39e9-491a-ba82-203e041b9d85","page_size":[595,841],"block_position":"18-5"},{"bbox":[72,238,161,250],"type":"title","angle":0,"index":6,"level":2,"text":"## 定义 2.1 (矩阵的逆)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"d2ff2045-d56e-4cec-acdc-98e72e214486","page_size":[595,841],"block_position":"18-6"},{"bbox":[69,254,274,268],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"若 $AB = BA = E$ ，则称 $B$ 为 $A$ 的逆，记 $A^{-1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"74fe1abe-aff1-45b4-8e86-73d17d492962","page_size":[595,841],"block_position":"18-7"},{"bbox":[73,287,201,300],"type":"title","angle":0,"index":8,"level":2,"text":"## 定理 2.1 (矩阵逆的基本性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"57b2f171-0aba-4cb7-ae95-44388c28bf47","page_size":[595,841],"block_position":"18-8"},{"bbox":[68,303,427,318],"type":"interline_equation","angle":0,"index":9,"text":"$$\n(1) (k A) ^ {- 1} = k ^ {- 1} A ^ {- 1} (2) (A B) ^ {- 1} = B ^ {- 1} A ^ {- 1} (3) \\left(A ^ {T}\\right) ^ {- 1} = \\left(A ^ {- 1}\\right) ^ {T} (4) | A ^ {- 1} | = | A | ^ {- 1}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":18,"id":"bd5116da-b7c8-451c-b004-3684f7be099c","page_size":[595,841],"block_position":"18-9"},{"bbox":[54,334,281,347],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"证明 (2) $(AB)^{-1}AB = E$ ，因此 $(AB)^{-1} = B^{-1}A^{-1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"fc9e411b-e876-4f6e-a345-8eb7b5ad8b84","page_size":[595,841],"block_position":"18-10"},{"bbox":[73,349,210,363],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"(3) $(A^{-1})^T A^T = (AA^{-1})^T = E$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"248deb9f-f340-48f4-84df-2905f07dbdb6","page_size":[595,841],"block_position":"18-11"},{"bbox":[73,365,287,379],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"(4) $|A^{-1}A|=|A^{-1}|\\cdot|A|=1$ ，因此 $|A^{-1}|=|A|^{-1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"a909cc5e-0a60-4212-8b29-0d0e9d423086","page_size":[595,841],"block_position":"18-12"},{"bbox":[73,388,181,400],"type":"title","angle":0,"index":13,"level":2,"text":"## 定理 2.2 (可逆充要条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"04e3b6f8-6728-42bc-971d-5c1e5410b34a","page_size":[595,841],"block_position":"18-13"},{"bbox":[69,405,198,417],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"矩阵 $A$ 可以的充要条件如下： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"2695160e-9b63-417a-8673-790856bc1ea6","page_size":[595,841],"block_position":"18-14"},{"bbox":[84,421,129,433],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"- $|A| \\neq 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"fe6a1600-d5e1-4113-9c3a-6c866c857df3","page_size":[595,841],"block_position":"18-15"},{"bbox":[83,436,207,448],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"- $A$ 可表示为初等矩阵的积 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"0e1a2505-94da-4817-b44d-73600d006bf4","page_size":[595,841],"block_position":"18-16"},{"bbox":[73,468,211,481],"type":"title","angle":0,"index":17,"level":2,"text":"## 定理 2.3 (利用初等行变换求逆)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"a79b0cad-6efc-4729-884e-2beb59520496","page_size":[595,841],"block_position":"18-17"},{"bbox":[68,485,467,498],"type":"text","angle":0,"index":18,"merge_prev":false,"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， $(A,B)\\rightarrow (E,A^{-1}B)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"80773751-14b1-4f81-9255-1082420531c1","page_size":[595,841],"block_position":"18-18"},{"bbox":[52,509,318,542],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"笔记 同样可以通过初等列变换得到： $\\begin{pmatrix} A \\\\ B \\end{pmatrix} \\rightarrow \\begin{pmatrix} E \\\\ BA^{-1} \\end{pmatrix}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"15aafffe-8b85-4351-9fdb-fe8280521805","page_size":[595,841],"block_position":"18-19"},{"bbox":[35,550,276,562],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"练习 2.1 利用初等变换求逆 (1) 计算下面矩阵的逆: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"1013786e-2d4c-471b-a571-03c58c1f11de","page_size":[595,841],"block_position":"18-20"},{"bbox":[260,565,333,614],"type":"interline_equation","angle":0,"index":21,"text":"$$\n\\left[ \\begin{array}{c c c} 1 & 1 & - 1 \\\\ 2 & 1 & 0 \\\\ 1 & - 1 & 0 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":18,"id":"3f5c62ef-6d1d-4dc3-abaa-c4763334fa36","page_size":[595,841],"block_position":"18-21"},{"bbox":[52,616,183,667],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"解结果为 $\\frac{1}{3}\\left[ \\begin{array}{ccc}0 & 1 & 1\\\\ 0 & 1 & -2\\\\ -3 & 2 & -1 \\end{array} \\right]$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"83507cbe-8687-46d7-a047-6686a8aaae64","page_size":[595,841],"block_position":"18-22"},{"bbox":[35,672,537,685],"type":"text","angle":0,"index":23,"merge_prev":false,"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 $(E-A)^{-1}=E+A+A^{2}+\\cdots+A^{k-1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"812f1fd2-b132-41a0-8a5f-80796d32e580","page_size":[595,841],"block_position":"18-23"},{"bbox":[73,689,392,701],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"(2) $A, B$ 为 $n$ 阶方阵，若 $A + B = AB$ ，证明： $A - E$ 可逆，并求逆矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"260cef1a-81fb-4a29-b5c5-0c76bcacafe7","page_size":[595,841],"block_position":"18-24"},{"bbox":[73,704,318,717],"type":"text","angle":0,"index":25,"merge_prev":false,"text":"(3) 若 $A + B$ 可逆，证明 $A^{-1} + B^{-1}$ 也可逆，并求其逆 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"4f40bc07-d2d8-44e7-97c8-4ac396828958","page_size":[595,841],"block_position":"18-25"},{"bbox":[53,720,158,732],"type":"text","angle":0,"index":26,"merge_prev":false,"text":"解(1)直接做乘法验证: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"bfeaa593-5525-4d25-a780-6ac09e0ef734","page_size":[595,841],"block_position":"18-26"},{"bbox":[76,739,517,751],"type":"interline_equation","angle":0,"index":27,"text":"$$\n(E - A) \\left(E + A + A ^ {2} + \\dots + A ^ {k - 1}\\right) = E + A + A ^ {2} + \\dots + A ^ {k - 1} - A - A ^ {2} - \\dots - A ^ {k} = E - A ^ {k} = E\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":18,"id":"c8aa84f0-34f5-4026-9b70-59ecd1330bfa","page_size":[595,841],"block_position":"18-27"},{"bbox":[73,759,502,772],"type":"text","angle":0,"index":28,"merge_prev":false,"text":"(2) 由于 $A + B = AB$ ，则 $(A - E)(B - E) = AB - (A + B) + E = E$ ，因此 $(A - E)^{-1} = B - E$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":18,"id":"138051d3-76e7-4bbe-8df6-6b6081d70ffe","page_size":[595,841],"block_position":"18-28"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99998,"text":"2.1 矩阵基础 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":19,"is_discarded":true,"id":"f0c612b9-9a61-4c84-8235-708191e331a6","page_size":[595,841],"block_position":"19-0"},{"bbox":[73,71,258,84],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"(3) 根据 $B(A^{-1} + B^{-1})A = (A + B)$ 得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"fd7e4548-fce5-40d7-acf3-7c19b0738582","page_size":[595,841],"block_position":"19-1"},{"bbox":[150,89,442,104],"type":"interline_equation","angle":0,"index":3,"text":"$$\n(A ^ {- 1} + B ^ {- 1}) B = A ^ {- 1} (A + B) \\Rightarrow (A ^ {- 1} + B ^ {- 1}) B (A + B) ^ {- 1} = A ^ {- 1}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":19,"id":"3cb42a01-54d1-4064-89b1-7899ae7ed75c","page_size":[595,841],"block_position":"19-2"},{"bbox":[55,109,223,123],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"因此 $(A^{-1} + B^{-1})^{-1} = B(A + B)^{-1}A$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"c3720ab7-2bab-4e09-8ee9-593d328ed710","page_size":[595,841],"block_position":"19-3"},{"bbox":[73,146,180,158],"type":"title","angle":0,"index":5,"level":2,"text":"## 定义 2.2 ((代数) 余子式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"f359d71c-7d48-4e1a-9627-f7a7c0821f0a","page_size":[595,841],"block_position":"19-4"},{"bbox":[68,162,473,177],"type":"text","angle":0,"index":6,"merge_prev":false,"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 $(-1)^{i + j}M_{ij}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"f48f9abc-e0d8-4e42-bed6-bf16beb9433d","page_size":[595,841],"block_position":"19-5"},{"bbox":[73,197,160,209],"type":"title","angle":0,"index":7,"level":2,"text":"## 定义 2.3 (伴随矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"57bb9454-9ece-4cdc-badd-f407e5b1c66e","page_size":[595,841],"block_position":"19-6"},{"bbox":[68,214,371,227],"type":"text","angle":0,"index":8,"merge_prev":false,"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!!): ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"559d9cae-2983-4bba-bf3b-1aef74c8e98e","page_size":[595,841],"block_position":"19-7"},{"bbox":[224,230,367,299],"type":"interline_equation","angle":0,"index":9,"text":"$$\nA ^ {*} = \\left[ \\begin{array}{c c c c} A _ {1 1} & A _ {2 1} & \\dots & A _ {n 1} \\\\ A _ {1 2} & A _ {2 2} & \\dots & A _ {n 2} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ A _ {1 n} & A _ {2 n} & \\dots & A _ {n n} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":19,"id":"d9ec6eee-bd8e-4452-a2b5-ae0fd59939fe","page_size":[595,841],"block_position":"19-8"},{"bbox":[73,320,180,332],"type":"title","angle":0,"index":10,"level":2,"text":"## 定理 2.4 (利用伴随求逆)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"94d078bb-278d-4b64-842e-70a4d03b0996","page_size":[595,841],"block_position":"19-9"},{"bbox":[68,336,368,362],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"$A$ 及其伴随矩阵 $A^{*}$ 满足 $AA^{*} = |A|E$ ，若 $A$ 可逆，则 $A^{-1} = \\frac{1}{|A|} A^{*}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"766b3c92-6cf8-467d-b409-82685ec1ffc3","page_size":[595,841],"block_position":"19-10"},{"bbox":[54,372,361,403],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"b42ccb75-68af-4d69-a5f8-353794c9cbea","page_size":[595,841],"block_position":"19-11"},{"bbox":[252,407,340,444],"type":"interline_equation","angle":0,"index":13,"text":"$$\nB _ {i j} = \\left\\{ \\begin{array}{l l} | A |, & i = j \\\\ 0, & i \\neq j \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":19,"id":"7f479f67-e6f8-46f7-be2b-dcc1ca674041","page_size":[595,841],"block_position":"19-12"},{"bbox":[55,449,134,461],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"因此 $AA^{*} = |A|E$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"996ad34e-8957-4fae-91e5-9c225c688cc4","page_size":[595,841],"block_position":"19-13"},{"bbox":[57,479,174,493],"type":"title","angle":0,"index":15,"level":2,"text":"## 2.1.3 伴随矩阵的性质","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"d4fbe0ad-676a-4a40-b54d-94bd4dc59299","page_size":[595,841],"block_position":"19-14"},{"bbox":[73,507,190,519],"type":"title","angle":0,"index":16,"level":2,"text":"## 定理 2.5 (伴随运算的性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"72fa3db0-abe0-4091-a588-bdf3600b9497","page_size":[595,841],"block_position":"19-15"},{"bbox":[69,523,152,535],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"$A$ 为 $n$ 阶方阵，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"540f5527-240c-4ada-8f85-34bb439d1c01","page_size":[595,841],"block_position":"19-16"},{"bbox":[82,538,163,551],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"1. $(A^{*})^{T} = (A^{T})^{*}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"ed30018f-67d2-4842-9637-f0b72d791153","page_size":[595,841],"block_position":"19-17"},{"bbox":[81,553,170,567],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"2. $(A^{*})^{-1} = (A^{-1})^{*}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"9ad00491-6211-4a48-b497-e19ceec19571","page_size":[595,841],"block_position":"19-18"},{"bbox":[81,570,162,582],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"3. $(AB)^{*} = B^{*}A^{*}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"e015b1d9-f3fc-45d6-927f-1f4f2d6b326a","page_size":[595,841],"block_position":"19-19"},{"bbox":[55,599,169,613],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"证明 (1) 设 $A = (a_{ij})$ ，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"47d219e9-d769-4eac-a550-50c4be55912c","page_size":[595,841],"block_position":"19-20"},{"bbox":[217,613,373,679],"type":"interline_equation","angle":0,"index":22,"text":"$$\n(A ^ {*}) ^ {T} = \\left[ \\begin{array}{c c c c} A _ {1 1} & A _ {1 2} & \\dots & A _ {1 n} \\\\ A _ {2 1} & A _ {2 2} & \\dots & A _ {2 n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ A _ {n 1} & A _ {n 2} & \\dots & A _ {n n} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":19,"id":"1b15a1da-005f-44f6-bb5c-6d5bbf39f5c0","page_size":[595,841],"block_position":"19-21"},{"bbox":[54,687,320,701],"type":"text","angle":0,"index":23,"merge_prev":false,"text":"同理设 $(A^T)^* = (B_{ij})$ ，则 $B_{ij} = A_{ji}^T$ ，因此行列式大小相同。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"cb336a5d-e7d4-4694-b0a9-e450d8a363ed","page_size":[595,841],"block_position":"19-22"},{"bbox":[73,703,257,724],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"(2) $A^{*} = |A|A^{-1}$ ，则 $(A^{*})^{-1} = \\frac{1}{|A|} A$ 因此 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"be7f8ddc-2079-4abe-872a-fb3e6dc0e3b5","page_size":[595,841],"block_position":"19-23"},{"bbox":[239,726,354,753],"type":"interline_equation","angle":0,"index":25,"text":"$$\n(A ^ {- 1}) ^ {*} = | A ^ {- 1} | A = \\frac {1}{| A |} A\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":19,"id":"90752e4c-f63d-472e-819b-ad530919ae67","page_size":[595,841],"block_position":"19-24"},{"bbox":[54,756,116,767],"type":"text","angle":0,"index":26,"merge_prev":false,"text":"因此结论成立 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":19,"id":"1e8fee5d-05ba-4692-8394-801ec1853a39","page_size":[595,841],"block_position":"19-25"},{"bbox":[292,803,302,812],"type":"page_number","angle":0,"index":999999,"text":"15 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":19,"is_discarded":true,"id":"bac2aff8-8b9f-4548-832d-2996c56827a9","page_size":[595,841],"block_position":"19-26"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99999,"text":"2.1 矩阵基础 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":20,"is_discarded":true,"id":"eee4098a-5beb-4dfb-8279-11e287b7e65c","page_size":[595,841],"block_position":"20-0"},{"bbox":[73,70,308,84],"type":"interline_equation","angle":0,"index":1,"text":"$$\n(3) (A B) ^ {*} = | A B | (A B) ^ {- 1} = | B | (B) ^ {- 1} | A | A ^ {- 1} = B ^ {*} A ^ {*}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":20,"id":"8ec4dc45-20be-4dbe-8940-3a0c22e72328","page_size":[595,841],"block_position":"20-1"},{"bbox":[73,94,200,106],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"定理 2.6 (伴随矩阵的行列式) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"e2b90572-1252-4e7a-947f-4accb341e68d","page_size":[595,841],"block_position":"20-2"},{"bbox":[69,111,130,124],"type":"interline_equation","angle":0,"index":3,"text":"$$\n| A ^ {*} | = | A | ^ {n - 1}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":20,"id":"533d9209-78e6-42e6-b0ac-6a0336b1d8f8","page_size":[595,841],"block_position":"20-3"},{"bbox":[54,140,255,154],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"证明 $|A^{*}| = \\left| |A| A^{-1} \\right| = |A|^{n} \\cdot |A^{-1}| = |A|^{n - 1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"95da56ff-5ec6-4a2e-91a7-5f07d5a81f0e","page_size":[595,841],"block_position":"20-4"},{"bbox":[73,163,181,176],"type":"title","angle":0,"index":5,"level":2,"text":"## 定理 2.7 (伴随矩阵的秩)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"fb78d6d3-0408-4906-be1f-8adce192ba51","page_size":[595,841],"block_position":"20-5"},{"bbox":[69,180,179,193],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"$A^{*}$ 为 $A$ 的伴随矩阵，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"d1c19746-b36d-4ee1-9aa9-5997f094d6dc","page_size":[595,841],"block_position":"20-6"},{"bbox":[233,192,359,248],"type":"interline_equation","angle":0,"index":7,"text":"$$\nr (A ^ {*}) = \\left\\{ \\begin{array}{l l} n, & r (A) = n \\\\ 1, & r (A) = n - 1 \\\\ 0, & r (A) <   n - 1 \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":20,"id":"94dc1c60-3c39-4b61-a890-626c2311e56e","page_size":[595,841],"block_position":"20-7"},{"bbox":[54,265,531,280],"type":"text","angle":0,"index":8,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"be17f7cf-ffa8-470c-941a-7ec098bf79a5","page_size":[595,841],"block_position":"20-8"},{"bbox":[177,285,416,299],"type":"interline_equation","angle":0,"index":9,"text":"$$\nr (A A ^ {*}) \\geq r (A) + r (A ^ {*}) - n = r (A ^ {*}) - 1 \\Rightarrow r (A ^ {*}) \\leq 1\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":20,"id":"8830856d-9683-4afb-9015-c48f16187072","page_size":[595,841],"block_position":"20-9"},{"bbox":[54,305,531,318],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"35b25a94-0afb-4b3b-b615-f998b71b2be5","page_size":[595,841],"block_position":"20-10"},{"bbox":[72,327,201,339],"type":"title","angle":0,"index":11,"level":2,"text":"## 定理 2.8 (伴随矩阵的特征值)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"d4f26856-c363-4bb5-8f55-f18bb3034a7e","page_size":[595,841],"block_position":"20-11"},{"bbox":[68,345,329,371],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"矩阵 A 的特征值为 $\\lambda_{1},\\cdots,\\lambda_{n}$ ，则 $A^{*}$ 的特征值 $\\lambda_{j}^{*}=\\prod_{i\\neq j}\\lambda_{i}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"4ef8232a-e343-416b-979a-43d1a3a6d2bc","page_size":[595,841],"block_position":"20-12"},{"bbox":[54,387,540,430],"type":"text","angle":0,"index":13,"merge_prev":false,"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， ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"69bb3318-cb63-4db4-91c0-93fa76548979","page_size":[595,841],"block_position":"20-13"},{"bbox":[55,428,195,439],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"但值整体相同)，因此结论成立。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"e6531bed-ecb9-4a18-a2a5-f42d565ad377","page_size":[595,841],"block_position":"20-14"},{"bbox":[56,459,151,473],"type":"title","angle":0,"index":15,"level":2,"text":"## 2.1.4 广义逆矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"7fdc72b7-d02d-4c91-a0fc-84e91cb2048b","page_size":[595,841],"block_position":"20-15"},{"bbox":[73,486,151,498],"type":"title","angle":0,"index":16,"level":2,"text":"## 定义 2.4 (广义逆)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"2fde401a-50cb-4c39-9345-0e41ed1a1f39","page_size":[595,841],"block_position":"20-16"},{"bbox":[69,503,273,516],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"$A_{m\\times n}$ 是数域 $\\mathbb{P}$ 上的矩阵，若存在 $G_{n\\times m}$ 使得 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"98ccf4bb-afcf-4d48-959f-3b193216a7fe","page_size":[595,841],"block_position":"20-17"},{"bbox":[273,523,320,534],"type":"interline_equation","angle":0,"index":18,"text":"$$\nA G A = A\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":20,"id":"f363a995-dfb9-45e5-b047-18100466ac34","page_size":[595,841],"block_position":"20-18"},{"bbox":[69,541,191,553],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"则称 $G$ 是 $A$ 的广义逆矩阵。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"b29f5626-bfa1-47be-a2ca-236abb16e96f","page_size":[595,841],"block_position":"20-19"},{"bbox":[35,565,47,581],"type":"image","angle":0,"index":20,"img_path":"/ae1bf8e0a1c16b0d253249710c607fe306d87bcf8e814a8530bbe1480fa8f84e.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":20,"id":"cd46500a-ada0-4301-9490-9b2193e4f034","page_size":[595,841],"block_position":"20-20"},{"bbox":[52,570,183,581],"type":"title","angle":0,"index":21,"level":2,"text":"## 笔记 广义逆矩阵一般不唯一。","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"0f492eee-d68a-42bb-b03a-ad0cc812b189","page_size":[595,841],"block_position":"20-21"},{"bbox":[73,591,191,603],"type":"title","angle":0,"index":22,"level":2,"text":"## 定理 2.9 (广义逆的表达式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"cc5ad0c5-1bae-46bf-ad56-7bddcf486905","page_size":[595,841],"block_position":"20-22"},{"bbox":[68,609,524,639],"type":"text","angle":0,"index":23,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"39d547af-b173-4cf6-90d5-0141e96f334e","page_size":[595,841],"block_position":"20-23"},{"bbox":[236,645,357,675],"type":"interline_equation","angle":0,"index":25,"text":"$$\nG = Q ^ {- 1} \\left( \\begin{array}{c c} E _ {r} & C \\\\ D & F \\end{array} \\right) P ^ {- 1}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":20,"id":"33c9110e-db3a-42af-8cd9-524921855084","page_size":[595,841],"block_position":"20-24"},{"bbox":[68,679,381,692],"type":"text","angle":0,"index":26,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"3393e96a-1e6e-4e2d-a8f4-62d8b3d7ac20","page_size":[595,841],"block_position":"20-25"},{"bbox":[54,709,325,722],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"证明 (1) 验证 $G$ 是广义逆: 直接乘起来就行, 得到 $AGA = A$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":20,"id":"1f8307ed-b33e-49f8-b51f-945b349adfcf","page_size":[595,841],"block_position":"20-26"},{"bbox":[292,803,303,812],"type":"page_number","angle":0,"index":1000023,"text":"16 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":20,"is_discarded":true,"id":"2d2d3578-82d8-445e-846d-a7863ed20aba","page_size":[595,841],"block_position":"20-27"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99999,"text":"2.1 矩阵基础 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":21,"is_discarded":true,"id":"dbb75043-3ca6-4218-837d-76ed8ca69985","page_size":[595,841],"block_position":"21-0"},{"bbox":[73,69,358,101],"type":"text","angle":0,"index":1,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"fd98f7fb-a62d-493d-a86d-64b3267b5fd4","page_size":[595,841],"block_position":"21-1"},{"bbox":[241,106,352,138],"type":"interline_equation","angle":0,"index":2,"text":"$$\nA G A = P \\left( \\begin{array}{c c} B & O \\\\ O & O \\end{array} \\right) Q\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":21,"id":"0f170662-afad-4036-a664-3ed2a61b5946","page_size":[595,841],"block_position":"21-2"},{"bbox":[55,142,132,153],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"因此得到 $B = E_{r}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"d0adcc38-3f7c-4ef6-b251-6a9f44e1b6a6","page_size":[595,841],"block_position":"21-3"},{"bbox":[73,163,255,175],"type":"title","angle":0,"index":5,"level":2,"text":"## 定理 2.10 (非方阵线性方程组解的表达式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"2d8b36a9-a38d-43d6-8562-5a3b84703deb","page_size":[595,841],"block_position":"21-4"},{"bbox":[69,179,309,191],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"$A_{m\\times n}$ ， $G$ 是 $A$ 的广义逆，则 $AX = b$ 的通解可表示为 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"6484283b-b9ac-4af9-8fca-e3ca42c3b5fd","page_size":[595,841],"block_position":"21-5"},{"bbox":[243,199,349,211],"type":"interline_equation","angle":0,"index":7,"text":"$$\nX = G b + (E _ {n} - G A) Y\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":21,"id":"ddf3fffd-f1b2-44b9-ad62-75a757c0a0c6","page_size":[595,841],"block_position":"21-6"},{"bbox":[68,218,176,230],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"这里 $Y$ 是任一 $n$ 维向量 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"0e9571e8-894d-4a85-a30e-89512547d3bf","page_size":[595,841],"block_position":"21-7"},{"bbox":[54,247,527,259],"type":"text","angle":0,"index":9,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"9b764f42-af76-4b43-8f7f-0e471caea5c3","page_size":[595,841],"block_position":"21-8"},{"bbox":[207,266,387,278],"type":"interline_equation","angle":0,"index":10,"text":"$$\nA X = A G b + (A - A G A) Y = b + O = b\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":21,"id":"cbdcfc1c-dbaf-4b72-9b2f-88b74e804080","page_size":[595,841],"block_position":"21-9"},{"bbox":[73,285,298,297],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"(2) 证明解都有该形式: 设 $X_{0}$ 是 $AX = b$ 的解, 则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"4b3ee296-a135-4602-a267-bcb2fd07ce80","page_size":[595,841],"block_position":"21-10"},{"bbox":[155,305,437,317],"type":"interline_equation","angle":0,"index":12,"text":"$$\nX _ {0} = G b + X _ {0} - G b = G b + X _ {0} - G A X _ {0} = G b + (E _ {n} - G A) X _ {0}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":21,"id":"ffb54881-654b-42b5-afbd-87c344e2881e","page_size":[595,841],"block_position":"21-11"},{"bbox":[56,340,207,354],"type":"title","angle":0,"index":13,"level":2,"text":"## 2.1.5 Moore-Penrose 广义逆","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"37fe7a62-127e-4a25-9087-4575824da1ec","page_size":[595,841],"block_position":"21-12"},{"bbox":[73,367,161,379],"type":"title","angle":0,"index":14,"level":2,"text":"## 定义 2.5 (共轭转置)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"3da0703c-60fb-44c0-b546-2137574e0853","page_size":[595,841],"block_position":"21-13"},{"bbox":[68,383,258,397],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"$A$ 是 $\\mathbb{C}$ 上 $n\\times m$ 矩阵，则定义 $A^{H} = (\\overline{A})^{T}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"9f52aaa3-9d89-46b8-9d1a-8414ce93d845","page_size":[595,841],"block_position":"21-14"},{"bbox":[73,419,218,431],"type":"title","angle":0,"index":16,"level":2,"text":"## 定义 2.6 (Moore-Penrose 广义逆)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"a56cea48-1fc8-406e-9a34-e36a9de32409","page_size":[595,841],"block_position":"21-15"},{"bbox":[69,436,337,448],"type":"text","angle":0,"index":17,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"9540fa9d-68c2-42e1-93d6-4ef7d1451cf5","page_size":[595,841],"block_position":"21-16"},{"bbox":[81,451,141,463],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"1. AGA = A ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"1c4d2279-3b1b-4af1-982c-d5c03b8c6e4f","page_size":[595,841],"block_position":"21-17"},{"bbox":[81,467,142,478],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"2. $GAG = G$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"925381ff-fd63-4f4b-8aa2-a3134d81c684","page_size":[595,841],"block_position":"21-18"},{"bbox":[81,481,157,495],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"3. $(AG)^{H} = AG$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"31ee4b97-7f56-4337-af53-6926fe503484","page_size":[595,841],"block_position":"21-19"},{"bbox":[81,497,157,510],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"4. $(GA)^{H} = GA$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"d262c0f1-a735-4046-bb63-d74de9ec2d74","page_size":[595,841],"block_position":"21-20"},{"bbox":[69,513,189,525],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"则称 $G$ 为 $A$ 的M-P广义逆 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"f540627e-2d5d-456d-ae7f-7ded62f38b70","page_size":[595,841],"block_position":"21-21"},{"bbox":[73,546,237,558],"type":"title","angle":0,"index":23,"level":2,"text":"## 定理 2.11 (M-P 广义逆的存在唯一性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"76315dde-fbf2-44c5-b92a-2cf7eef2c2da","page_size":[595,841],"block_position":"21-22"},{"bbox":[68,562,525,591],"type":"text","angle":0,"index":24,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"95122a20-44b9-41c3-93cf-7483d41c5940","page_size":[595,841],"block_position":"21-23"},{"bbox":[224,596,370,610],"type":"interline_equation","angle":0,"index":25,"text":"$$\nA ^ {+} = C ^ {H} (C C ^ {H}) ^ {- 1} (B ^ {H} B) ^ {- 1} B ^ {H}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":21,"id":"efeda2b3-fe15-43ba-a12e-742a391a13fa","page_size":[595,841],"block_position":"21-24"},{"bbox":[54,628,413,640],"type":"text","angle":0,"index":26,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"524dd1f2-9b43-4dc1-bc40-331b2d8b4b68","page_size":[595,841],"block_position":"21-25"},{"bbox":[166,645,427,693],"type":"interline_equation","angle":0,"index":27,"text":"$$\nA = (\\alpha_ {1}, \\dots , \\alpha_ {n}) = (\\beta_ {1}, \\dots , \\beta_ {r}) \\left( \\begin{array}{c c c} c _ {1 1} & \\dots & c _ {1 n} \\\\ \\vdots & & \\vdots \\\\ c _ {r 1} & \\dots & c _ {r n} \\end{array} \\right) = B C\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":21,"id":"942ee21a-b2bb-413e-bab0-d8c4775c3f83","page_size":[595,841],"block_position":"21-26"},{"bbox":[54,698,537,727],"type":"text","angle":0,"index":28,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"43e08eac-597d-442d-aa4a-bfa1dea38d6d","page_size":[595,841],"block_position":"21-27"},{"bbox":[226,731,366,746],"type":"interline_equation","angle":0,"index":29,"text":"$$\nG = C ^ {H} (C C ^ {H}) ^ {- 1} (B ^ {H} B) ^ {- 1} B ^ {H}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":21,"id":"8597fd3d-d7d3-4db5-a8c4-20bce941754e","page_size":[595,841],"block_position":"21-28"},{"bbox":[55,753,180,766],"type":"text","angle":0,"index":30,"merge_prev":false,"text":"可验证 $G$ 满足上述所有条件 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":21,"id":"b3848d0e-163e-471f-aa19-f0f3e9835a61","page_size":[595,841],"block_position":"21-29"},{"bbox":[292,803,302,811],"type":"page_number","angle":0,"index":1000003,"text":"17 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":21,"is_discarded":true,"id":"49ed61a6-375e-4f03-8327-49835bf550b2","page_size":[595,841],"block_position":"21-30"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99999,"text":"2.2 分块矩阵 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":22,"is_discarded":true,"id":"bcc62f00-a818-4ac4-aa0d-7ab0befa4ee5","page_size":[595,841],"block_position":"22-0"},{"bbox":[73,71,421,84],"type":"text","angle":0,"index":1,"merge_prev":false,"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 $X$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":22,"id":"f9a4d559-f062-4bad-bab2-5cb316004cfa","page_size":[595,841],"block_position":"22-1"},{"bbox":[172,89,421,177],"type":"interline_equation","angle":0,"index":2,"text":"$$\n\\begin{array}{l} \\boldsymbol {X} = \\boldsymbol {X A X} = \\boldsymbol {X} (\\boldsymbol {A X}) ^ {\\mathrm{H}} = \\boldsymbol {X X} ^ {\\mathrm{H}} \\boldsymbol {A} ^ {\\mathrm{H}} = \\boldsymbol {X X} ^ {\\mathrm{H}} (\\boldsymbol {A G A}) ^ {\\mathrm{H}} \\\\ \\quad = \\boldsymbol {X X} ^ {\\mathrm{H}} \\boldsymbol {A} ^ {\\mathrm{H}} (\\boldsymbol {A G}) ^ {\\mathrm{H}} = \\boldsymbol {X} (\\boldsymbol {A X}) ^ {\\mathrm{H}} (\\boldsymbol {A G}) ^ {\\mathrm{H}} = \\boldsymbol {X A X A G} \\\\ \\quad = \\boldsymbol {X A G} = \\boldsymbol {X} (\\boldsymbol {A G A}) \\boldsymbol {G} = (\\boldsymbol {X A}) ^ {\\mathrm{H}} (\\boldsymbol {G A}) ^ {\\mathrm{H}} \\boldsymbol {G} \\\\ \\quad = (\\boldsymbol {G A X A}) ^ {\\mathrm{H}} \\boldsymbol {G} = (\\boldsymbol {G A}) ^ {\\mathrm{H}} \\boldsymbol {G} = (\\boldsymbol {G A}) \\boldsymbol {G} \\\\ \\quad = \\boldsymbol {G}. \\end{array}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":22,"id":"f07855e6-2dc7-4c51-9f55-f9350521d7f7","page_size":[595,841],"block_position":"22-2"},{"bbox":[35,193,301,243],"type":"text","angle":0,"index":3,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":22,"id":"46f4606c-51fc-43cd-86ae-a74bfdc74f8d","page_size":[595,841],"block_position":"22-3"},{"bbox":[54,243,164,255],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"解(1) 先进行分解，得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":22,"id":"28bebb54-fa7e-4892-8891-227b3b9d7181","page_size":[595,841],"block_position":"22-4"},{"bbox":[213,253,379,302],"type":"interline_equation","angle":0,"index":5,"text":"$$\nB = \\left( \\begin{array}{c c} 1 & 0 \\\\ 1 & 1 \\\\ 0 & 1 \\end{array} \\right), C = \\left( \\begin{array}{c c c} 1 & 0 & - 1 \\\\ 0 & 2 & 1 \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":22,"id":"7490a830-ba8f-4130-90ff-358c05a69039","page_size":[595,841],"block_position":"22-5"},{"bbox":[55,310,117,322],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"最终结果得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":22,"id":"5c91e0dc-0ede-4c84-b1a2-7bdcdc93c7f0","page_size":[595,841],"block_position":"22-6"},{"bbox":[233,320,358,370],"type":"interline_equation","angle":0,"index":7,"text":"$$\nA ^ {+} = \\frac {1}{9} \\left( \\begin{array}{c c c} 3 & 2 & - 1 \\\\ 0 & 2 & 2 \\\\ - 3 & - 1 & 2 \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":22,"id":"ec7fcc0f-4fa7-47c8-9445-714259dc54dc","page_size":[595,841],"block_position":"22-7"},{"bbox":[73,379,215,391],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"定理 2.12 (最小二乘解的表达式) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":22,"id":"3b44b007-47e6-4bfb-b549-7a0c06ed2340","page_size":[595,841],"block_position":"22-8"},{"bbox":[69,396,453,409],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":22,"id":"6e4bccc1-cc7c-4f6d-aac2-73553297a84e","page_size":[595,841],"block_position":"22-9"},{"bbox":[56,446,143,462],"type":"title","angle":0,"index":10,"level":2,"text":"## 2.2 分块矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":22,"id":"fbe5e874-22f2-4748-b1fa-a6ebdd574d2f","page_size":[595,841],"block_position":"22-10"},{"bbox":[57,479,186,493],"type":"title","angle":0,"index":12,"level":2,"text":"## 2.2.1 常见矩阵拆分技巧","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":22,"id":"2a26c7bd-886c-49ac-a612-2fd38ed967fd","page_size":[595,841],"block_position":"22-11"},{"bbox":[73,507,166,518],"type":"title","angle":0,"index":13,"level":2,"text":"## 定理 2.13 (点态分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":22,"id":"33f82ee3-5eea-4694-8849-b36275f408e9","page_size":[595,841],"block_position":"22-12"},{"bbox":[70,523,161,534],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"$A$ 是 $n \\times n$ 矩阵，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":22,"id":"2c97b54d-b27c-48ec-a924-c384b6a6978c","page_size":[595,841],"block_position":"22-13"},{"bbox":[139,538,454,606],"type":"interline_equation","angle":0,"index":15,"text":"$$\nA = \\left[ \\begin{array}{c c c c} a _ {1} b _ {1} & a _ {1} b _ {2} & \\dots & a _ {1} b _ {n} \\\\ a _ {2} b _ {1} & a _ {2} b _ {2} & \\dots & a _ {2} b _ {n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a _ {n} b _ {1} & a _ {n} b _ {2} & \\dots & a _ {n} b _ {n} \\end{array} \\right] \\Rightarrow A = \\left( \\begin{array}{c} a _ {1} \\\\ a _ {2} \\\\ \\vdots \\\\ a _ {n} \\end{array} \\right) \\left( \\begin{array}{c c c c} b _ {1} & b _ {2} & \\dots & b _ {n} \\end{array} \\right)\\tag{❤️}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":22,"id":"c11d3db5-24fc-4017-a604-3b5b869b34d1","page_size":[595,841],"block_position":"22-14"},{"bbox":[73,627,199,640],"type":"title","angle":0,"index":16,"level":2,"text":"## 定理 2.14 ( $A^{T}A$ 的和式表达)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":22,"id":"81477afc-466d-4794-a33d-983fc382918c","page_size":[595,841],"block_position":"22-15"},{"bbox":[69,645,341,675],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"$A$ 为 $n\\times m$ 矩阵，将 $A$ 做下面的行划分，则 $A^T A = \\sum_{i = 1}^{n}\\alpha_i^T\\alpha_i$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":22,"id":"577e32cc-0e79-4720-9ded-c18607c2dba4","page_size":[595,841],"block_position":"22-16"},{"bbox":[265,678,327,746],"type":"interline_equation","angle":0,"index":18,"text":"$$\nA = \\left( \\begin{array}{c} \\alpha_ {1} \\\\ \\alpha_ {2} \\\\ \\vdots \\\\ \\alpha_ {s} \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":22,"id":"ec79800c-0a0e-4f29-a2cb-56773adc7f4d","page_size":[595,841],"block_position":"22-17"},{"bbox":[292,803,302,812],"type":"page_number","angle":0,"index":1000010,"text":"18 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":22,"is_discarded":true,"id":"e7e0b66f-3a25-4a87-86e4-58c9740c2102","page_size":[595,841],"block_position":"22-18"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99998,"text":"2.2 分块矩阵 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":23,"is_discarded":true,"id":"e1300916-b736-4593-b644-b7e00263778a","page_size":[595,841],"block_position":"23-0"},{"bbox":[525,600,536,611],"type":"text","angle":0,"index":0,"text":"# ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"d5cd99ae-065c-49e9-9919-52da83456ba6","page_size":[595,841],"block_position":"23-1"},{"bbox":[56,72,163,86],"type":"title","angle":0,"index":2,"level":2,"text":"## 2.2.2 分块初等变换","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"f0ecf366-bbce-4622-8f95-d647c39f990e","page_size":[595,841],"block_position":"23-2"},{"bbox":[73,99,201,112],"type":"title","angle":0,"index":3,"level":2,"text":"## 定义 2.7 (分块矩阵初等变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"43e5589f-09f8-4a67-b0fc-a151050dc300","page_size":[595,841],"block_position":"23-3"},{"bbox":[69,116,246,128],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"分块矩阵初等变换分为行变换和列变换： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"b7f28e1d-c0e7-4765-9f4a-eb714793eac5","page_size":[595,841],"block_position":"23-4"},{"bbox":[83,132,454,145],"type":"text","angle":0,"index":5,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"036dd999-6f7b-4122-8498-fb0c89292b30","page_size":[595,841],"block_position":"23-5"},{"bbox":[83,148,453,160],"type":"text","angle":0,"index":6,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"6c132692-8372-428b-bdf0-ef45e17ddf4b","page_size":[595,841],"block_position":"23-6"},{"bbox":[73,180,181,193],"type":"title","angle":0,"index":7,"level":2,"text":"## 定义 2.8 (分块初等矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"c330ba5f-a168-491d-9fda-74ed6defc065","page_size":[595,841],"block_position":"23-7"},{"bbox":[69,198,448,211],"type":"text","angle":0,"index":8,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"710c65dd-5707-4d0c-b3e0-ab291cb9035d","page_size":[595,841],"block_position":"23-8"},{"bbox":[260,213,330,245],"type":"interline_equation","angle":0,"index":9,"text":"$$\nA = \\left[ \\begin{array}{c c} E & 0 \\\\ P & E \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":23,"id":"5bc383a8-fb76-4928-80c2-c9ff1f1242d3","page_size":[595,841],"block_position":"23-9"},{"bbox":[73,267,255,280],"type":"title","angle":0,"index":10,"level":2,"text":"## 定理 2.15 (初等变换的分块初等矩阵表达)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"2e3027c2-a123-46de-84d4-668347ba12ac","page_size":[595,841],"block_position":"23-10"},{"bbox":[69,284,326,296],"type":"text","angle":0,"index":11,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"7a2915b4-4c38-44f2-84cb-feed96578dad","page_size":[595,841],"block_position":"23-11"},{"bbox":[178,298,415,332],"type":"interline_equation","angle":0,"index":12,"text":"$$\n\\left[ \\begin{array}{c c} E & 0 \\\\ P & E \\end{array} \\right] \\left[ \\begin{array}{c c} A _ {1} & A _ {2} \\\\ A _ {3} & A _ {4} \\end{array} \\right] = \\left[ \\begin{array}{c c} A _ {1} & A _ {2} \\\\ P A _ {1} + A _ {3} & P A _ {2} + A _ {4} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":23,"id":"6f831c13-d6fd-4637-aaa9-275969417faf","page_size":[595,841],"block_position":"23-12"},{"bbox":[56,364,186,379],"type":"title","angle":0,"index":13,"level":2,"text":"## 2.2.3 分块打洞技巧总结","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"d53c75ab-7ec6-4244-9c18-c5f58f485fd5","page_size":[595,841],"block_position":"23-13"},{"bbox":[73,391,166,404],"type":"title","angle":0,"index":14,"level":2,"text":"## 定理 2.16 (分块打洞)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"39c93a0a-05ff-484a-94a9-8884a9622465","page_size":[595,841],"block_position":"23-14"},{"bbox":[68,408,309,442],"type":"text","angle":0,"index":15,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"2aa6ff37-c5b4-4cd9-b4e4-6b6455620fd9","page_size":[595,841],"block_position":"23-15"},{"bbox":[195,444,397,477],"type":"interline_equation","angle":0,"index":16,"text":"$$\n\\left[ \\begin{array}{c c} A & B \\\\ O & D - C A ^ {- 1} B \\end{array} \\right] \\quad \\left[ \\begin{array}{c c} A - B D ^ {- 1} C & O \\\\ O & D \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":23,"id":"84d46dd1-d9ac-417b-8f66-f29a527a477f","page_size":[595,841],"block_position":"23-16"},{"bbox":[69,481,346,493],"type":"text","angle":0,"index":17,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"77f52db9-024a-4f08-8c0e-74adae3d5a8c","page_size":[595,841],"block_position":"23-17"},{"bbox":[73,513,180,526],"type":"title","angle":0,"index":18,"level":2,"text":"## 推论 2.1 (两种特殊形式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"329e8a97-4138-4e66-ad55-c6be0e659768","page_size":[595,841],"block_position":"23-18"},{"bbox":[69,530,166,541],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"对称形式和归纳形式： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"c3c6ddc4-e1f7-4967-ad50-7e52cee6d381","page_size":[595,841],"block_position":"23-19"},{"bbox":[83,548,141,563],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"- 对称形式: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"1043b2f6-e55e-42c1-b8b9-d77998292261","page_size":[595,841],"block_position":"23-20"},{"bbox":[146,542,315,572],"type":"interline_equation","angle":0,"index":21,"text":"$$\n\\left[ \\begin{array}{c c} A & B \\\\ B ^ {T} & D \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A & O \\\\ O & D - B ^ {T} A ^ {- 1} B \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":23,"id":"5fa790f0-b9ed-4a0f-913e-f9b7cf6ce01d","page_size":[595,841],"block_position":"23-21"},{"bbox":[83,581,141,595],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"归纳形式： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"02152fa3-ca0c-4a50-a106-25c4911498a1","page_size":[595,841],"block_position":"23-22"},{"bbox":[146,574,323,606],"type":"interline_equation","angle":0,"index":23,"text":"$$\n\\left[ \\begin{array}{c c} A & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A & \\alpha \\\\ 0 & a _ {n n} - \\beta^ {T} A ^ {- 1} \\alpha \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":23,"id":"22dbfe83-04f4-4ca1-97dc-4916a251b586","page_size":[595,841],"block_position":"23-23"},{"bbox":[35,638,539,666],"type":"text","angle":0,"index":24,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"f796750c-38d5-43cd-a04b-40d50126059c","page_size":[595,841],"block_position":"23-24"},{"bbox":[73,669,508,682],"type":"text","angle":0,"index":25,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"0c9f59d5-8dac-4a8f-b433-242c4915fc09","page_size":[595,841],"block_position":"23-25"},{"bbox":[54,684,539,713],"type":"text","angle":0,"index":26,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"42cc8d03-57a0-4b84-8560-cc6ceb30cb50","page_size":[595,841],"block_position":"23-26"},{"bbox":[79,715,511,748],"type":"interline_equation","angle":0,"index":27,"text":"$$\nA = \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] \\Rightarrow \\left[ \\begin{array}{c c} B _ {1} & \\\\ & 1 \\end{array} \\right] \\cdot \\left[ \\begin{array}{c c} E _ {n - 1} & O \\\\ - \\beta^ {T} A _ {1} ^ {- 1} & 1 \\end{array} \\right] \\cdot \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] = \\left[ \\begin{array}{c c} B _ {1} A _ {1} & B _ {1} \\alpha \\\\ O & a _ {n n} - \\beta^ {T} A _ {1} ^ {- 1} \\alpha \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":23,"id":"758dc423-95e4-407e-b102-ccd6a1b0f41f","page_size":[595,841],"block_position":"23-27"},{"bbox":[54,751,324,763],"type":"text","angle":0,"index":29,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":23,"id":"3d68c9ea-8cdc-499c-a9a5-445b4eb1bfb3","page_size":[595,841],"block_position":"23-28"},{"bbox":[292,803,302,811],"type":"page_number","angle":0,"index":1000027,"text":"19 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":23,"is_discarded":true,"id":"6071a6ff-39cb-4328-b7e5-dc8bd4bcb5cc","page_size":[595,841],"block_position":"23-29"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99995,"text":"2.2 分块矩阵 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":24,"is_discarded":true,"id":"d10b88be-1dc4-4ba0-9092-b9321332a9bc","page_size":[595,841],"block_position":"24-0"},{"bbox":[526,538,536,548],"type":"text","angle":0,"index":0,"text":"# ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"8a677d6c-4949-4a0b-9697-685b15b6d2fc","page_size":[595,841],"block_position":"24-1"},{"bbox":[526,679,536,690],"type":"text","angle":0,"index":1,"text":"# ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"7e7653db-7338-4c8b-b735-6632acdf4fe5","page_size":[595,841],"block_position":"24-2"},{"bbox":[525,379,536,390],"type":"text","angle":0,"index":2,"text":"❤️ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"937ab93b-1f70-4748-8154-3e5959c50e68","page_size":[595,841],"block_position":"24-3"},{"bbox":[73,72,399,84],"type":"text","angle":0,"index":5,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"dc0781ca-dc35-432a-8502-5b46ab070c4b","page_size":[595,841],"block_position":"24-4"},{"bbox":[73,87,520,121],"type":"interline_equation","angle":0,"index":6,"text":"$$\nA = \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] \\Rightarrow \\left[ \\begin{array}{c c} E _ {n - 1} & O \\\\ - \\beta^ {T} A _ {1} ^ {- 1} & 1 \\end{array} \\right] \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\beta^ {T} & a _ {n n} \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {n - 1} & - A _ {1} ^ {- 1} \\alpha \\\\ O & 1 \\end{array} \\right] = \\left[ \\begin{array}{c c} A _ {1} & O \\\\ O & a _ {n n} - \\beta^ {T} A _ {1} ^ {- 1} \\alpha \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":24,"id":"856926af-011b-4701-bb32-1332cc7c9d95","page_size":[595,841],"block_position":"24-5"},{"bbox":[54,123,539,152],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"c76ca258-08c7-486b-8f1c-5dd16d88e0bf","page_size":[595,841],"block_position":"24-6"},{"bbox":[56,170,174,185],"type":"title","angle":0,"index":8,"level":2,"text":"## 2.2.4 分块矩阵行列式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"89cb9847-8d5e-41d4-9110-0ad66bec43b6","page_size":[595,841],"block_position":"24-7"},{"bbox":[73,198,226,211],"type":"title","angle":0,"index":9,"level":2,"text":"## 定理 2.17 (分块对角矩阵的行列式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"0502ea0e-0a5e-4c55-ae09-8a3da0e84539","page_size":[595,841],"block_position":"24-8"},{"bbox":[69,215,232,227],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"根据 Laplace 多行展开定理可以得到: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"2b9c3727-b2a8-49ba-9f8f-87c0758dc919","page_size":[595,841],"block_position":"24-9"},{"bbox":[145,248,233,280],"type":"interline_equation","angle":0,"index":11,"text":"$$\n\\left| \\begin{array}{c c} A & 0 \\\\ B & C \\end{array} \\right| = | A | | B |,\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":24,"id":"294402d6-529e-47c1-addc-92b895256169","page_size":[595,841],"block_position":"24-10"},{"bbox":[248,231,449,298],"type":"interline_equation","angle":0,"index":12,"text":"$$\n\\left| \\begin{array}{c c c c} A _ {1 1} & 0 & \\dots & 0 \\\\ A _ {2 1} & A _ {2 2} & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ A _ {n 1} & A _ {n 2} & \\dots & A _ {n n} \\end{array} \\right| = | A _ {1 1} | | A _ {2 2} | \\dots | A _ {n n} |\\tag{❤️}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":24,"id":"53b69cde-bea7-428b-b367-38e192714309","page_size":[595,841],"block_position":"24-11"},{"bbox":[73,320,226,332],"type":"title","angle":0,"index":13,"level":2,"text":"## 定理 2.18 (一般分块矩阵的行列式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"75cfa350-75ea-466c-89d3-1d1040a9b4a3","page_size":[595,841],"block_position":"24-12"},{"bbox":[69,337,258,349],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"设 $A, D$ 分别为 $r, s$ 阶矩阵， $A, D$ 可逆，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"10c70c5e-988a-4d54-aa65-89a5b05fa5cd","page_size":[595,841],"block_position":"24-13"},{"bbox":[189,352,406,384],"type":"interline_equation","angle":0,"index":15,"text":"$$\n\\left| \\begin{array}{c c} A & B \\\\ C & D \\end{array} \\right| = | D | | A - B D ^ {- 1} C | = | A | | D - C A ^ {- 1} B |\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":24,"id":"da485f7f-5d84-4762-ac40-f1ac5b4e29f1","page_size":[595,841],"block_position":"24-14"},{"bbox":[54,401,340,414],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"证明 (1) 如果 $D$ 可逆，则把 $B$ 消掉 (左乘 $BD^{-1}$ )，取行列式即可 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"6133e01b-d0f7-4e51-adb8-fb7912a30f9c","page_size":[595,841],"block_position":"24-15"},{"bbox":[217,417,376,449],"type":"interline_equation","angle":0,"index":17,"text":"$$\n\\left[ \\begin{array}{c c} A & B \\\\ C & D \\end{array} \\right] \\Rightarrow \\left[ \\begin{array}{c c} A - B D ^ {- 1} C & 0 \\\\ C & D \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":24,"id":"13124e42-5ce7-483c-a92c-ba90faec506f","page_size":[595,841],"block_position":"24-16"},{"bbox":[73,463,273,476],"type":"title","angle":0,"index":18,"level":2,"text":"## 定理 2.19 (行列式计算技巧 1: 配凑分块矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"550f40c1-ae6a-409a-a2ca-9c4c14e9b718","page_size":[595,841],"block_position":"24-17"},{"bbox":[69,480,363,492],"type":"text","angle":0,"index":19,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"fa78f6f1-b4ef-4445-83cf-323922185f06","page_size":[595,841],"block_position":"24-18"},{"bbox":[133,497,461,528],"type":"interline_equation","angle":0,"index":20,"text":"$$\n\\left| \\begin{array}{c c} J & u \\\\ u ^ {T} & 1 \\end{array} \\right| = | 1 | \\cdot | J - u ^ {T} u | = | J | \\cdot | 1 - u J ^ {- 1} u ^ {T} | \\Rightarrow | A | = | J | \\cdot | 1 - u J ^ {- 1} u ^ {T} |\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":24,"id":"6f93064a-8f16-41fa-9fb5-80dcf3a0902e","page_size":[595,841],"block_position":"24-19"},{"bbox":[69,532,162,544],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"注意这里 $u$ 是列向量 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"cb90a7fb-19dd-4923-8da3-5fb92d7ece46","page_size":[595,841],"block_position":"24-20"},{"bbox":[57,576,163,589],"type":"title","angle":0,"index":22,"level":2,"text":"## 2.2.5 分块矩阵求逆","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"d5fc017a-0b6a-49ea-80c2-54c81d9a8a21","page_size":[595,841],"block_position":"24-21"},{"bbox":[74,599,321,612],"type":"text","angle":0,"index":23,"merge_prev":false,"text":"一般分块矩阵求逆都先转换为分块对角矩阵，具体如下。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"3f1b3278-6d13-4869-a454-73a0d98d9e70","page_size":[595,841],"block_position":"24-22"},{"bbox":[73,619,215,633],"type":"title","angle":0,"index":24,"level":2,"text":"## 定理 2.20 (上三角分块矩阵求逆)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"699f181e-1231-4cc8-aa10-e1fd3947c038","page_size":[595,841],"block_position":"24-23"},{"bbox":[69,637,414,649],"type":"text","angle":0,"index":25,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"237d5ce8-d9aa-4cb7-b06e-70f53dc63a53","page_size":[595,841],"block_position":"24-24"},{"bbox":[180,652,411,685],"type":"interline_equation","angle":0,"index":26,"text":"$$\nA = \\left[ \\begin{array}{c c} A _ {1} & A _ {3} \\\\ O & A _ {2} \\end{array} \\right] \\Rightarrow A ^ {- 1} = \\left[ \\begin{array}{c c} A _ {1} ^ {- 1} & - A _ {1} ^ {- 1} A _ {3} A _ {2} ^ {- 1} \\\\ 0 & A _ {2} ^ {- 1} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":24,"id":"20853df1-90bc-4293-ac13-c59cbabcb279","page_size":[595,841],"block_position":"24-25"},{"bbox":[54,702,202,714],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"证明 根据分块矩阵初等变换可知 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"b173ce9a-fe72-4b66-b1d8-9e7c8df5f405","page_size":[595,841],"block_position":"24-26"},{"bbox":[68,717,524,752],"type":"interline_equation","angle":0,"index":28,"text":"$$\n\\left[ \\begin{array}{c c} E & - A _ {2} ^ {- 1} A _ {3} \\\\ O & E \\end{array} \\right] \\left[ \\begin{array}{c c} A _ {1} & A _ {3} \\\\ O & A _ {2} \\end{array} \\right] = \\left[ \\begin{array}{c c} A _ {1} & O \\\\ O & A _ {2} \\end{array} \\right] \\Rightarrow \\left[ \\begin{array}{c c} E & - A _ {2} ^ {- 1} A _ {3} \\\\ O & E \\end{array} \\right] ^ {- 1} \\left[ \\begin{array}{c c} A _ {1} & A _ {3} \\\\ O & A _ {2} \\end{array} \\right] ^ {- 1} = \\left[ \\begin{array}{c c} A _ {1} ^ {- 1} & O \\\\ O & A _ {2} ^ {- 1} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":24,"id":"26163ddd-f4bf-4b40-9dbb-2bfe13e04df5","page_size":[595,841],"block_position":"24-27"},{"bbox":[55,754,142,766],"type":"text","angle":0,"index":29,"merge_prev":false,"text":"因此可以得到结论。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":24,"id":"c3223154-5d96-43e7-8d4d-df2cbc79f082","page_size":[595,841],"block_position":"24-28"},{"bbox":[291,803,303,811],"type":"page_number","angle":0,"index":1000002,"text":"20 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":24,"is_discarded":true,"id":"a1a69b8e-f96e-44ff-8b08-6d68747e638d","page_size":[595,841],"block_position":"24-29"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99999,"text":"2.3 矩阵的秩 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":25,"is_discarded":true,"id":"75622a8f-6068-4ee0-af65-3e713f6ae2b4","page_size":[595,841],"block_position":"25-0"},{"bbox":[73,73,226,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 定理 2.21 (一般分块矩阵可逆条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"795de122-a4b5-49a1-bebc-c01dd4ed69b5","page_size":[595,841],"block_position":"25-1"},{"bbox":[68,90,411,123],"type":"text","angle":0,"index":2,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"eda56b04-daaa-440e-b876-9bcf491631c4","page_size":[595,841],"block_position":"25-2"},{"bbox":[35,149,379,182],"type":"text","angle":0,"index":3,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"78e77b1a-0204-4b1c-aee2-0d3e53efa0fb","page_size":[595,841],"block_position":"25-3"},{"bbox":[52,180,410,216],"type":"text","angle":0,"index":4,"merge_prev":false,"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: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"acb037d3-e2f2-479a-a3be-32ed3195ded0","page_size":[595,841],"block_position":"25-4"},{"bbox":[160,218,431,253],"type":"interline_equation","angle":0,"index":5,"text":"$$\n\\left[ \\begin{array}{c c} O & A \\\\ B & O \\end{array} \\right] ^ {- 1} = \\left[ \\begin{array}{c c} A ^ {- 1} & O \\\\ O & B ^ {- 1} \\end{array} \\right] \\left[ \\begin{array}{c c} O & E \\\\ E & O \\end{array} \\right] = \\left[ \\begin{array}{c c} O & B ^ {- 1} \\\\ A ^ {- 1} & O \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":25,"id":"e9b6ef9a-d49f-4a1d-bf6a-a5dca7c7a5c8","page_size":[595,841],"block_position":"25-5"},{"bbox":[56,272,143,288],"type":"title","angle":0,"index":6,"level":2,"text":"## 2.3 矩阵的秩","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"8bbf8472-21a9-468f-b4a5-3900cd3d728d","page_size":[595,841],"block_position":"25-6"},{"bbox":[56,305,138,319],"type":"title","angle":0,"index":7,"level":2,"text":"## 2.3.1 秩的概念","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"30ddc889-d291-4194-9e97-86d2b850ebc3","page_size":[595,841],"block_position":"25-7"},{"bbox":[73,332,161,344],"type":"title","angle":0,"index":8,"level":2,"text":"## 定义 2.9 (矩阵的秩)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"ac9d8b19-4abb-458f-82f6-3c3c337a59d9","page_size":[595,841],"block_position":"25-8"},{"bbox":[69,349,136,361],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"矩阵的秩等于: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"c3367f4a-0ffb-446d-8999-4812e3b70acf","page_size":[595,841],"block_position":"25-9"},{"bbox":[83,364,177,376],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"- 矩阵行列向量的秩 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"18e0a81a-5378-4608-b324-5e99dc6f8121","page_size":[595,841],"block_position":"25-10"},{"bbox":[83,380,186,392],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"- 最大非零子式的阶数 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"9fa534a9-8980-46ba-b717-28b412005de1","page_size":[595,841],"block_position":"25-11"},{"bbox":[56,424,235,439],"type":"title","angle":0,"index":12,"level":2,"text":"## 2.3.2 初等变换标准型与矩阵分解","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"18376353-2d8e-4eae-9539-9b15597c2df4","page_size":[595,841],"block_position":"25-12"},{"bbox":[73,451,195,464],"type":"title","angle":0,"index":13,"level":2,"text":"## 定理 2.22 (初等变换标准型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"5691f539-9a7a-4c60-9db3-914e3b53ae21","page_size":[595,841],"block_position":"25-13"},{"bbox":[69,469,407,481],"type":"text","angle":0,"index":14,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"150ec965-f4cc-4a7e-a83c-fff8cde53fce","page_size":[595,841],"block_position":"25-14"},{"bbox":[249,485,344,516],"type":"interline_equation","angle":0,"index":15,"text":"$$\nA = P \\left[ \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right] Q\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":25,"id":"2303b3d5-d912-4a7a-8a42-bcc902b9a229","page_size":[595,841],"block_position":"25-15"},{"bbox":[53,533,458,546],"type":"text","angle":0,"index":16,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"d3851cde-53f3-41af-9bde-dafc081f07c5","page_size":[595,841],"block_position":"25-16"},{"bbox":[73,555,206,568],"type":"title","angle":0,"index":17,"level":2,"text":"## 定理 2.23 (行列满秩的标准型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"b8a90f73-6837-4a42-9b9f-844b498d7035","page_size":[595,841],"block_position":"25-17"},{"bbox":[67,571,527,629],"type":"text","angle":0,"index":18,"merge_prev":false,"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 $A = \\left[ \\begin{array}{ll}E_m & O \\end{array} \\right]Q$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"2fd49b11-4342-4b13-8bff-17c48281a4e7","page_size":[595,841],"block_position":"25-18"},{"bbox":[53,640,539,670],"type":"text","angle":0,"index":19,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":25,"id":"a4b61377-ab9c-47ad-bca6-04c4d624e838","page_size":[595,841],"block_position":"25-19"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":1000019,"text":"21 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":25,"is_discarded":true,"id":"ea01553d-40d2-4d62-9843-011e354e2a49","page_size":[595,841],"block_position":"25-20"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99999,"text":"2.3 矩阵的秩 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":26,"is_discarded":true,"id":"0156f4c9-401b-4756-95ac-e6d9ff7a0080","page_size":[595,841],"block_position":"26-0"},{"bbox":[55,72,270,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 2.3.3 利用初等变换标准型进行矩阵分解","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"946455cb-55df-40ae-bf91-32e8a7de5fa5","page_size":[595,841],"block_position":"26-1"},{"bbox":[73,100,166,112],"type":"title","angle":0,"index":2,"level":2,"text":"## 定理 2.24 (矩阵分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"6da175b3-5cf6-412e-bd6d-68080c2e476a","page_size":[595,841],"block_position":"26-2"},{"bbox":[68,118,213,148],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"$s\\times n$ 矩阵 $\\left[ \\begin{array}{cc}E_r & O\\\\ O & O \\end{array} \\right]$ 可分解为 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"31c65a11-6265-421b-96ae-a254fd114a35","page_size":[595,841],"block_position":"26-3"},{"bbox":[221,152,372,186],"type":"interline_equation","angle":0,"index":4,"text":"$$\n\\left[ \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right] = \\left[ \\begin{array}{c} E _ {r} \\\\ O \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {r} & O \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":26,"id":"f6817ddd-e751-413b-bb5d-8345da077e62","page_size":[595,841],"block_position":"26-4"},{"bbox":[69,189,186,200],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"若为方阵时，可以分解为： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"3682f4e9-02a9-4218-9e62-1f4959f2c65f","page_size":[595,841],"block_position":"26-5"},{"bbox":[212,198,381,233],"type":"interline_equation","angle":0,"index":6,"text":"$$\n{\\left[ \\begin{array}{l l} E _ {r} & O \\\\ O & O \\end{array} \\right]} = {\\left[ \\begin{array}{l l} E _ {r} & O \\\\ O & O \\end{array} \\right]} {\\left[ \\begin{array}{l l} E _ {r} & O \\\\ O & O \\end{array} \\right]}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":26,"id":"3fee2109-2fd9-4a01-afb0-14c40c77d942","page_size":[595,841],"block_position":"26-6"},{"bbox":[35,249,443,262],"type":"text","angle":0,"index":7,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"60a1899c-0647-445e-a9aa-c430799213fe","page_size":[595,841],"block_position":"26-7"},{"bbox":[54,265,539,293],"type":"text","angle":0,"index":8,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"ea8e6769-b820-45ae-89ea-c243523e4fbf","page_size":[595,841],"block_position":"26-8"},{"bbox":[73,296,423,309],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"9298cd75-2426-4f23-8d19-449287ac385d","page_size":[595,841],"block_position":"26-9"},{"bbox":[54,312,171,324],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"证明 (1) 使用等价标准型: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"cdb6f65e-0bf4-4e69-8759-1bf1dd160261","page_size":[595,841],"block_position":"26-10"},{"bbox":[190,322,402,356],"type":"interline_equation","angle":0,"index":11,"text":"$$\nA = P \\left[ \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right] Q = P \\left[ \\begin{array}{c} E _ {r} \\\\ O \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {r} & O \\end{array} \\right] Q\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":26,"id":"8ab6bcaa-7d18-4684-993f-4692c971667b","page_size":[595,841],"block_position":"26-11"},{"bbox":[73,362,514,396],"type":"text","angle":0,"index":12,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"b112e80c-1151-4a1c-b7ff-9ce59d170ce5","page_size":[595,841],"block_position":"26-12"},{"bbox":[73,394,138,407],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"(3) 和 (2) 一样 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"3fbecd38-7475-40da-b84d-a85fa8f96c09","page_size":[595,841],"block_position":"26-13"},{"bbox":[35,425,403,438],"type":"text","angle":0,"index":14,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"2ec1060e-007e-413e-a40a-76c581f4d0cc","page_size":[595,841],"block_position":"26-14"},{"bbox":[54,441,190,453],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"证明 (1) 利用等价标准型得到: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"2413be55-992b-4bb9-abaf-b36c186c5eae","page_size":[595,841],"block_position":"26-15"},{"bbox":[189,456,404,488],"type":"interline_equation","angle":0,"index":16,"text":"$$\nA = P \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) Q = P Q Q ^ {- 1} \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) Q\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":26,"id":"a6ac2d9c-c066-4385-b09b-b854c1cfd623","page_size":[595,841],"block_position":"26-16"},{"bbox":[54,491,185,504],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"PQ 为可逆矩阵，后面为幂等 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"31f9d2a9-a8d1-4602-9570-c352c6db5441","page_size":[595,841],"block_position":"26-17"},{"bbox":[56,523,139,537],"type":"title","angle":0,"index":18,"level":2,"text":"## 2.3.4 秩不等式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"d68d7ebe-9e81-47e5-a5fd-8b5d7773638c","page_size":[595,841],"block_position":"26-18"},{"bbox":[73,550,211,562],"type":"title","angle":0,"index":19,"level":2,"text":"## 引理 2.1 (乘可逆矩阵不改变秩)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"bd8bddd4-43e5-4cf4-b733-f3fdb1cd4044","page_size":[595,841],"block_position":"26-19"},{"bbox":[69,566,234,580],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"$P, Q$ 为可逆矩阵，则 $r(A) = r(PAQ)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"964836b8-7564-4466-8825-31f5c3982b95","page_size":[595,841],"block_position":"26-20"},{"bbox":[54,596,272,608],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"证明 相当于做了一系列初等变换，因此不改变秩 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"8d4113e0-75ad-4af7-a474-749512120713","page_size":[595,841],"block_position":"26-21"},{"bbox":[35,628,314,640],"type":"text","angle":0,"index":22,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"6f33c384-e9ff-4fd5-8b87-dc5c5e64e406","page_size":[595,841],"block_position":"26-22"},{"bbox":[73,644,227,656],"type":"text","angle":0,"index":23,"merge_prev":false,"text":"(1) 若 $r(A) = n$ ，则 $r(AB) = r(B)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"a44558cc-a213-4e23-83dc-926672317dd2","page_size":[595,841],"block_position":"26-23"},{"bbox":[73,659,227,671],"type":"text","angle":0,"index":25,"merge_prev":false,"text":"(2) 若 $r(B) = n$ ，则 $r(AB) = r(A)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"2447c929-4587-4ee9-8849-4ad755487878","page_size":[595,841],"block_position":"26-24"},{"bbox":[52,671,380,703],"type":"text","angle":0,"index":26,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"0eb175a5-2f2e-4058-9c02-6c009c2348e5","page_size":[595,841],"block_position":"26-25"},{"bbox":[172,707,420,740],"type":"interline_equation","angle":0,"index":27,"text":"$$\nr (A B) = r (\\left[ \\begin{array}{c} A _ {1} \\\\ O \\end{array} \\right] B) = r \\binom{A _ {1} B}{O} = r (A _ {1} B) = r (B)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":26,"id":"9258a457-1286-402d-8b16-861fbbe2d818","page_size":[595,841],"block_position":"26-26"},{"bbox":[73,742,161,755],"type":"text","angle":0,"index":28,"merge_prev":false,"text":"(2) 同理 (转置即可) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":26,"id":"67521f3e-e1f9-4d0c-b7da-8242a438d226","page_size":[595,841],"block_position":"26-27"},{"bbox":[291,802,303,811],"type":"page_number","angle":0,"index":1000023,"text":"22 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":26,"is_discarded":true,"id":"b81eaa01-5b9e-4cdb-8749-5a82d633dfd7","page_size":[595,841],"block_position":"26-28"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99999,"text":"2.3 矩阵的秩 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":27,"is_discarded":true,"id":"1c4fe024-75ee-4174-b361-0085d48f94be","page_size":[595,841],"block_position":"27-0"},{"bbox":[74,73,181,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 引理 2.2 (分块矩阵的秩)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"b0de3ced-a9d9-4526-8245-59325d926c0b","page_size":[595,841],"block_position":"27-1"},{"bbox":[83,90,222,155],"type":"interline_equation","angle":0,"index":2,"text":"$$\n\\begin{array}{l} \\bullet r \\left( \\begin{array}{c c} A & O \\\\ O & B \\end{array} \\right) = r (A) + r (B) \\\\ \\bullet r \\left( \\begin{array}{c c} A & O \\\\ C & B \\end{array} \\right) \\geq r (A) + r (B) \\end{array}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":27,"id":"bb1434b0-ce74-422c-a34d-edce0412d6af","page_size":[595,841],"block_position":"27-2"},{"bbox":[54,166,462,198],"type":"text","angle":0,"index":3,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"ceb2f255-9ab9-4169-9f63-02755a193e15","page_size":[595,841],"block_position":"27-3"},{"bbox":[150,204,442,264],"type":"interline_equation","angle":0,"index":4,"text":"$$\n\\left( \\begin{array}{c c} P _ {1} & O \\\\ O & P _ {2} \\end{array} \\right) \\left( \\begin{array}{c c} A & O \\\\ O & B \\end{array} \\right) \\left( \\begin{array}{c c} Q _ {1} & O \\\\ O & Q _ {2} \\end{array} \\right) = \\left( \\begin{array}{c c c c} E _ {r _ {1}} & O & O & O \\\\ O & O & O & O \\\\ O & O & E _ {r _ {2}} & O \\\\ O & O & O & O \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":27,"id":"bc8331dd-47ce-4e65-9baa-ed5a82f1acc1","page_size":[595,841],"block_position":"27-4"},{"bbox":[73,269,456,301],"type":"text","angle":0,"index":5,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"fd749a3b-5d47-44b8-9d83-4b576a055280","page_size":[595,841],"block_position":"27-5"},{"bbox":[83,306,509,369],"type":"interline_equation","angle":0,"index":6,"text":"$$\n\\left( \\begin{array}{c c} P _ {1} & O \\\\ O & P _ {2} \\end{array} \\right) \\left( \\begin{array}{c c} A & O \\\\ C & B \\end{array} \\right) \\left( \\begin{array}{c c} Q _ {1} & O \\\\ O & Q _ {2} \\end{array} \\right) = \\left( \\begin{array}{c c c c} E _ {r _ {1}} & O & O & O \\\\ O & O & O & O \\\\ C _ {1 1} & C _ {1 2} & E _ {r _ {2}} & O \\\\ C _ {2 1} & C _ {2 2} & O & O \\end{array} \\right) \\to \\left( \\begin{array}{c c c c} E _ {r _ {1}} & O & O & O \\\\ O & O & O & O \\\\ O & O & E _ {r _ {2}} & O \\\\ O & C _ {2 2} & O & O \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":27,"id":"07e7852e-45e1-4223-80dc-f00f1c2913bf","page_size":[595,841],"block_position":"27-6"},{"bbox":[74,382,206,395],"type":"title","angle":0,"index":7,"level":2,"text":"## 定理 2.25 (矩阵秩基本不等式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"900732a8-150f-429f-ae71-051e5d17030a","page_size":[595,841],"block_position":"27-7"},{"bbox":[83,399,210,412],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"• $r(AB) \\leq \\min\\{r(A), r(B)\\}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"8eee408d-9368-4dcb-9846-2ea7ea318764","page_size":[595,841],"block_position":"27-8"},{"bbox":[83,414,203,427],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"• $r(A+B) \\leq r(A) + r(B)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"6778c4eb-e679-4c83-84c8-9b82e6875a1a","page_size":[595,841],"block_position":"27-9"},{"bbox":[83,429,384,443],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"- Sylvester 不等式: $A, B$ 是 $n$ 阶方阵, 则 $r(A) + r(B) \\leq r(AB) + n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"ef0d2fc5-2739-4fbe-aa86-901001aa32bb","page_size":[595,841],"block_position":"27-10"},{"bbox":[83,445,437,458],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"- Frobenius 不等式: $A, B, C$ 是 $n$ 阶方阵, 则 $r(ABC) \\geq r(AB) + r(BC) - r(B)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"b20c2dc4-e0ec-4ad8-87dd-5f910692b251","page_size":[595,841],"block_position":"27-11"},{"bbox":[83,461,360,473],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"- $A, B$ 为 $s \\times n, n \\times m$ 阶矩阵， $AB = O$ ，则 $r(A) + r(B) \\neq n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"5c258a5a-8616-4537-9098-f420e7e57274","page_size":[595,841],"block_position":"27-12"},{"bbox":[83,476,278,488],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"- $A$ 是 $m \\times n$ 的实矩阵，则 $r(A^T A) = r(A)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"60161fba-5aac-46c5-8136-4cac9c1c102f","page_size":[595,841],"block_position":"27-13"},{"bbox":[54,501,279,567],"type":"text","angle":0,"index":14,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"208cc699-5d13-4f63-9201-72ee00eaa91d","page_size":[595,841],"block_position":"27-14"},{"bbox":[151,571,439,622],"type":"interline_equation","angle":0,"index":15,"text":"$$\nA B = \\left( \\begin{array}{c c c} a _ {1 1} & \\dots & a _ {1 n} \\\\ \\vdots & & \\vdots \\\\ a _ {n 1} & \\dots & a _ {n n} \\end{array} \\right) \\left( \\begin{array}{c} \\beta_ {1} \\\\ \\vdots \\\\ \\beta_ {n} \\end{array} \\right) = \\left( \\begin{array}{c} a _ {1 1} \\beta_ {1} + \\dots + a _ {1 n} \\beta_ {n} \\\\ \\vdots \\\\ a _ {n 1} \\beta_ {1} + \\dots + a _ {n n} \\beta_ {n} \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":27,"id":"c41523f5-0ce1-44e5-ae48-28feca4a1408","page_size":[595,841],"block_position":"27-15"},{"bbox":[54,626,468,640],"type":"text","angle":0,"index":16,"merge_prev":false,"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 $r(AB) \\leq r(A)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"298841ce-1fe9-4158-990c-cab8fac8d5af","page_size":[595,841],"block_position":"27-16"},{"bbox":[73,642,460,655],"type":"text","angle":0,"index":17,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"73d00811-1f40-4699-81e6-0f08263eb0f1","page_size":[595,841],"block_position":"27-17"},{"bbox":[73,657,148,669],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"(3) 是 (4) 的推论 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"f134395e-d6df-4ea1-aee1-f76c3a514343","page_size":[595,841],"block_position":"27-18"},{"bbox":[73,673,322,686],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"(4)Frobenius: 即构造 $r(ABC) + r(B) \\geq r(AB) + r(BC)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"fc20ff2d-bc50-41bb-b836-c2517eb825dc","page_size":[595,841],"block_position":"27-19"},{"bbox":[182,689,410,722],"type":"interline_equation","angle":0,"index":20,"text":"$$\n\\left[ \\begin{array}{c c} A B C & \\\\ & B \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A B C & O \\\\ B C & B \\end{array} \\right] \\to \\left[ \\begin{array}{c c} O & - A B \\\\ B C & B \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":27,"id":"e44b3255-206f-47bf-9fe3-a6b682b1369e","page_size":[595,841],"block_position":"27-20"},{"bbox":[54,724,177,736],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"根据分块矩阵秩不等式可知 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"95b29875-9c87-44eb-8d9d-2e53d8881b23","page_size":[595,841],"block_position":"27-21"},{"bbox":[35,756,443,768],"type":"text","angle":0,"index":23,"merge_prev":false,"text":"练习 2.9 秩不等式的推广 (1) A, B 为 n 阶方阵，则 $r(A - ABA) = r(A) + r(E - BA) - n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":27,"id":"bdc05494-fde3-4225-8047-e8289438e7b6","page_size":[595,841],"block_position":"27-22"},{"bbox":[291,802,302,812],"type":"page_number","angle":0,"index":1000021,"text":"23 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":27,"is_discarded":true,"id":"b04eab5e-8d06-423d-9dd0-d4a526da398a","page_size":[595,841],"block_position":"27-23"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99998,"text":"2.3 矩阵的秩 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":28,"is_discarded":true,"id":"b1b8108e-1392-48dd-90e0-d7e9a97c9f66","page_size":[595,841],"block_position":"28-0"},{"bbox":[52,71,446,100],"type":"text","angle":0,"index":2,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"a17cad97-d994-465b-a609-b174ea947bed","page_size":[595,841],"block_position":"28-1"},{"bbox":[115,101,477,135],"type":"interline_equation","angle":0,"index":3,"text":"$$\n\\left[ \\begin{array}{c c} A & \\\\ & E _ {n} - B A \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A & O \\\\ B A & E _ {n} - B A \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A & A \\\\ B A & E _ {n} \\end{array} \\right] \\to \\left[ \\begin{array}{c c} A - A B A & O \\\\ O & E _ {n} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":28,"id":"087c463c-09ae-490b-b309-b132e56cba4e","page_size":[595,841],"block_position":"28-2"},{"bbox":[54,137,424,150],"type":"text","angle":0,"index":4,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"e732b7af-a2a0-45ca-b720-4c20284eb079","page_size":[595,841],"block_position":"28-3"},{"bbox":[73,153,343,165],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"(2) 注意从复杂的一侧 $A + B$ 开始，否则 $A, B$ 不知道怎么搞， ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"c505db6c-c134-41ec-ac94-5bb694f7c459","page_size":[595,841],"block_position":"28-4"},{"bbox":[123,169,470,200],"type":"interline_equation","angle":0,"index":6,"text":"$$\n\\left( \\begin{array}{c c} A + B & O \\\\ O & O \\end{array} \\right) \\to \\left( \\begin{array}{c c} A + B & O \\\\ A ^ {2} + A B & O \\end{array} \\right) = \\left( \\begin{array}{c c} A + B & O \\\\ A ^ {2} & O \\end{array} \\right) \\to \\left( \\begin{array}{c c} A + B & A ^ {2} \\\\ A ^ {2} & A ^ {3} \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":28,"id":"59aead17-f048-4c37-beda-9fd1667b3113","page_size":[595,841],"block_position":"28-5"},{"bbox":[54,204,521,217],"type":"text","angle":0,"index":7,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"4d797e10-6f6c-4386-90bb-254e2ceff4a5","page_size":[595,841],"block_position":"28-6"},{"bbox":[222,221,371,253],"type":"interline_equation","angle":0,"index":8,"text":"$$\n\\left( \\begin{array}{c c} A + B & A ^ {2} \\\\ A ^ {2} & A ^ {3} \\end{array} \\right) \\to \\left( \\begin{array}{c c} B & A ^ {2} \\\\ O & A ^ {3} \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":28,"id":"d9185738-859e-49ee-9b0c-f83f43430234","page_size":[595,841],"block_position":"28-7"},{"bbox":[54,256,458,269],"type":"text","angle":0,"index":9,"merge_prev":false,"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. ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"a96587ce-8225-43f7-bcc8-e800a4cfda2b","page_size":[595,841],"block_position":"28-8"},{"bbox":[35,287,531,300],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"1d178e8e-4a44-43d5-95d5-8ed1ecb5ed2f","page_size":[595,841],"block_position":"28-9"},{"bbox":[52,303,421,332],"type":"text","angle":0,"index":11,"merge_prev":false,"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 $A = O$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"d828b926-c735-4675-aa6c-b1517e060ce2","page_size":[595,841],"block_position":"28-10"},{"bbox":[72,334,411,347],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"eb936c0f-3557-4af0-82f0-ab59a45334b7","page_size":[595,841],"block_position":"28-11"},{"bbox":[180,354,412,367],"type":"interline_equation","angle":0,"index":13,"text":"$$\nr (A B - B A) \\leq r (A B) + r (B A) \\leq r (A) + r (A) \\leq 2 k\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":28,"id":"f417ddc9-ef56-479f-a038-c9e838872a63","page_size":[595,841],"block_position":"28-12"},{"bbox":[54,373,107,385],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"因此不可逆 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"0f1f195a-e4b4-40e0-8eb1-6edec077884e","page_size":[595,841],"block_position":"28-13"},{"bbox":[55,419,211,433],"type":"title","angle":0,"index":15,"level":2,"text":"## 2.3.5 空间分解法研究矩阵秩","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"7e69279c-9fc1-486c-9a2b-3d3ad6267863","page_size":[595,841],"block_position":"28-14"},{"bbox":[71,447,217,459],"type":"title","angle":0,"index":16,"level":2,"text":"## 定理 2.26 (多项式矩阵的秩关系)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"9aa7be2b-27a7-4084-8b6e-2107dc2305c5","page_size":[595,841],"block_position":"28-15"},{"bbox":[67,464,524,491],"type":"text","angle":0,"index":17,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"5f2a8119-be3b-423c-bf68-88823685d323","page_size":[595,841],"block_position":"28-16"},{"bbox":[83,495,267,507],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"- $A^2 = A$ 当且仅当 $r(A - E) + r(A) = n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"572df1dc-2a84-48b4-b52e-ac07ef791018","page_size":[595,841],"block_position":"28-17"},{"bbox":[83,510,287,523],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"- $A^2 = E$ 当且仅当 $r(A - E) + r(A + E) = n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"35d567b8-2dfa-4efc-b493-3498d68229fe","page_size":[595,841],"block_position":"28-18"},{"bbox":[52,540,538,568],"type":"text","angle":0,"index":20,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"8459a945-63dc-43aa-b293-8ef47a99a26c","page_size":[595,841],"block_position":"28-19"},{"bbox":[121,570,471,603],"type":"interline_equation","angle":0,"index":21,"text":"$$\n\\left[ \\begin{array}{c c} f _ {1} (A) & \\\\ & f _ {2} (A) \\end{array} \\right] \\to \\left[ \\begin{array}{c c} f _ {1} (A) & f _ {1} (A) u (A) + v (A) f _ {2} (A) \\\\ O & f _ {2} (A) \\end{array} \\right] = \\left[ \\begin{array}{c c} f _ {1} (A) & E _ {n} \\\\ O & f _ {2} (A) \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":28,"id":"89b67253-5432-48e5-9319-e9a3966b6f4c","page_size":[595,841],"block_position":"28-20"},{"bbox":[54,606,121,618],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"再做初等变换： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"338c1d50-6b0b-4e03-8bd8-8e66198fafa9","page_size":[595,841],"block_position":"28-21"},{"bbox":[146,617,446,650],"type":"interline_equation","angle":0,"index":23,"text":"$$\n\\left[ \\begin{array}{c c} f _ {1} (A) & E _ {n} \\\\ O & f _ {2} (A) \\end{array} \\right] \\to \\left[ \\begin{array}{c c} O & E _ {n} \\\\ - f _ {2} (A) f _ {1} (A) & f _ {2} (A) \\end{array} \\right] \\to \\left[ \\begin{array}{c c} O & E _ {n} \\\\ - f (A) & O \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":28,"id":"abb2b597-a20f-4141-8f33-4ff531d03907","page_size":[595,841],"block_position":"28-22"},{"bbox":[54,656,251,669],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"因此得到 $r(f_{1}(A))+r(f_{2}(A))=n+r(f(A))$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"32d8aec5-cf58-42b7-ab26-145e4b20a32a","page_size":[595,841],"block_position":"28-23"},{"bbox":[35,671,304,684],"type":"text","angle":0,"index":25,"merge_prev":false,"text":"笔记 分块矩阵、多项式互素、空间直和本质上是一体的。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"49af3e79-432c-40ff-bfdb-b2790f00bc1b","page_size":[595,841],"block_position":"28-24"},{"bbox":[35,703,538,747],"type":"text","angle":0,"index":26,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":28,"id":"f2986548-a3d7-4aea-9d0c-1b9e47cbdada","page_size":[595,841],"block_position":"28-25"},{"bbox":[145,754,447,766],"type":"interline_equation","angle":0,"index":27,"text":"$$\nd i m V _ {1} + d i m V _ {2} = d i m (V _ {1} + V _ {2}) + d i m (V _ {1} \\cap V _ {2}) \\leq d i m W _ {1} + d i m W _ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":28,"id":"7c693168-efbd-4708-8999-73fe1de57949","page_size":[595,841],"block_position":"28-26"},{"bbox":[291,803,303,811],"type":"page_number","angle":0,"index":999999,"text":"24 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":28,"is_discarded":true,"id":"1c338299-c210-4762-b6b6-c7d28616b345","page_size":[595,841],"block_position":"28-27"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99999,"text":"2.4 矩阵的迹 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":29,"is_discarded":true,"id":"18f88eeb-e0e2-4931-818e-5c1359f58d05","page_size":[595,841],"block_position":"29-0"},{"bbox":[55,72,101,84],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"因此推出： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"d1a4857d-c0f6-449a-8a5a-7ce7424f562e","page_size":[595,841],"block_position":"29-1"},{"bbox":[170,87,423,100],"type":"interline_equation","angle":0,"index":2,"text":"$$\n(n - r (A)) + (n - r (B)) \\leq (n - r (A B)) + (n - r (A + B))\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":29,"id":"4c50d128-65b6-4c84-8983-99c390f92d68","page_size":[595,841],"block_position":"29-2"},{"bbox":[54,111,120,122],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"化简得到结论。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"696f23a8-78c4-4da5-a9fa-a85e42744687","page_size":[595,841],"block_position":"29-3"},{"bbox":[35,121,47,137],"type":"image","angle":0,"index":4,"img_path":"/ce3a524e4cfc4924fcc83503ac164d9bab0392e84942ca5de982e270059db9a0.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":29,"id":"ea30ddd1-2fe2-4276-b2a1-dd346c7aef29","page_size":[595,841],"block_position":"29-4"},{"bbox":[52,126,436,138],"type":"text","angle":0,"index":5,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"c938813f-d7a6-40e4-a050-87f736ec28db","page_size":[595,841],"block_position":"29-5"},{"bbox":[56,161,143,178],"type":"title","angle":0,"index":6,"level":2,"text":"## 2.4 矩阵的迹","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"c516720b-7cc4-4ed9-b9ab-95ab00ac83b1","page_size":[595,841],"block_position":"29-6"},{"bbox":[73,195,186,208],"type":"title","angle":0,"index":7,"level":2,"text":"## 定理 2.27 (矩阵迹的性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"822f0cbe-53fd-4116-9a26-a7a86e15b03d","page_size":[595,841],"block_position":"29-7"},{"bbox":[69,213,165,225],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"$A, B$ 为 $n$ 阶方阵，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"a170d94e-6aa6-44eb-80bf-d44eda66e11f","page_size":[595,841],"block_position":"29-8"},{"bbox":[83,228,212,272],"type":"interline_equation","angle":0,"index":9,"text":"$$\n\\begin{array}{l} \\bullet \\operatorname{tr} (A) = \\operatorname{tr} (k A) \\\\ \\bullet \\operatorname{tr} (A + B) = \\operatorname{tr} (A) + \\operatorname{tr} (B) \\\\ \\bullet \\operatorname{tr} (A B) = \\operatorname{tr} (B A) \\end{array}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":29,"id":"7fb492ff-da73-4629-b5ec-1051d112e79e","page_size":[595,841],"block_position":"29-9"},{"bbox":[83,275,310,287],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"若 $A$ 为实方阵，则 $A = O$ 当且仅当 $\\operatorname{tr}(A^T A) = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"f7b27f87-6a09-46e0-b80e-7e189cd5bcaa","page_size":[595,841],"block_position":"29-10"},{"bbox":[54,304,102,317],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"证明 显然 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"f51bd98d-cdf3-489f-8c1f-4400053f4cdc","page_size":[595,841],"block_position":"29-11"},{"bbox":[56,339,172,356],"type":"title","angle":0,"index":12,"level":2,"text":"## 2.5 几种技巧矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"660b0d4a-2b07-4bfc-a881-c916d34e3f6a","page_size":[595,841],"block_position":"29-12"},{"bbox":[57,372,139,386],"type":"title","angle":0,"index":13,"level":2,"text":"## 2.5.1 初等矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"cf2df352-490b-4711-ae9e-bc09b8ce5643","page_size":[595,841],"block_position":"29-13"},{"bbox":[73,400,166,412],"type":"title","angle":0,"index":14,"level":2,"text":"## 定义 2.10 (初等矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"6a7cedbe-eb4d-47c2-8581-5626eabad956","page_size":[595,841],"block_position":"29-14"},{"bbox":[68,417,526,446],"type":"text","angle":0,"index":15,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"016aaec8-79fa-493a-b211-0a68728d9486","page_size":[595,841],"block_position":"29-15"},{"bbox":[73,467,186,480],"type":"title","angle":0,"index":16,"level":2,"text":"## 定理 2.28 (初等矩阵乘法)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"bed9fc74-a582-4919-838e-722194c19cd0","page_size":[595,841],"block_position":"29-16"},{"bbox":[69,483,340,497],"type":"text","angle":0,"index":17,"merge_prev":false,"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， ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"0936009b-0fb4-4e6c-b887-e1ef19dfdbac","page_size":[595,841],"block_position":"29-17"},{"bbox":[57,528,270,543],"type":"title","angle":0,"index":18,"level":2,"text":"## 2.5.2 基本矩阵：仅一个元素为1的矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"1d6ad69e-08a1-4a04-bd0b-f6b637a5351d","page_size":[595,841],"block_position":"29-18"},{"bbox":[73,555,166,568],"type":"title","angle":0,"index":19,"level":2,"text":"## 定义 2.11 (基本矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"b3eeff83-88b4-4f6e-b020-04fff578b964","page_size":[595,841],"block_position":"29-19"},{"bbox":[69,572,417,586],"type":"text","angle":0,"index":20,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"cc25d537-d362-49a6-bc4b-5c6c18ae6d68","page_size":[595,841],"block_position":"29-20"},{"bbox":[73,608,186,619],"type":"title","angle":0,"index":21,"level":2,"text":"## 定理 2.29 (基本矩阵乘法)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"8cf6bfa6-ffc9-46e5-9f1b-f35c72032c14","page_size":[595,841],"block_position":"29-21"},{"bbox":[69,624,186,635],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"基本矩阵乘法有以下性质： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"b3272c7e-4f3c-4e3d-9007-37f7b3374ebe","page_size":[595,841],"block_position":"29-22"},{"bbox":[83,640,245,652],"type":"text","angle":0,"index":23,"merge_prev":false,"text":"- $AE_{ij}$ 将 $A$ 的第 $i$ 列移动到第 $j$ 列， ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"c112eb10-77f1-4d1b-a974-1a33704cdc49","page_size":[595,841],"block_position":"29-23"},{"bbox":[83,655,246,667],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"- $E_{ij}A$ 将 $A$ 的第 $j$ 行移动到第 $i$ 行。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"8b6169d3-663f-41b0-9ce5-492e44be8f27","page_size":[595,841],"block_position":"29-24"},{"bbox":[84,671,213,684],"type":"text","angle":0,"index":25,"merge_prev":false,"text":"- 基础矩阵分解： $E_{ij} = e_i e_j^T$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"a366074e-3e1b-4a3a-9ea8-865d3d701b95","page_size":[595,841],"block_position":"29-25"},{"bbox":[83,687,320,700],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"- 基本矩阵相乘： $E_{ik}E_{kj}=E_{ij}$ ， $k\\neq l$ 时 $E_{ik}E_{lj}=0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":29,"id":"4a604104-fdc9-4312-9f11-07e5a37386c6","page_size":[595,841],"block_position":"29-26"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":1000025,"text":"25 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":29,"is_discarded":true,"id":"bdcf7b99-6ba4-44a0-9057-81ff36415133","page_size":[595,841],"block_position":"29-27"}],[{"bbox":[459,47,539,58],"type":"header","angle":0,"index":-99999,"text":"2.5 几种技巧矩阵 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":30,"is_discarded":true,"id":"977b3620-4584-4ced-923a-ecf1deeb512c","page_size":[595,841],"block_position":"30-0"},{"bbox":[35,72,480,151],"type":"text","angle":0,"index":1,"merge_prev":false,"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： $A^k = \\left( \\begin{array}{cc}O & E_{n - k}\\\\ E_k & O \\end{array} \\right)$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"b7ac98ef-021a-47c1-bac5-0d6ce18e1ee2","page_size":[595,841],"block_position":"30-1"},{"bbox":[73,153,334,164],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"(2) $A$ 是 $n$ 阶上三角阵且主对角上元素均为0，证明 $A^n = O$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"5334592b-0dd2-45b5-a65a-3df04304d8bb","page_size":[595,841],"block_position":"30-2"},{"bbox":[53,167,501,181],"type":"text","angle":0,"index":3,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"4f7ef069-68b2-4594-b5bf-79d72d4424e9","page_size":[595,841],"block_position":"30-3"},{"bbox":[234,186,359,200],"type":"interline_equation","angle":0,"index":4,"text":"$$\nA ^ {2} = (e _ {n - 1}, e _ {n}, e _ {1} \\dots , e _ {n - 2})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":30,"id":"8278096a-6e84-41b7-9ec2-573ceb78a7b5","page_size":[595,841],"block_position":"30-4"},{"bbox":[54,206,157,218],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"以此类推容易得到结论 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"7ed391a2-09df-4641-af3a-dedbcd244f45","page_size":[595,841],"block_position":"30-5"},{"bbox":[53,221,539,259],"type":"text","angle":0,"index":6,"merge_prev":false,"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 $A^{n}=O$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"92698fa0-8c2f-4b9b-901e-ad7e10846920","page_size":[595,841],"block_position":"30-6"},{"bbox":[36,276,498,289],"type":"text","angle":0,"index":7,"merge_prev":false,"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 $\\alpha^T A\\alpha = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"2c989901-bb02-4af8-ab4c-34fcb9601450","page_size":[595,841],"block_position":"30-7"},{"bbox":[72,292,372,304],"type":"text","angle":0,"index":8,"merge_prev":false,"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 $\\alpha^T A\\alpha = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"e5364f82-a42b-4a56-ab0c-b004b7628bf2","page_size":[595,841],"block_position":"30-8"},{"bbox":[54,307,400,320],"type":"text","angle":0,"index":9,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"c64dc7ad-6f53-4131-8ef1-0ba875ae4644","page_size":[595,841],"block_position":"30-9"},{"bbox":[192,327,401,341],"type":"interline_equation","angle":0,"index":10,"text":"$$\n0 = (e _ {i} + e _ {j}) ^ {T} A (e _ {i} + e _ {j}) = a _ {i i} + a _ {j j} + a _ {i j} + a _ {j i}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":30,"id":"f568bc9b-1b43-411f-98e8-04595a580c0d","page_size":[595,841],"block_position":"30-10"},{"bbox":[54,346,539,391],"type":"text","angle":0,"index":11,"merge_prev":false,"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 $A^T = -A$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"ac513f3a-347b-4617-a420-91e9e5f930e3","page_size":[595,841],"block_position":"30-11"},{"bbox":[56,408,138,423],"type":"title","angle":0,"index":12,"level":2,"text":"## 2.5.3 置换矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"322e0504-aa9c-4284-b0b5-4116bb379923","page_size":[595,841],"block_position":"30-12"},{"bbox":[73,435,166,449],"type":"title","angle":0,"index":13,"level":2,"text":"## 定义 2.12 (置换矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"7032fd61-7b6f-491b-ba04-601f7a3fe1f6","page_size":[595,841],"block_position":"30-13"},{"bbox":[68,451,534,483],"type":"text","angle":0,"index":14,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"57dd7b44-3841-41dc-86a8-ee81886dbc95","page_size":[595,841],"block_position":"30-14"},{"bbox":[73,503,215,516],"type":"title","angle":0,"index":15,"level":2,"text":"## 定理 2.30 (置换矩阵是正交矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"dc080d7d-4bd9-48ce-9c4d-e0c2e8fbf975","page_size":[595,841],"block_position":"30-15"},{"bbox":[70,520,166,532],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"置换矩阵是正交矩阵。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"bcdb2a3e-5a89-445e-b2c8-a0d1b5ca2b4f","page_size":[595,841],"block_position":"30-16"},{"bbox":[54,548,145,560],"type":"title","angle":0,"index":17,"level":2,"text":"## 证明 证明 $P^T P = I$","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"75783177-ec59-46fa-8d4c-9849c8d489ce","page_size":[595,841],"block_position":"30-17"},{"bbox":[73,569,196,581],"type":"title","angle":0,"index":18,"level":2,"text":"## 定理 2.31 (置换矩阵的作用)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"69e77434-b987-4b3a-89bc-c4c50a760f14","page_size":[595,841],"block_position":"30-18"},{"bbox":[68,586,528,615],"type":"text","angle":0,"index":19,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"9d840f76-3f5e-4bd4-a4c2-d7e750b685ed","page_size":[595,841],"block_position":"30-19"},{"bbox":[73,635,205,648],"type":"title","angle":0,"index":20,"level":2,"text":"## 定理 2.32 (置换矩阵正交相似)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"853217f9-33de-494a-968a-d9121428a749","page_size":[595,841],"block_position":"30-20"},{"bbox":[68,652,346,666],"type":"text","angle":0,"index":21,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"636db1d1-2d11-4bb4-8669-8d7f7f566275","page_size":[595,841],"block_position":"30-21"},{"bbox":[186,669,404,736],"type":"interline_equation","angle":0,"index":23,"text":"$$\nP ^ {T} A P = P ^ {- 1} A P = \\left[ \\begin{array}{c c c c} a _ {i _ {1} i _ {1}} & a _ {i _ {1} i _ {2}} & \\dots & a _ {i _ {1} i _ {n}} \\\\ a _ {i _ {2} i _ {1}} & a _ {i _ {2} i _ {2}} & \\dots & a _ {i _ {2} i _ {n}} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a _ {i _ {n} i _ {1}} & a _ {i _ {n} i _ {2}} & \\dots & a _ {i _ {n} i _ {n}} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":30,"id":"441ac11d-f2b6-444a-b71b-7c2596f514e4","page_size":[595,841],"block_position":"30-22"},{"bbox":[69,739,536,753],"type":"text","angle":0,"index":24,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":30,"id":"f7e1f060-1c2a-4ba3-a1de-73da53643bb2","page_size":[595,841],"block_position":"30-23"},{"bbox":[291,803,303,812],"type":"page_number","angle":0,"index":1000021,"text":"26 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":30,"is_discarded":true,"id":"b715f9d7-2d55-4d1d-a536-697a2d48957b","page_size":[595,841],"block_position":"30-24"}],[{"bbox":[439,47,539,58],"type":"header","angle":0,"index":-99999,"text":"2.6 特殊矩阵性质总结 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":31,"is_discarded":true,"id":"29fb0c23-f90c-44c0-aff3-a0475d51a8d1","page_size":[595,841],"block_position":"31-0"},{"bbox":[36,71,539,100],"type":"text","angle":0,"index":1,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"c4b1b01d-4135-4948-8cde-e3ad8c653537","page_size":[595,841],"block_position":"31-1"},{"bbox":[54,102,196,116],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"证明 取 $P = [\\epsilon_{i_{1}}, \\cdots, \\epsilon_{i_{n}}]$ 即可。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"8c71fd4b-bc41-4b36-bf5c-2b736a8f138a","page_size":[595,841],"block_position":"31-2"},{"bbox":[73,124,196,137],"type":"title","angle":0,"index":3,"level":2,"text":"## 定义 2.13 (特殊的置换矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"19ded7db-0f85-47ae-846f-20477b4b5ec8","page_size":[595,841],"block_position":"31-3"},{"bbox":[67,141,459,211],"type":"text","angle":0,"index":4,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"e213e27a-36bd-4563-b08d-85599926057b","page_size":[595,841],"block_position":"31-4"},{"bbox":[56,246,201,263],"type":"title","angle":0,"index":5,"level":2,"text":"## 2.6 特殊矩阵性质总结","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"fd30a0bf-3447-4dff-af90-aec7d0c751a3","page_size":[595,841],"block_position":"31-5"},{"bbox":[56,279,138,293],"type":"title","angle":0,"index":6,"level":2,"text":"## 2.6.1 对角矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"580bbb36-0014-4470-a770-00897eecba45","page_size":[595,841],"block_position":"31-6"},{"bbox":[72,306,186,318],"type":"title","angle":0,"index":7,"level":2,"text":"## 定理 2.33 (对角矩阵乘法)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"1dede727-8260-4905-b27d-92edee0ee500","page_size":[595,841],"block_position":"31-7"},{"bbox":[68,322,487,336],"type":"text","angle":0,"index":8,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"95bdc4c0-e9fc-4bd7-a34a-a6f855aa8159","page_size":[595,841],"block_position":"31-8"},{"bbox":[73,355,186,368],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"定理 2.34 (对角矩阵求逆) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"a53c769a-545a-4a5f-bb60-e8b0b6d1f390","page_size":[595,841],"block_position":"31-9"},{"bbox":[68,372,249,387],"type":"interline_equation","angle":0,"index":10,"text":"$$\n\\operatorname{diag} \\left\\{d _ {1}, \\dots , d _ {n} \\right\\} ^ {- 1} = \\operatorname{diag} \\left\\{d _ {1} ^ {- 1}, \\dots , d _ {n} ^ {- 1} \\right\\}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":31,"id":"69f42fe5-97ac-4a03-b368-415e555fe9a7","page_size":[595,841],"block_position":"31-10"},{"bbox":[56,418,174,432],"type":"title","angle":0,"index":11,"level":2,"text":"## 2.6.2 上、下三角矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"4f9b5828-b33a-4549-8457-25522c39ffe2","page_size":[595,841],"block_position":"31-11"},{"bbox":[72,445,236,458],"type":"title","angle":0,"index":12,"level":2,"text":"## 定理 2.35 (上下三角乘法、逆、伴随)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"f149d414-d4f6-4d7e-b376-adccc2bb2931","page_size":[595,841],"block_position":"31-12"},{"bbox":[68,462,174,474],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"$A, B$ 是上三角矩阵，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"f2930458-d41b-4128-aef0-bcec0371b491","page_size":[595,841],"block_position":"31-13"},{"bbox":[83,478,357,491],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"- $AB$ 仍然是上三角矩阵，且 $AB$ 对角元为 $a_{11}b_{11},\\dots ,a_{nn}b_{nn}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"b203d80a-2acb-4577-9bd9-f3e44bdda0f0","page_size":[595,841],"block_position":"31-14"},{"bbox":[83,492,318,507],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"- $A^{-1}$ 也是上三角阵，且 $A^{-1}$ 对角元为 $a_{11}^{-1}, \\cdots, a_{nn}^{-1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"baf9c3e0-6448-4278-bade-40311b17bffc","page_size":[595,841],"block_position":"31-15"},{"bbox":[83,508,526,538],"type":"text","angle":0,"index":16,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"2ad5b325-9bab-4d32-92ca-fefaa092e6d9","page_size":[595,841],"block_position":"31-16"},{"bbox":[52,547,539,612],"type":"text","angle":0,"index":17,"merge_prev":false,"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 $c_{ii} = \\sum_{k=1}^{i-1} a_{ik} b_{ki} + a_{ii} b_{ii} + \\sum_{k=i+1}^{n} a_{ik} b_{ki} = a_{ii} b_{ii}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"6ba86da6-a4a8-41c1-a6e9-d5a382ec39f0","page_size":[595,841],"block_position":"31-17"},{"bbox":[54,612,540,640],"type":"text","angle":0,"index":18,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"00e13b70-785f-44fe-aee4-2838ac659d73","page_size":[595,841],"block_position":"31-18"},{"bbox":[73,643,192,655],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"(3) 根据 $A^{*} = |A|A^{-1}$ 即可 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"3b382d44-9618-4235-99a4-a505cf03056c","page_size":[595,841],"block_position":"31-19"},{"bbox":[73,665,224,677],"type":"title","angle":0,"index":21,"level":2,"text":"## 定理 2.36 (n 个上三角矩阵的乘积)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"ee827bef-628d-46c0-a7ed-527e0c32b17c","page_size":[595,841],"block_position":"31-20"},{"bbox":[68,682,380,694],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"$A_{1},\\cdots,A_{n}$ 时 n 个对角为 0 的 n 阶上三角矩阵，证明 $A_{1}A_{2}\\cdots A_{n}=O$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"c6349040-3a33-4790-a282-908ee1174e9c","page_size":[595,841],"block_position":"31-21"},{"bbox":[53,704,358,737],"type":"text","angle":0,"index":23,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":31,"id":"066d5aa3-f0fa-40c5-9103-255cfe324d41","page_size":[595,841],"block_position":"31-22"},{"bbox":[133,740,461,775],"type":"interline_equation","angle":0,"index":24,"text":"$$\nA _ {1} \\dots A _ {n} = \\left[ \\begin{array}{c c} B _ {1} \\dots B _ {n - 1} & * \\\\ 0 & 0 \\end{array} \\right] \\cdot \\left[ \\begin{array}{c c} B _ {n} & * \\\\ 0 & 0 \\end{array} \\right] = \\left[ \\begin{array}{c c} 0 & * \\\\ 0 & 0 \\end{array} \\right] \\left[ \\begin{array}{c c} B _ {n} & * \\\\ 0 & 0 \\end{array} \\right] = O\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":31,"id":"17f04968-d99c-474f-a917-7b4a8ce60c43","page_size":[595,841],"block_position":"31-23"},{"bbox":[291,802,302,811],"type":"page_number","angle":0,"index":1000019,"text":"27 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":31,"is_discarded":true,"id":"fdf1befd-62e6-4042-a5c6-4dc0b0045fad","page_size":[595,841],"block_position":"31-24"}],[{"bbox":[439,47,539,58],"type":"header","angle":0,"index":-99998,"text":"2.6 特殊矩阵性质总结 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":32,"is_discarded":true,"id":"b25d0fc8-9bf4-4277-85f2-e6fe3d5a647c","page_size":[595,841],"block_position":"32-0"},{"bbox":[56,72,151,86],"type":"title","angle":0,"index":2,"level":2,"text":"## 2.6.3 实对称矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"3bb4e47f-c671-4c1b-9467-17be89acd575","page_size":[595,841],"block_position":"32-1"},{"bbox":[73,100,176,112],"type":"title","angle":0,"index":3,"level":2,"text":"## 定义 2.14 (实对称矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"fcaad0c4-cd34-48cf-b058-5499d9d13b78","page_size":[595,841],"block_position":"32-2"},{"bbox":[69,117,343,130],"type":"text","angle":0,"index":4,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"424f925c-8f4b-40bc-b8c8-1d478808c94f","page_size":[595,841],"block_position":"32-3"},{"bbox":[73,151,226,163],"type":"title","angle":0,"index":5,"level":2,"text":"## 定理 2.37 (实对称矩阵的矩阵性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"6393913c-df1c-435f-b5ff-388cb7d447ae","page_size":[595,841],"block_position":"32-4"},{"bbox":[69,168,198,179],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"实对称矩阵 A 具有以下性质： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"24c49802-eafe-43b1-8b46-790bd600c14d","page_size":[595,841],"block_position":"32-5"},{"bbox":[84,183,188,195],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"- $A^{-1}$ 也是实对称矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"cc12b2ca-7c80-4c21-908c-d54e9560698e","page_size":[595,841],"block_position":"32-6"},{"bbox":[54,211,535,224],"type":"text","angle":0,"index":8,"merge_prev":false,"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 $(A^{-1})^T = A^{-1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"e2150554-ccbf-4d9c-91d4-e36d7a917447","page_size":[595,841],"block_position":"32-7"},{"bbox":[73,233,236,246],"type":"title","angle":0,"index":9,"level":2,"text":"## 定理 2.38 (实对称矩阵的特征值性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"6390489e-10c2-49af-9d69-1f7dda3dc416","page_size":[595,841],"block_position":"32-8"},{"bbox":[69,250,176,262],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"实对称矩阵有如下性质： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"4b3c9f29-ef47-4857-bd99-23013a263670","page_size":[595,841],"block_position":"32-9"},{"bbox":[83,266,186,277],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"- 每个特征值都是实数 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"92d1abce-048e-40a7-aaa6-2e2219b78b32","page_size":[595,841],"block_position":"32-10"},{"bbox":[83,281,216,293],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"- 不同特征值的特征向量正交 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"2e5b01c9-fc85-440e-a3c6-8831b26e9a2f","page_size":[595,841],"block_position":"32-11"},{"bbox":[83,297,267,309],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"- 一定正交相似 (相似且合同) 于对角矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"684fd6aa-8271-4173-ad6f-170f107d7c9c","page_size":[595,841],"block_position":"32-12"},{"bbox":[53,323,525,339],"type":"text","angle":0,"index":14,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"353a4786-ea48-4bcd-9804-0a19b3cf6c59","page_size":[595,841],"block_position":"32-13"},{"bbox":[265,343,327,359],"type":"interline_equation","angle":0,"index":15,"text":"$$\n\\overline {{\\xi}} ^ {T} A \\xi = \\lambda \\overline {{\\xi}} ^ {T} \\xi\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":32,"id":"3def88fd-8e34-478e-b68b-65237a1c5a3f","page_size":[595,841],"block_position":"32-14"},{"bbox":[54,364,308,377],"type":"text","angle":0,"index":16,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"7ad637f1-77a8-4fd6-9880-3e00c1de441f","page_size":[595,841],"block_position":"32-15"},{"bbox":[232,380,360,398],"type":"interline_equation","angle":0,"index":17,"text":"$$\n\\overline {{\\xi}} ^ {T} A = \\overline {{\\xi}} ^ {T} \\overline {{\\lambda}} \\Rightarrow \\overline {{\\xi}} ^ {T} A \\xi = \\overline {{\\lambda}} \\overline {{\\xi}} ^ {T} \\xi\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":32,"id":"d7b0df90-1aa7-4395-9fa2-57876fd4c77e","page_size":[595,841],"block_position":"32-16"},{"bbox":[54,404,145,416],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"因此比较得到 $\\lambda = \\overline{\\lambda}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"12542d0f-afd6-4354-9d70-a197134fb505","page_size":[595,841],"block_position":"32-17"},{"bbox":[73,419,524,432],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"a5fb520a-5a4a-47f6-ac30-55b644c9a449","page_size":[595,841],"block_position":"32-18"},{"bbox":[170,439,423,452],"type":"interline_equation","angle":0,"index":20,"text":"$$\n\\lambda_ {1} (\\xi_ {1}, \\xi_ {2}) = (\\lambda_ {1} \\xi_ {1}) ^ {T} \\xi_ {2} = (A \\xi_ {1}) ^ {T} \\xi_ {2} = \\xi_ {1} ^ {T} (A \\xi_ {2}) = \\lambda_ {2} (\\xi_ {1}, \\xi_ {2})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":32,"id":"68c3a12d-7864-41e1-9b02-0cfc73b43a01","page_size":[595,841],"block_position":"32-19"},{"bbox":[54,459,326,470],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"从而得到 $(\\lambda_{2} - \\lambda_{1})(\\xi_{1},\\xi_{2}) = 0$ ，而 $\\lambda_1\\neq \\lambda_2$ ，从而 $(\\xi_1,\\xi_2) = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"9b25d453-2b5e-4f4b-b433-5ad16c55caa2","page_size":[595,841],"block_position":"32-20"},{"bbox":[54,474,539,502],"type":"text","angle":0,"index":22,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"4716a5cb-3ec1-4cbc-aac6-8d2db90da3e4","page_size":[595,841],"block_position":"32-21"},{"bbox":[184,505,408,537],"type":"interline_equation","angle":0,"index":23,"text":"$$\nA T = T \\left( \\begin{array}{c c} \\lambda & \\alpha^ {T} \\\\ 0 & A _ {n - 1} \\end{array} \\right) \\Rightarrow T ^ {T} A T = \\left( \\begin{array}{c c} \\lambda & \\alpha^ {T} \\\\ 0 & A _ {n - 1} \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":32,"id":"b272de03-d312-4b4b-a437-d13d653c15d8","page_size":[595,841],"block_position":"32-22"},{"bbox":[54,540,539,570],"type":"text","angle":0,"index":24,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"bc5e7f93-81d3-4a18-842a-f395720159dc","page_size":[595,841],"block_position":"32-23"},{"bbox":[219,570,371,601],"type":"interline_equation","angle":0,"index":25,"text":"$$\nP ^ {T} T ^ {T} A T P = \\left( \\begin{array}{c c} \\lambda & 0 \\\\ 0 & P ^ {T} A _ {n - 1} P \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":32,"id":"351939eb-7525-42be-9bb9-5aa5d4d83bc8","page_size":[595,841],"block_position":"32-24"},{"bbox":[55,608,203,621],"type":"text","angle":0,"index":26,"merge_prev":false,"text":"显然 $TP = Q$ ，则 $Q$ 也是正交阵。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"4550d429-f6fd-4431-b09c-7ce87fcace81","page_size":[595,841],"block_position":"32-25"},{"bbox":[56,640,151,654],"type":"title","angle":0,"index":27,"level":2,"text":"## 2.6.4 反对称矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"a7171653-48da-4f76-8b3b-125f0804b802","page_size":[595,841],"block_position":"32-26"},{"bbox":[73,667,216,679],"type":"title","angle":0,"index":28,"level":2,"text":"## 定理 2.39 (反对称矩阵的行列式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"282b1113-14d6-43c2-8a40-e6ac529ba9cb","page_size":[595,841],"block_position":"32-27"},{"bbox":[70,684,223,696],"type":"text","angle":0,"index":29,"merge_prev":false,"text":"奇数阶反对称矩阵的行列式必为 0。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"68b95068-303c-4464-805a-055e05564e39","page_size":[595,841],"block_position":"32-28"},{"bbox":[54,712,422,725],"type":"text","angle":0,"index":30,"merge_prev":false,"text":"证明 根据 $A^T = -A$ 得到 $|A| = (-1)^n |A|$ ，当 $n$ 为奇数时 $|A| = -|A|$ ，因此 $|A| = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":32,"id":"b85eff93-1611-4665-8e17-5db024bf20fd","page_size":[595,841],"block_position":"32-29"},{"bbox":[291,802,302,812],"type":"page_number","angle":0,"index":999999,"text":"28 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":32,"is_discarded":true,"id":"05485491-f653-43a1-99f4-caad3dbe9d01","page_size":[595,841],"block_position":"32-30"}],[{"bbox":[439,47,539,58],"type":"header","angle":0,"index":-99999,"text":"2.6 特殊矩阵性质总结 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":33,"is_discarded":true,"id":"1d1f7e3f-2b39-4aa7-832c-8c290ec5d6c9","page_size":[595,841],"block_position":"33-0"},{"bbox":[73,74,225,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 定理 2.40 (反对称矩阵的矩阵性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"c5778b41-7a88-46ec-96ba-9ee3277d01be","page_size":[595,841],"block_position":"33-1"},{"bbox":[69,90,162,101],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"$A$ 为反对称矩阵，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"4eaf4660-09f8-4913-a792-24ca69739573","page_size":[595,841],"block_position":"33-2"},{"bbox":[83,105,177,117],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"- $A^{-1}$ 为反对称矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"79f70dab-f84e-48ac-b396-7cabc4f40c2b","page_size":[595,841],"block_position":"33-3"},{"bbox":[55,134,176,146],"type":"title","angle":0,"index":4,"level":2,"text":"## 证明 (1) 类似实对称矩阵的","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"5f8d709f-5e6a-40c6-a52a-868c82dd414d","page_size":[595,841],"block_position":"33-4"},{"bbox":[73,155,215,168],"type":"title","angle":0,"index":5,"level":2,"text":"## 定理 2.41 (反对称矩阵的二次型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"20d3149d-f6ed-4351-9298-f9567293e27d","page_size":[595,841],"block_position":"33-5"},{"bbox":[68,173,311,185],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"$A$ 为反对称矩阵当且仅当任意 $X \\in \\mathbb{R}^n$ ，有 $X^T A X = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"753ea406-62a7-4bbd-9342-5f8a85065676","page_size":[595,841],"block_position":"33-6"},{"bbox":[54,201,232,214],"type":"title","angle":0,"index":7,"level":2,"text":"## 证明 在基础矩阵技巧一节中给出过证明","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"15b09b1e-941e-4bd4-abe2-ed4655699d73","page_size":[595,841],"block_position":"33-7"},{"bbox":[73,222,265,235],"type":"title","angle":0,"index":8,"level":2,"text":"## 定理 2.42 (反对称矩阵的特征值、特征向量)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"95b4e557-0200-4805-a0f0-4c79d25a51a2","page_size":[595,841],"block_position":"33-8"},{"bbox":[69,239,506,252],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"e0307df2-b26d-4eb9-8ff6-3eefb9efb37a","page_size":[595,841],"block_position":"33-9"},{"bbox":[54,268,539,297],"type":"text","angle":0,"index":10,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"a5d47df2-087a-4915-b222-357f1dc343d1","page_size":[595,841],"block_position":"33-10"},{"bbox":[130,302,464,315],"type":"interline_equation","angle":0,"index":11,"text":"$$\nA u = a u - b v, A v = b u + a v \\Rightarrow u ^ {T} A u = a u ^ {T} u - b u ^ {T} v, v ^ {T} A v = b v ^ {T} u + a v ^ {T} v\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":33,"id":"3198e16e-b2d9-419f-acae-619d7d7531e8","page_size":[595,841],"block_position":"33-11"},{"bbox":[54,322,380,335],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"aea3dc03-6eaa-48bc-a804-777d428d758f","page_size":[595,841],"block_position":"33-12"},{"bbox":[180,341,413,354],"type":"interline_equation","angle":0,"index":13,"text":"$$\nu ^ {T} A u = (u ^ {T} A u) ^ {T} = u ^ {T} A ^ {T} u = - u ^ {T} A u \\Rightarrow u ^ {T} A u = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":33,"id":"165d5ac4-e251-49a8-937d-4f2a2d9f9208","page_size":[595,841],"block_position":"33-13"},{"bbox":[55,360,460,374],"type":"text","angle":0,"index":14,"merge_prev":false,"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 $\\lambda = a + bi = bi$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"e7f3d0a1-1513-40c9-a7c8-a48e054423b1","page_size":[595,841],"block_position":"33-14"},{"bbox":[73,376,461,390],"type":"text","angle":0,"index":15,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"6412af80-a802-4766-91ee-d45c4071dbae","page_size":[595,841],"block_position":"33-15"},{"bbox":[249,396,344,409],"type":"interline_equation","angle":0,"index":16,"text":"$$\nA \\alpha = - k \\beta , A \\beta = k \\alpha\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":33,"id":"c669c71c-87d8-4da8-8300-aea80a314e15","page_size":[595,841],"block_position":"33-16"},{"bbox":[55,414,421,428],"type":"text","angle":0,"index":17,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"910585fc-1232-49cf-b8de-4a5b99ab47bc","page_size":[595,841],"block_position":"33-17"},{"bbox":[196,433,386,470],"type":"interline_equation","angle":0,"index":18,"text":"$$\n\\alpha^ {T} A \\beta + \\beta^ {T} A \\alpha = \\left\\{ \\begin{array}{l} \\alpha^ {T} A \\beta - (\\alpha^ {T} A \\beta) ^ {T} = 0 \\\\ k (\\alpha^ {T} \\alpha - \\beta^ {T} \\beta) \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":33,"id":"4f19940c-4b70-4d9b-a2fa-43d220df2820","page_size":[595,841],"block_position":"33-18"},{"bbox":[54,473,207,486],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"得到 $\\alpha^T\\alpha -\\beta^T\\beta = 0$ ，即模长相等 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"c85fa7fa-f5d7-4280-9696-33494a3975da","page_size":[595,841],"block_position":"33-19"},{"bbox":[73,496,215,508],"type":"title","angle":0,"index":20,"level":2,"text":"## 定理 2.43 (反对称矩阵的标准型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"df7e9d97-7dd8-4f64-8396-fceebcf632c6","page_size":[595,841],"block_position":"33-20"},{"bbox":[70,513,182,524],"type":"title","angle":0,"index":21,"level":2,"text":"## $A$ 为 $n$ 阶反对称矩阵，则","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"793854f5-0954-425f-a4cf-7ad4786492c0","page_size":[595,841],"block_position":"33-21"},{"bbox":[83,528,215,540],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"- $A$ 正交相似于如下准对角阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"ea40c4d3-ab7a-4906-ab2a-9b06793e4126","page_size":[595,841],"block_position":"33-22"},{"bbox":[195,542,421,576],"type":"interline_equation","angle":0,"index":23,"text":"$$\n\\operatorname{diag} \\left\\{\\left( \\begin{array}{c c} 0 & b _ {1} \\\\ - b _ {1} & 0 \\end{array} \\right), \\dots , \\left( \\begin{array}{c c} 0 & b _ {s} \\\\ - b _ {s} & 0 \\end{array} \\right), 0, \\dots , 0 \\right\\}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":33,"id":"28f2567d-2064-4f54-88e7-5351a4e4e988","page_size":[595,841],"block_position":"33-23"},{"bbox":[84,578,167,590],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"- $A$ 合同于标准型 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"37fff709-f498-4a4e-93fb-44b7a751ce71","page_size":[595,841],"block_position":"33-24"},{"bbox":[202,588,415,622],"type":"interline_equation","angle":0,"index":25,"text":"$$\n\\operatorname{diag} \\left\\{\\left( \\begin{array}{c c} 0 & 1 \\\\ - 1 & 0 \\end{array} \\right), \\dots , \\left( \\begin{array}{c c} 0 & 1 \\\\ - 1 & 0 \\end{array} \\right), 0, \\dots , 0 \\right\\}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":33,"id":"c09b8e93-2c15-4a59-a89b-1f979258912c","page_size":[595,841],"block_position":"33-25"},{"bbox":[73,642,236,655],"type":"title","angle":0,"index":26,"level":2,"text":"## 定理 2.44 (将矩阵拆为对称 + 反对称)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":33,"id":"7a7dda6a-8f82-47de-92d2-65ad14de9f60","page_size":[595,841],"block_position":"33-26"},{"bbox":[69,660,202,683],"type":"interline_equation","angle":0,"index":28,"text":"$$\nA = \\frac {1}{2} (A + A ^ {T}) + \\frac {1}{2} (A - A ^ {T})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":33,"id":"99191177-05a3-4ebe-89d7-1075210be22a","page_size":[595,841],"block_position":"33-27"},{"bbox":[291,802,302,811],"type":"page_number","angle":0,"index":1000026,"text":"29 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":33,"is_discarded":true,"id":"3f876841-70a8-4d3a-8408-c7676875ae3e","page_size":[595,841],"block_position":"33-28"}],[{"bbox":[439,46,539,58],"type":"header","angle":0,"index":-99999,"text":"2.6 特殊矩阵性质总结 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":34,"is_discarded":true,"id":"a5b0ba68-105d-4dd5-8685-fc44ee8fc888","page_size":[595,841],"block_position":"34-0"},{"bbox":[57,73,160,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 2.6.5 Hermite 矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"1f463ea4-2769-47ba-92a5-8b17391fb6d8","page_size":[595,841],"block_position":"34-1"},{"bbox":[73,100,184,112],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"定义 2.15 (Hermite 矩阵) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"8503e7b7-7605-43f0-9eec-eeec5aa36e99","page_size":[595,841],"block_position":"34-2"},{"bbox":[69,118,216,132],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"$\\overline{A}^T = A$ 的矩阵称为Hermite矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"4f3b4d5f-8c92-4ef0-ae11-cf5110db987e","page_size":[595,841],"block_position":"34-3"},{"bbox":[73,153,214,164],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"定理 2.45 (Hermite 矩阵的性质) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"856c5cc7-8c19-4221-bcd5-1cf3d680af4a","page_size":[595,841],"block_position":"34-4"},{"bbox":[69,169,207,181],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"Hermite 矩阵的特征值都是实数 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"ef585667-8d8f-44b7-b4d4-af42991ee1b1","page_size":[595,841],"block_position":"34-5"},{"bbox":[57,213,138,227],"type":"title","angle":0,"index":6,"level":2,"text":"## 2.6.6 正交矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"89c57983-24bd-4a6c-9847-53414f6ab2e4","page_size":[595,841],"block_position":"34-6"},{"bbox":[73,240,166,253],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"定义 2.16 (正交矩阵) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"669dc49f-fdac-4cdc-a9e5-4f6b47bdfa12","page_size":[595,841],"block_position":"34-7"},{"bbox":[69,258,272,270],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"若矩阵 $T$ 满足 $TT^{T} = I$ ，则称 $T$ 为正交矩阵。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"c2f54fa8-19e5-4f6f-9382-f3ba9dc902a3","page_size":[595,841],"block_position":"34-8"},{"bbox":[73,291,195,304],"type":"title","angle":0,"index":9,"level":2,"text":"## 定理 2.46 (正交矩阵的性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"45b3c93b-4ef6-47ee-8765-c8fe24fd6071","page_size":[595,841],"block_position":"34-9"},{"bbox":[69,308,185,320],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"$A, B$ 为 $n$ 阶正交矩阵，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"0c5180ea-ecd7-4099-a752-8ab6d91991f7","page_size":[595,841],"block_position":"34-10"},{"bbox":[84,322,171,335],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"- 求逆: $A^{-1} = A^{T}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"19c49c6b-3595-47f0-b069-a3941d100755","page_size":[595,841],"block_position":"34-11"},{"bbox":[84,338,215,351],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"- $A^{-1}, A^T, AB$ 都是正交矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"b0f01e7a-7e1e-4334-95cd-3ce54b599360","page_size":[595,841],"block_position":"34-12"},{"bbox":[84,352,226,381],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"- $S = \\left[ \\begin{array}{ll}A & 0\\\\ 0 & B \\end{array} \\right]$ 也是正交矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"4a43127d-f2bb-480e-8f6c-3d271b5f9ae6","page_size":[595,841],"block_position":"34-13"},{"bbox":[83,382,330,394],"type":"text","angle":0,"index":14,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"eeeff3f7-25c1-4ce7-9fb1-b25922c6e2d2","page_size":[595,841],"block_position":"34-14"},{"bbox":[55,410,116,423],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"证明 (1) 显然 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"31801c8c-ea18-4f35-9c81-2db26bc79f55","page_size":[595,841],"block_position":"34-15"},{"bbox":[74,426,153,439],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"(2) $(AB)^T AB = I$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"1f0f468c-d41f-4b7d-93ce-a62c8fc011a2","page_size":[595,841],"block_position":"34-16"},{"bbox":[73,441,186,454],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"(3) 直接用 $S^T S$ 验证即可 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"caa22614-b1cc-44e4-a2cf-10613f44702c","page_size":[595,841],"block_position":"34-17"},{"bbox":[73,457,290,470],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"(4) $A=\\left[\\alpha_{1},\\cdots,\\alpha_{n}\\right]$ ，计算 $A^{T}A=I$ 可得出结论。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"5620fb1c-e365-43f4-80ea-784d5917e0d2","page_size":[595,841],"block_position":"34-18"},{"bbox":[35,488,472,516],"type":"text","angle":0,"index":20,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"cd7998da-c58d-4f61-bb38-15ccb79da0a7","page_size":[595,841],"block_position":"34-19"},{"bbox":[74,539,215,551],"type":"title","angle":0,"index":21,"level":2,"text":"## 定理 2.47 (分块上三角正交矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"eba6ace4-c0e9-4e0a-9eac-80682bc8acf2","page_size":[595,841],"block_position":"34-20"},{"bbox":[69,556,355,589],"type":"text","angle":0,"index":22,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"f913a7c7-fc6e-4c8f-8244-fcd0dd4f9528","page_size":[595,841],"block_position":"34-21"},{"bbox":[54,606,261,618],"type":"text","angle":0,"index":23,"merge_prev":false,"text":"证明 根据分块矩阵转置以及乘法可以得到结论 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"3366d333-10a5-4b99-9eb9-e7a2f118e898","page_size":[595,841],"block_position":"34-22"},{"bbox":[219,620,372,654],"type":"interline_equation","angle":0,"index":24,"text":"$$\nA ^ {T} A = \\left[ \\begin{array}{c c} A _ {1} ^ {T} & 0 \\\\ A _ {2} ^ {T} & A _ {4} ^ {T} \\end{array} \\right] \\left[ \\begin{array}{c c} A _ {1} & A _ {2} \\\\ 0 & A _ {4} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":34,"id":"2f79c4d3-3682-4d1a-8274-199ae75300b7","page_size":[595,841],"block_position":"34-23"},{"bbox":[74,667,246,680],"type":"title","angle":0,"index":25,"level":2,"text":"## 定理 2.48 (正交矩阵的行列式与特征值)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"307f4816-d55a-4e72-9d31-baebb4677b81","page_size":[595,841],"block_position":"34-24"},{"bbox":[68,684,451,696],"type":"text","angle":0,"index":26,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"388b41f5-10e8-46f4-813e-87ff25b45067","page_size":[595,841],"block_position":"34-25"},{"bbox":[83,699,256,712],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"若 $|A| = -1: -1$ 一定是 $A$ 的特征值 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"9cb72774-b51c-451c-97f9-b2a8aa4841d9","page_size":[595,841],"block_position":"34-26"},{"bbox":[83,715,335,727],"type":"text","angle":0,"index":28,"merge_prev":false,"text":"若 $A$ 是奇数阶的，且 $|A| = 1$ ，则1一定是 $A$ 的特征值 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"280bb038-ebba-4c73-808d-926358686f13","page_size":[595,841],"block_position":"34-27"},{"bbox":[83,730,306,743],"type":"text","angle":0,"index":29,"merge_prev":false,"text":"- 复特征值的特征向量实部和虚部模长相等且正交 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":34,"id":"8e6fdf0e-0fd1-458b-ac9e-e07c6bed602d","page_size":[595,841],"block_position":"34-28"},{"bbox":[292,803,303,812],"type":"page_number","angle":0,"index":1000017,"text":"30 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":34,"is_discarded":true,"id":"d88ded27-c6e9-4ca5-b9bc-0ad10a10468d","page_size":[595,841],"block_position":"34-29"}],[{"bbox":[439,47,539,58],"type":"header","angle":0,"index":-99999,"text":"2.6 特殊矩阵性质总结 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":35,"is_discarded":true,"id":"dd869807-972d-494a-b07b-15f715d46c70","page_size":[595,841],"block_position":"35-0"},{"bbox":[52,71,539,100],"type":"text","angle":0,"index":1,"merge_prev":false,"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 $\\pm 1$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"b2a83c6c-035d-4296-aaa6-1a5f18ecc637","page_size":[595,841],"block_position":"35-1"},{"bbox":[73,102,343,116],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"(2) $|A|=-1$ 情况：根据 $|A|$ 是特征值乘积可知至少有 1 个 -1 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"51da4464-7549-42bc-8e77-02c27834ee3a","page_size":[595,841],"block_position":"35-2"},{"bbox":[73,118,173,131],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"(3) $|A|=1$ 情况：同理 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"f57d442f-b98d-4b97-b19f-6fc73f2168ae","page_size":[595,841],"block_position":"35-3"},{"bbox":[53,132,539,161],"type":"text","angle":0,"index":4,"merge_prev":false,"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)。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"3f1f9814-c8de-4352-8484-bc62b16f442a","page_size":[595,841],"block_position":"35-4"},{"bbox":[73,170,224,183],"type":"title","angle":0,"index":5,"level":2,"text":"## 定理 2.49 (n 阶正交矩阵的标准型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"9e3f6262-288c-4526-baea-13f81d6de3f5","page_size":[595,841],"block_position":"35-5"},{"bbox":[68,187,255,199],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"$A$ 是 $n$ 阶正交矩阵，则 $A$ 总是正交相似于 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"7dafb0c4-e0b5-46c3-b6fd-4aa0187227bb","page_size":[595,841],"block_position":"35-6"},{"bbox":[145,201,447,235],"type":"interline_equation","angle":0,"index":7,"text":"$$\n\\operatorname{diag} \\left\\{\\lambda_ {1}, \\dots , \\lambda_ {r}, \\left( \\begin{array}{c c} \\cos \\theta_ {1} & - \\sin \\theta_ {1} \\\\ \\sin \\theta_ {1} & \\cos \\theta_ {1} \\end{array} , \\dots , \\left( \\begin{array}{c c} \\cos \\theta_ {m} & - \\sin \\theta_ {m} \\\\ \\sin \\theta_ {m} & \\cos \\theta_ {m} \\end{array} \\right)\\right) \\right\\}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":35,"id":"867f58b3-38e2-4ad5-bb2c-9b6c36956e55","page_size":[595,841],"block_position":"35-7"},{"bbox":[57,268,157,282],"type":"title","angle":0,"index":8,"level":2,"text":"## 2.6.7 Unitary 矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"d9f2454f-3aca-4e01-a708-0f2f3e960f94","page_size":[595,841],"block_position":"35-8"},{"bbox":[73,295,211,307],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"定理 2.50 (Unitary 矩阵的性质) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"f2c6be88-3276-46ab-8bd0-7ddef91a4ce5","page_size":[595,841],"block_position":"35-9"},{"bbox":[69,312,212,324],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"Unitary 矩阵的特征值模长均为 1 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"07081656-d6dd-4d54-8c3a-a30767d99750","page_size":[595,841],"block_position":"35-10"},{"bbox":[57,357,138,371],"type":"title","angle":0,"index":11,"level":2,"text":"## 2.6.8 幂零矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"190f66ed-0b0e-4614-aed0-545a676cae60","page_size":[595,841],"block_position":"35-11"},{"bbox":[73,384,166,396],"type":"title","angle":0,"index":12,"level":2,"text":"## 定义 2.17 (幂零矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"9aeca6b6-b1fe-477e-a384-f1566a9e465c","page_size":[595,841],"block_position":"35-12"},{"bbox":[69,401,287,415],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"若存在整数 l，使得 $A^{l}=O$ ，则称 A 为幂零矩阵。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"fd68d1ef-c6a0-414a-a411-2e9eef8c51f6","page_size":[595,841],"block_position":"35-13"},{"bbox":[73,435,205,448],"type":"title","angle":0,"index":14,"level":2,"text":"## 定理 2.51 (幂零矩阵充要条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"87f1ddcd-6f94-4e7f-b687-23d7952744d0","page_size":[595,841],"block_position":"35-14"},{"bbox":[69,452,170,464],"type":"title","angle":0,"index":15,"level":2,"text":"## $A$ 是幂零矩阵当且仅当","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"7ffabbf0-6a81-45e8-ad21-5a5e46767d44","page_size":[595,841],"block_position":"35-15"},{"bbox":[81,468,215,479],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"1. 特征值: $A$ 的特征值只有 0 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"c43755b7-8039-43de-ac13-8408f9f16b70","page_size":[595,841],"block_position":"35-16"},{"bbox":[80,483,249,494],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"2. 特征多项式: $A$ 的特征多项式为 $x^{n}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"eda1499e-982e-40aa-a7b1-970861fbc35b","page_size":[595,841],"block_position":"35-17"},{"bbox":[80,498,279,510],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"3. 最小多项式： $A$ 的最小多项式为 $x^{k}, k \\leq n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"29d496da-f100-47da-a66a-a40787b1cc04","page_size":[595,841],"block_position":"35-18"},{"bbox":[80,514,244,527],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"4. 迹 (重点): 对 $\\forall k > 0$ , $\\operatorname{tr}(A^{k}) = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"9e62dd82-0824-4216-8566-f045c9c1b951","page_size":[595,841],"block_position":"35-19"},{"bbox":[57,560,138,573],"type":"title","angle":0,"index":20,"level":2,"text":"## 2.6.9 幂等矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"2a895b36-66c3-450c-9e14-0beee38fce68","page_size":[595,841],"block_position":"35-20"},{"bbox":[73,587,165,599],"type":"title","angle":0,"index":21,"level":2,"text":"## 定义 2.18 (幂等矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"53042536-248f-49c0-b997-b4b40156810e","page_size":[595,841],"block_position":"35-21"},{"bbox":[69,604,282,617],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"若 $n$ 阶矩阵 $A$ 满足 $A^2 = A$ ，则称 $A$ 为幂等矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"74c041dd-d661-4fa9-9802-9b58b4dd075b","page_size":[595,841],"block_position":"35-22"},{"bbox":[73,637,205,650],"type":"title","angle":0,"index":23,"level":2,"text":"## 定理 2.52 (幂等矩阵充要条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"4430a6c1-feee-407e-b8cc-945ee5c30b14","page_size":[595,841],"block_position":"35-23"},{"bbox":[68,654,307,666],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"$n$ 阶方阵 $A$ 是幂等矩阵当且仅当 $r(A) + r(A - E) = n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"c7d28cbd-f5c9-4a25-aa01-ad988608d8d0","page_size":[595,841],"block_position":"35-24"},{"bbox":[54,684,515,697],"type":"text","angle":0,"index":25,"merge_prev":false,"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） ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"088078fc-c94d-4f72-afc6-a5df37608542","page_size":[595,841],"block_position":"35-25"},{"bbox":[230,704,362,717],"type":"interline_equation","angle":0,"index":27,"text":"$$\nn = n - r (A) + n - r (A - E)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":35,"id":"0f4540c7-8989-460a-bac6-4aa0c6a05981","page_size":[595,841],"block_position":"35-26"},{"bbox":[55,724,171,735],"type":"text","angle":0,"index":28,"merge_prev":false,"text":"因此 $r(A) + r(A - E) = n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":35,"id":"a1c714fd-92fd-41da-9e1a-fe7eb57af908","page_size":[595,841],"block_position":"35-27"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":1000025,"text":"31 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":35,"is_discarded":true,"id":"d1675964-5334-44ea-9a1d-f40dd817dc9f","page_size":[595,841],"block_position":"35-28"}],[{"bbox":[439,47,539,58],"type":"header","angle":0,"index":-99999,"text":"2.6 特殊矩阵性质总结 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":36,"is_discarded":true,"id":"e715deb3-4455-495a-bf13-cfc3451918b4","page_size":[595,841],"block_position":"36-0"},{"bbox":[73,74,195,85],"type":"title","angle":0,"index":1,"level":2,"text":"## 定理 2.53 (幂等矩阵的性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"c29f5eb6-c2cc-4b69-9eff-abf65424ec23","page_size":[595,841],"block_position":"36-1"},{"bbox":[69,90,173,101],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"$A$ 为 $n$ 阶幂等矩阵，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"71d79546-2acd-421e-8214-fa0cf3c683dd","page_size":[595,841],"block_position":"36-2"},{"bbox":[83,105,195,117],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"- $A$ 的特征值只有1和0 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"64c00baf-6c4f-400a-a487-ca82d3f34eb7","page_size":[595,841],"block_position":"36-3"},{"bbox":[83,121,290,134],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"- $A$ 一定可以对角化，且标准型为 $\\mathrm{diag}\\{E_r, O\\}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"72289b67-1943-4ed7-8cb6-9485e53bc4b4","page_size":[595,841],"block_position":"36-4"},{"bbox":[83,137,153,149],"type":"interline_equation","angle":0,"index":5,"text":"$$\nr (A) = \\operatorname{tr} (A)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":36,"id":"9088cb4b-ae34-40f3-8d1c-8c7e0829421e","page_size":[595,841],"block_position":"36-5"},{"bbox":[54,166,493,179],"type":"text","angle":0,"index":6,"merge_prev":false,"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 $\\lambda = 0,1$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"906c0464-ded5-4808-aa80-21be431c19a4","page_size":[595,841],"block_position":"36-6"},{"bbox":[73,181,503,195],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"a72104af-0fa0-4a0f-90a5-6d51e89b549a","page_size":[595,841],"block_position":"36-7"},{"bbox":[73,198,241,211],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"(3) 根据秩、迹在相似下的不变性可知 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"7180e9dd-a30b-44b4-abf0-3af08cf5bbb0","page_size":[595,841],"block_position":"36-8"},{"bbox":[73,232,205,245],"type":"title","angle":0,"index":9,"level":2,"text":"## 定理 2.54 (加减乘法的幂等性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"ed3388dd-0fb6-41d0-8807-4a950498f850","page_size":[595,841],"block_position":"36-9"},{"bbox":[69,250,205,261],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"$A, B$ 为两个 $n$ 阶幂等矩阵，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"8e9407ae-4bfe-4d60-825b-654fc3c64486","page_size":[595,841],"block_position":"36-10"},{"bbox":[80,265,285,277],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"1. $A + B$ 为幂等矩阵当且仅当 $AB = BA = O$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"c0589f87-b79f-48a2-94e7-526214d50a3b","page_size":[595,841],"block_position":"36-11"},{"bbox":[80,281,286,292],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"2. $A - B$ 为幂等矩阵当且仅当 $AB = BA = B$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"0a587f40-d0ca-46e2-a74c-b3bb5f93700f","page_size":[595,841],"block_position":"36-12"},{"bbox":[80,296,303,308],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"3. 若 $AB = BA$ ，则 $AB$ 为幂等矩阵。反之不一定 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"9ad998d9-4d19-4b95-8db1-03a41b10d763","page_size":[595,841],"block_position":"36-13"},{"bbox":[80,312,256,324],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"4. 如 E-A-B 可逆，则 $r(A)=r(B)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"215afc25-4249-44f7-beda-716ddabfc397","page_size":[595,841],"block_position":"36-14"},{"bbox":[52,341,539,370],"type":"text","angle":0,"index":15,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"6f582f01-37ef-4568-b343-0a2a5f9c1cfa","page_size":[595,841],"block_position":"36-15"},{"bbox":[185,376,408,389],"type":"interline_equation","angle":0,"index":16,"text":"$$\nA B = A B ^ {2} = (- B A) B = - B A B = B (B A) = B A\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":36,"id":"f4517710-6dc1-4afd-8faf-080b4ab86478","page_size":[595,841],"block_position":"36-16"},{"bbox":[35,412,427,423],"type":"text","angle":0,"index":17,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"9797e1bb-1624-422f-8937-2ef58ec9c798","page_size":[595,841],"block_position":"36-17"},{"bbox":[54,427,190,439],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"证明 (1) 利用等价标准型得到: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"7b89efc9-56e9-4578-bb12-e7d7084f21da","page_size":[595,841],"block_position":"36-18"},{"bbox":[189,442,404,475],"type":"interline_equation","angle":0,"index":19,"text":"$$\nA = P \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) Q = P Q Q ^ {- 1} \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) Q\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":36,"id":"deb4fa36-1933-42f6-b2b9-f454dab524de","page_size":[595,841],"block_position":"36-19"},{"bbox":[54,478,184,491],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"PQ 为可逆矩阵，后面为幂等 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"32203e1f-d73e-4083-912d-6da003ef3e1f","page_size":[595,841],"block_position":"36-20"},{"bbox":[74,513,205,525],"type":"title","angle":0,"index":21,"level":2,"text":"## 定理 2.55 (幂等矩阵列的性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"7b12ef9a-4cca-4a2c-b111-1cdcc8de9ede","page_size":[595,841],"block_position":"36-21"},{"bbox":[69,530,415,543],"type":"text","angle":0,"index":22,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"6f786288-3cbb-4d3f-bdb4-aa8c523319d1","page_size":[595,841],"block_position":"36-22"},{"bbox":[83,545,200,558],"type":"text","angle":0,"index":23,"merge_prev":false,"text":"- $A_{1}, \\cdots, A_{m}$ 为幂等矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"bc750ba5-3d3e-4a3e-ac50-787b2cc3d5b8","page_size":[595,841],"block_position":"36-23"},{"bbox":[83,561,202,574],"type":"interline_equation","angle":0,"index":24,"text":"$$\n\\bullet \\mathrm{r} (A _ {1}) + \\dots + \\mathrm{r} (A _ {m}) = n\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":36,"id":"b3c2a7b2-4ceb-45e9-a110-931403525a99","page_size":[595,841],"block_position":"36-24"},{"bbox":[83,577,180,590],"type":"text","angle":0,"index":25,"merge_prev":false,"text":"- $\\forall i \\neq j$ 有 $A_{i}A_{j} = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"927341c5-0ae8-4d52-8a19-24819fa9da21","page_size":[595,841],"block_position":"36-25"},{"bbox":[54,607,249,619],"type":"text","angle":0,"index":26,"merge_prev":false,"text":"证明 $(1) \\Rightarrow (2)$ : 由于 $r(A_{i}) = \\operatorname{tr}(A_{i})$ , 因此 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"d1ce1bee-8167-4e1f-9c56-ecefc334e4ba","page_size":[595,841],"block_position":"36-26"},{"bbox":[120,627,473,640],"type":"interline_equation","angle":0,"index":27,"text":"$$\nr \\left(A _ {1}\\right) + \\dots + r \\left(A _ {m}\\right) = \\operatorname{tr} \\left(A _ {1}\\right) + \\dots + \\operatorname{tr} \\left(A _ {m}\\right) = \\operatorname{tr} \\left(A _ {1} + \\dots + A _ {m}\\right) = \\operatorname{tr} \\left(E _ {n}\\right) = n\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":36,"id":"59272a34-2c83-4d0c-bdd7-9dcfb08f9caa","page_size":[595,841],"block_position":"36-27"},{"bbox":[73,646,332,658],"type":"text","angle":0,"index":29,"merge_prev":false,"text":"(2) $\\Leftarrow$ (3): 比较经典，先通过分块矩阵的初等变换凑出 E: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":36,"id":"dda58024-0957-40f4-8085-ce90fa1d99a7","page_size":[595,841],"block_position":"36-28"},{"bbox":[90,662,502,744],"type":"interline_equation","angle":0,"index":30,"text":"$$\n\\left( \\begin{array}{c c c c c} A _ {1} & & & & \\\\ & A _ {2} & & & \\\\ & & A _ {3} & & \\\\ & & & \\ddots & \\\\ & & & & A _ {m} \\end{array} \\right) \\left( \\begin{array}{c c c c c} A _ {1} & A _ {2} & A _ {3} & \\dots & A _ {m} \\\\ A _ {2} & & & & \\\\ & & A _ {3} & & \\\\ & & & \\ddots & \\\\ & & & & A _ {m} \\end{array} \\right) \\to \\left( \\begin{array}{c c c c c} E & A _ {2} & A _ {3} & \\dots & A _ {m} \\\\ A _ {2} & A _ {2} & & \\\\ A _ {3} & & A _ {3} & \\\\ \\vdots & & & \\ddots \\\\ A _ {m} & & & A _ {m} \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":36,"id":"86089300-2baf-4492-b0c8-d8b136d4f2b0","page_size":[595,841],"block_position":"36-29"},{"bbox":[291,802,303,812],"type":"page_number","angle":0,"index":1000027,"text":"32 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":36,"is_discarded":true,"id":"e30dfa64-c6b4-4b2a-af77-2834e5068fce","page_size":[595,841],"block_position":"36-30"}],[{"bbox":[439,47,539,58],"type":"header","angle":0,"index":-99999,"text":"2.6 特殊矩阵性质总结 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":37,"is_discarded":true,"id":"c606b242-07f9-432c-abb9-4a623b701f8e","page_size":[595,841],"block_position":"37-0"},{"bbox":[54,72,164,84],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"再通过 E 消去其他部分: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":37,"id":"15a6977c-643a-4101-b13a-d9494dca17f2","page_size":[595,841],"block_position":"37-1"},{"bbox":[76,88,515,169],"type":"interline_equation","angle":0,"index":2,"text":"$$\n\\rightarrow \\left(\\begin{array}{c c c c c}E&O&O&\\dots&O\\\\A _ {2}&A _ {2} - A _ {2} ^ {2}&- A _ {2} A _ {3}&\\dots&- A _ {2} A _ {m}\\\\A _ {3}&- A _ {3} A _ {2}&A _ {3} - A _ {3} ^ {2}&\\dots&- A _ {3} A _ {m}\\\\\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\A _ {m}&- A _ {m} A _ {2}&- A _ {m} A _ {3}&\\dots&A _ {m} - A _ {m} ^ {2}\\end{array}\\right)\\rightarrow \\left(\\begin{array}{c c c c c}E&O&O&\\dots&O\\\\O&A _ {2} - A _ {2} ^ {2}&- A _ {2} A _ {3}&\\dots&- A _ {2} A _ {m}\\\\O&- A _ {3} A _ {2}&A _ {3} - A _ {3} ^ {2}&\\dots&- A _ {3} A _ {m}\\\\\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\O&- A _ {m} A _ {2}&- A _ {m} A _ {3}&\\dots&A _ {m} - A _ {m} ^ {2}\\end{array}\\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":37,"id":"a76c5346-dd7c-460b-9067-b1a4c0140048","page_size":[595,841],"block_position":"37-2"},{"bbox":[54,173,254,185],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"根据 $r(A_{1}) + r(A_{2}) + \\cdots + r(A_{m}) = n$ ，得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":37,"id":"48baa499-a25a-4a6d-97ca-13b362b46966","page_size":[595,841],"block_position":"37-3"},{"bbox":[193,189,401,255],"type":"interline_equation","angle":0,"index":4,"text":"$$\n\\left( \\begin{array}{c c c c} A _ {2} - A _ {2} ^ {2} & - A _ {2} A _ {3} & \\dots & - A _ {2} A _ {m} \\\\ - A _ {3} A _ {2} & A _ {3} - A _ {3} ^ {2} & \\dots & - A _ {3} A _ {m} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ - A _ {m} A _ {2} & - A _ {m} A _ {3} & \\dots & A _ {m} - A _ {m} ^ {2} \\end{array} \\right) = O\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":37,"id":"61da64f4-78f4-412d-b6cd-e6559d2b715c","page_size":[595,841],"block_position":"37-4"},{"bbox":[54,259,513,291],"type":"text","angle":0,"index":5,"merge_prev":false,"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 $A_{1}A_{i} + A_{i} = A_{i}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":37,"id":"444f4971-f105-4efe-b044-488edc0fd37f","page_size":[595,841],"block_position":"37-5"},{"bbox":[54,298,174,310],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"因此 $A_{1}A_{i} = O$ ，结论成立 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":37,"id":"d9458576-3693-40fe-90f4-bd61ec1a69d5","page_size":[595,841],"block_position":"37-6"},{"bbox":[73,314,328,327],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"(3) $\\Rightarrow(1)$ ：直接对 $A_{1}+\\cdots+A_{m}=E_{n}$ 两边右乘 $A_{i}$ 得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":37,"id":"204a9966-95e1-4bad-b075-63a271e2ec51","page_size":[595,841],"block_position":"37-7"},{"bbox":[210,333,383,346],"type":"interline_equation","angle":0,"index":8,"text":"$$\nA _ {1} A _ {i} + A _ {2} A _ {i} + \\dots + A _ {m} A _ {i} = A _ {i} ^ {2} = A _ {i}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":37,"id":"92bc28d9-a4a5-4c3b-9788-5d5c1bbfc4f8","page_size":[595,841],"block_position":"37-8"},{"bbox":[54,353,138,364],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"因此都是幂等矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":37,"id":"cef05f7b-db1e-4046-b13b-8d5bad435b86","page_size":[595,841],"block_position":"37-9"},{"bbox":[35,363,48,379],"type":"image","angle":0,"index":10,"img_path":"/116dae9c4751e133d34c8783fe26e5db95777c31f641faaaf1b2315ed7ef9957.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":37,"id":"2d456bd9-9831-4815-bcce-0d587041deb0","page_size":[595,841],"block_position":"37-10"},{"bbox":[52,368,192,380],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"笔记 此定理在 ZJU2020 中考过 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":37,"id":"69daa107-7fc7-42ed-8eeb-5c114b6ef1a0","page_size":[595,841],"block_position":"37-11"},{"bbox":[55,399,169,412],"type":"title","angle":0,"index":12,"level":2,"text":"## 2.6.10 对角占优矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":37,"id":"12ae30e8-1396-49e9-a5ad-2f0a5710091c","page_size":[595,841],"block_position":"37-12"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":1000012,"text":"33 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":37,"is_discarded":true,"id":"ba70899e-6e9d-408f-8a77-1fff1c6c803e","page_size":[595,841],"block_position":"37-13"}],[{"bbox":[245,74,359,94],"type":"title","angle":0,"index":0,"level":2,"text":"## 第3章 行列式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"39ad09e9-e507-4db7-aa37-ec0c88a43949","page_size":[595,841],"block_position":"38-0"},{"bbox":[56,122,229,139],"type":"title","angle":0,"index":1,"level":2,"text":"## 3.1 行列式基本概念与性质","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"9cde0195-896b-4879-956f-d5f092140088","page_size":[595,841],"block_position":"38-1"},{"bbox":[73,157,190,169],"type":"title","angle":0,"index":2,"level":2,"text":"## 定理 3.1 (行列式初等变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"63b76e7c-7a9e-4ab4-8f7a-e48fef0bd433","page_size":[595,841],"block_position":"38-2"},{"bbox":[69,174,226,185],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"矩阵初等变换相对行列式的变换为: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"36cdb3dd-1fe1-4473-aa24-b18fae346298","page_size":[595,841],"block_position":"38-3"},{"bbox":[83,190,189,201],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"- 互换两行/列：取反号 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"b93932d0-dd48-43d8-a98d-d9a731bb40e2","page_size":[595,841],"block_position":"38-4"},{"bbox":[83,205,210,217],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"- 同行/列可同时提出公因式 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"02a3fbd1-6f9e-4a15-a97d-7916ede40679","page_size":[595,841],"block_position":"38-5"},{"bbox":[83,221,207,232],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"- 倍加：不改变行列式大小 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"7485ca2f-3c1f-49cb-986c-ef26f21edf04","page_size":[595,841],"block_position":"38-6"},{"bbox":[74,253,170,265],"type":"title","angle":0,"index":7,"level":2,"text":"## 定理 3.2 (行列式分拆)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"ee766268-cfee-4970-b7ce-59449646ef9e","page_size":[595,841],"block_position":"38-7"},{"bbox":[70,269,261,281],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"行列式可以关于一行或者一列进行分拆，即 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"ea3c960d-cba8-4087-8545-09cee0d67894","page_size":[595,841],"block_position":"38-8"},{"bbox":[95,285,500,373],"type":"interline_equation","angle":0,"index":9,"text":"$$\n\\left| \\begin{array}{c c c c} a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ b _ {s 1} + c _ {s 1} & b _ {s 2} + c _ {s 2} & \\dots & b _ {s n} + c _ {s n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right| = \\left| \\begin{array}{c c c c} a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ b _ {s 1} & b _ {s 2} & \\dots & b _ {s n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right| + \\left| \\begin{array}{c c c c} a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ c _ {s 1} & c _ {s 2} & \\dots & c _ {s n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right|.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":38,"id":"2be09833-f069-459a-9952-17a14f5a4946","page_size":[595,841],"block_position":"38-9"},{"bbox":[57,409,157,426],"type":"title","angle":0,"index":10,"level":2,"text":"## 3.2 行列式展开","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"0b0ca14e-907b-4ee2-b967-843d5aafc0b9","page_size":[595,841],"block_position":"38-10"},{"bbox":[73,443,211,456],"type":"title","angle":0,"index":11,"level":2,"text":"## 定义 3.1 (余子式与代数余子式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"b7e81d8a-805c-41bd-9969-f5bdf52da29b","page_size":[595,841],"block_position":"38-11"},{"bbox":[68,460,511,476],"type":"text","angle":0,"index":12,"merge_prev":false,"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 $A_{ij} := (-1)^{i+j} M_{ij}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"ed8c9810-23b5-477e-9613-d7e70208575a","page_size":[595,841],"block_position":"38-12"},{"bbox":[73,496,170,508],"type":"title","angle":0,"index":13,"level":2,"text":"## 定理 3.3 (行列式展开)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"f07be3ae-b83e-4072-af81-9bd1cf72cfd1","page_size":[595,841],"block_position":"38-13"},{"bbox":[69,512,437,545],"type":"text","angle":0,"index":14,"merge_prev":false,"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 $|A| = \\sum_{j=1}^{n} a_{ij} A_{ij}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"3b55f813-be70-4445-9ea3-bbe1fadfcf4c","page_size":[595,841],"block_position":"38-14"},{"bbox":[74,565,188,579],"type":"title","angle":0,"index":15,"level":2,"text":"## 定理 3.4 (不同行相乘为 0)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"0f2c1c79-2f3f-4fa8-9bc6-76c8c4521ae4","page_size":[595,841],"block_position":"38-15"},{"bbox":[68,582,351,615],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"$A$ 的第 $i$ 行元素与 $k$ 行的代数余子式乘积和为0: $\\sum_{j=1}^{n} a_{ij} A_{kj} = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"90af98bc-3d22-41bf-9fbc-9bbe6bc54261","page_size":[595,841],"block_position":"38-16"},{"bbox":[57,652,172,669],"type":"title","angle":0,"index":17,"level":2,"text":"## 3.3 特殊的行列式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"1cca9040-5a27-492f-b1e5-c6aa1c8792ab","page_size":[595,841],"block_position":"38-17"},{"bbox":[56,685,151,700],"type":"title","angle":0,"index":18,"level":2,"text":"## 3.3.1 爪型行列式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"dbf9961f-4f32-4ab2-9a25-9e09c62f07f2","page_size":[595,841],"block_position":"38-18"},{"bbox":[73,713,180,725],"type":"title","angle":0,"index":19,"level":2,"text":"## 定理 3.5 (正爪型行列式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"3f8a2799-2d84-424e-981b-a9c5036fa574","page_size":[595,841],"block_position":"38-19"},{"bbox":[70,729,330,743],"type":"text","angle":0,"index":20,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":38,"id":"b1c4672c-6b1e-4137-a955-3fc2ebc48706","page_size":[595,841],"block_position":"38-20"}],[{"bbox":[459,47,539,58],"type":"header","angle":0,"index":-99999,"text":"3.3 特殊的行列式 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":39,"is_discarded":true,"id":"1485d6d6-d96e-4be9-a4a1-786d7658f051","page_size":[595,841],"block_position":"39-0"},{"bbox":[73,73,181,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 定理 3.6 (异爪型行列式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"6ff16bed-7d35-4f2a-89b8-89a0bea1688e","page_size":[595,841],"block_position":"39-1"},{"bbox":[70,89,257,101],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"异爪型行列式(一横两斜)分以下几种情况: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"bbb693ae-0ba6-4b81-bf47-b3fcb51f23ee","page_size":[595,841],"block_position":"39-2"},{"bbox":[83,105,296,117],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"- 行和为0型：全部加到第一行，按第一列展开 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"831aa405-a49b-4746-8637-782cf7a24035","page_size":[595,841],"block_position":"39-3"},{"bbox":[83,121,327,133],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"- 两对角各自相等型：按斜直夹边展开，获得递推公式 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"20c4f294-0128-4e5d-bf7d-77fd28fc5c43","page_size":[595,841],"block_position":"39-4"},{"bbox":[35,146,350,225],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"练习3.1异爪型行列式(1)计算 $|A| = \\left| \\begin{array}{cccccc}0 & -1 & -1 & \\dots & -1 & -1\\\\ 1 & 1 & 0 & \\dots & 0 & 0\\\\ 0 & 1 & 1 & \\dots & 0 & 0\\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 0 & 0 & 0 & \\dots & 1 & 1 \\end{array} \\right|$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"9adcf916-043d-42e8-9a5a-cdbf4afc82ae","page_size":[595,841],"block_position":"39-5"},{"bbox":[73,227,373,267],"type":"text","angle":0,"index":6,"merge_prev":false,"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 $|A|$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"0a642a69-505b-40b2-b532-14686356ec87","page_size":[595,841],"block_position":"39-6"},{"bbox":[53,269,492,283],"type":"text","angle":0,"index":7,"merge_prev":false,"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 $D_{n} = 1$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"0c01ef87-0add-47d8-9602-901405de47b1","page_size":[595,841],"block_position":"39-7"},{"bbox":[73,285,176,297],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"(2) 上下相减后变为 (1) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"25905b90-8a6a-4d77-9ff6-c328bba2d572","page_size":[595,841],"block_position":"39-8"},{"bbox":[55,316,185,330],"type":"title","angle":0,"index":9,"level":2,"text":"## 3.3.2 大对角型 (三斜线)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"975c03a5-1c35-43e6-b367-5769d11e0d3d","page_size":[595,841],"block_position":"39-9"},{"bbox":[55,340,151,355],"type":"title","angle":0,"index":10,"level":2,"text":"## 3.3.3 行和相同型","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"77e2647b-0c99-4f3c-921c-62e32dee7221","page_size":[595,841],"block_position":"39-10"},{"bbox":[73,368,170,380],"type":"title","angle":0,"index":11,"level":2,"text":"## 定理 3.7 (行和相同型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"35784efe-7189-4567-b577-c952ccd0d5a9","page_size":[595,841],"block_position":"39-11"},{"bbox":[69,385,226,396],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"行和相同型行列式分以下两种情况： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"126e59f8-521e-46fe-8ee7-bfa92a810bb1","page_size":[595,841],"block_position":"39-12"},{"bbox":[83,401,365,412],"type":"text","angle":0,"index":13,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"d6586276-a2c7-4e77-8a4c-d6c2c7be8191","page_size":[595,841],"block_position":"39-13"},{"bbox":[83,417,266,428],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"- 循环型：上下相减 (例题见循环型一节) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"417575a5-1e5c-4ee2-ac09-3cc1ff947a92","page_size":[595,841],"block_position":"39-14"},{"bbox":[35,489,218,502],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"练习 3.2 内部大量相同型 计算行列式 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"9f67c97f-178c-493f-b537-13ba4d9fa8f6","page_size":[595,841],"block_position":"39-15"},{"bbox":[220,456,309,536],"type":"interline_equation","angle":0,"index":16,"text":"$$\n\\left| \\begin{array}{c c c c c} a & b & b & \\dots & b \\\\ b & a & b & \\dots & b \\\\ b & b & a & \\dots & b \\\\ \\vdots & \\vdots & & & \\vdots \\\\ b & b & b & \\dots & a \\end{array} \\right|\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":39,"id":"bae6d0f1-e3da-4c78-b3cb-a80e3386f6b4","page_size":[595,841],"block_position":"39-16"},{"bbox":[53,538,495,550],"type":"text","angle":0,"index":17,"merge_prev":false,"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 $(a - b)^{n - 1}[a + (n - 1)b]$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"bf14e8fb-ecbf-4824-8529-4381a1efd80e","page_size":[595,841],"block_position":"39-17"},{"bbox":[56,568,127,582],"type":"title","angle":0,"index":18,"level":2,"text":"## 3.3.4 循环型","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"98266539-f119-4630-814f-889926bac184","page_size":[595,841],"block_position":"39-18"},{"bbox":[73,592,401,605],"type":"text","angle":0,"index":19,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"269159dd-0a42-4235-80dd-21af047ba0f0","page_size":[595,841],"block_position":"39-19"},{"bbox":[36,608,183,620],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"练习3.3循环型 计算 $|A|, |B|$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":39,"id":"b2a028cf-2a71-42d4-bb19-5bad70ddcb2b","page_size":[595,841],"block_position":"39-20"},{"bbox":[85,624,503,722],"type":"interline_equation","angle":0,"index":21,"text":"$$\n| A | = \\left| \\begin{array}{c c c c c c} 0 & 1 & 2 & \\dots & n - 2 & n - 1 \\\\ 1 & 0 & 1 & \\dots & n - 3 & n - 2 \\\\ 2 & 1 & 0 & \\dots & n - 4 & n - 3 \\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ n - 2 & n - 3 & n - 4 & \\dots & 0 & 1 \\\\ n - 1 & n - 2 & n - 3 & \\dots & 1 & 0 \\end{array} \\right| \\quad | B | = \\left| \\begin{array}{c c c c c c} 1 & 2 & 3 & \\dots & n - 1 & n \\\\ n & 1 & 2 & \\dots & n - 2 & n - 1 \\\\ n - 1 & n & 1 & \\dots & n - 3 & n - 2 \\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 3 & 4 & 5 & \\dots & 1 & 2 \\\\ 2 & 3 & 4 & \\dots & n & 1 \\end{array} \\right|\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":39,"id":"9e52fc9d-4260-4455-92b4-2092ecd5f158","page_size":[595,841],"block_position":"39-21"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":1000021,"text":"35 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":39,"is_discarded":true,"id":"ac0ee9e7-d9ad-4496-b5eb-a6cfce8026f1","page_size":[595,841],"block_position":"39-22"}],[{"bbox":[467,47,539,58],"type":"header","angle":0,"index":-99999,"text":"3.4 Cramer 法则 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":40,"is_discarded":true,"id":"6c318e62-3050-4830-8bd0-11548930ca0d","page_size":[595,841],"block_position":"40-0"},{"bbox":[53,72,393,84],"type":"text","angle":0,"index":1,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":40,"id":"f3f7fa28-8f5b-4d50-9363-fdd753c33ecb","page_size":[595,841],"block_position":"40-1"},{"bbox":[214,88,380,185],"type":"interline_equation","angle":0,"index":2,"text":"$$\n\\left| \\begin{array}{c c c c c c} - 1 & 1 & 1 & \\dots & 1 & 1 \\\\ - 1 & - 1 & 1 & \\dots & 1 & 1 \\\\ - 1 & - 1 & - 1 & \\dots & 1 & 1 \\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ - 1 & - 1 & - 1 & \\dots & - 1 & 1 \\\\ n - 1 & n - 2 & n - 3 & \\dots & 1 & 0 \\end{array} \\right|\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":40,"id":"0cd5747f-aa10-41f7-af9a-ea7906dcaf3c","page_size":[595,841],"block_position":"40-2"},{"bbox":[73,189,373,200],"type":"text","angle":0,"index":3,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":40,"id":"7397aa44-020f-4a2c-9f15-8de2354b05f6","page_size":[595,841],"block_position":"40-3"},{"bbox":[56,220,163,233],"type":"title","angle":0,"index":4,"level":2,"text":"## 3.3.5 行列式加边法","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":40,"id":"460e8420-72bc-48d2-8eb2-d512418fe1a6","page_size":[595,841],"block_position":"40-4"},{"bbox":[73,247,181,259],"type":"title","angle":0,"index":5,"level":2,"text":"## 定理 3.8 (行列式加边法)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":40,"id":"6c6b0869-e595-4cad-b855-d29bb304498d","page_size":[595,841],"block_position":"40-5"},{"bbox":[69,264,257,275],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"计算矩阵 $A$ 的行列式 $|A|$ 时，可加上一边： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":40,"id":"b687405d-eb8a-4875-a934-f0b8876e5b79","page_size":[595,841],"block_position":"40-6"},{"bbox":[164,282,423,362],"type":"interline_equation","angle":0,"index":7,"text":"$$\n| A | = \\left| \\begin{array}{c c c c} a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ a _ {2 1} & a _ {2 2} & \\dots & a _ {2 n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right| = \\left| \\begin{array}{c c c c c} 1 & 1 & 1 & 1 & 1 \\\\ 0 & a _ {1 1} & a _ {1 2} & \\dots & a _ {1 n} \\\\ 0 & a _ {2 1} & a _ {2 2} & \\dots & a _ {2 n} \\\\ \\vdots & \\vdots & \\vdots & & \\vdots \\\\ 0 & a _ {n 1} & a _ {n 2} & \\dots & a _ {n n} \\end{array} \\right|\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":40,"id":"2fc8347a-059f-4dd8-ba4a-31dd051e8bec","page_size":[595,841],"block_position":"40-7"},{"bbox":[35,375,47,391],"type":"image","angle":0,"index":8,"img_path":"/5237bc175ab7cc857bef0b9fce8ebb278fd06b25162610870ed179f93aa53519.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":40,"id":"4ddc48f1-e343-4895-bb6b-142e057591fc","page_size":[595,841],"block_position":"40-8"},{"bbox":[52,380,343,392],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":40,"id":"05e4bf7a-1f05-43a1-9edf-1429f9614fee","page_size":[595,841],"block_position":"40-9"},{"bbox":[56,417,167,432],"type":"title","angle":0,"index":10,"level":2,"text":"## 3.4 Cramer 法则","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":40,"id":"fc639af8-6984-4db2-bbc8-b74fab5d7d36","page_size":[595,841],"block_position":"40-10"},{"bbox":[73,450,177,462],"type":"title","angle":0,"index":11,"level":2,"text":"## 定理 3.9 (Cramer 法则)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":40,"id":"07ad4911-fa5e-4cae-a410-37fc7e19bc59","page_size":[595,841],"block_position":"40-11"},{"bbox":[68,467,414,479],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":40,"id":"1821b66a-54be-4288-b023-be8f21697c85","page_size":[595,841],"block_position":"40-12"},{"bbox":[244,482,349,509],"type":"interline_equation","angle":0,"index":13,"text":"$$\n\\left(\\frac {| B _ {1} |}{| A |}, \\frac {| B _ {2} |}{| A |}, \\dots , \\frac {| B _ {n} |}{| A |}\\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":40,"id":"0c7a2b7d-941e-42d8-aadd-43b14e878a61","page_size":[595,841],"block_position":"40-13"},{"bbox":[69,512,257,523],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"其中 $B_{i}$ 为 $A$ 将第 $i$ 列换为 $B$ 组成的矩阵。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":40,"id":"d425a21a-5651-432e-9857-f75989c1a282","page_size":[595,841],"block_position":"40-14"},{"bbox":[292,803,303,812],"type":"page_number","angle":0,"index":1000014,"text":"36 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":40,"is_discarded":true,"id":"2edc9356-d568-4600-bc79-cfacf5fe8267","page_size":[595,841],"block_position":"40-15"}],[{"bbox":[227,74,377,94],"type":"title","angle":0,"index":0,"level":2,"text":"## 第 4 章 线性方程组","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"22d43c33-d132-4f5d-bc35-716cb8088fde","page_size":[595,841],"block_position":"41-0"},{"bbox":[56,122,244,138],"type":"title","angle":0,"index":1,"level":2,"text":"## 4.1 线性方程组解的判定定理","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"5344a427-b513-4e03-bb42-afc14aea40f9","page_size":[595,841],"block_position":"41-1"},{"bbox":[56,155,174,169],"type":"title","angle":0,"index":2,"level":2,"text":"## 4.1.1 解存在判定定理","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"727f21f9-efd4-4d3e-8799-ba6f733e287b","page_size":[595,841],"block_position":"41-2"},{"bbox":[72,183,190,195],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"定理 4.1 (解存在充要条件) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"3bbc293d-c090-43b4-b771-0f4ee3d6652c","page_size":[595,841],"block_position":"41-3"},{"bbox":[68,200,392,213],"type":"text","angle":0,"index":4,"merge_prev":false,"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 $\\overline{A} := [A, b]$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"91f5e0bc-ee38-48c0-9953-b83eed268d52","page_size":[595,841],"block_position":"41-4"},{"bbox":[56,245,174,260],"type":"title","angle":0,"index":5,"level":2,"text":"## 4.1.2 线性方程组同解","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"592e4ebf-5e8d-4859-9b75-218b3746b0ff","page_size":[595,841],"block_position":"41-5"},{"bbox":[73,273,161,285],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"定理 4.2 (同解定理) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"4db603ec-e81f-4471-ac5d-fcf4086a2e67","page_size":[595,841],"block_position":"41-6"},{"bbox":[68,290,411,303],"type":"text","angle":0,"index":7,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"394484ff-87fa-4e9d-b1c9-8011a138ae09","page_size":[595,841],"block_position":"41-7"},{"bbox":[35,335,401,348],"type":"text","angle":0,"index":8,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"3a3feac3-76c7-4a59-ae16-0e069fc1e7e5","page_size":[595,841],"block_position":"41-8"},{"bbox":[168,352,415,407],"type":"interline_equation","angle":0,"index":9,"text":"$$\n\\left\\{ \\begin{array}{l} x _ {1} + 2 x _ {2} + 3 x _ {3} = 0 \\\\ 2 x _ {1} + 3 x _ {2} + 5 x _ {3} = 0 \\\\ x _ {1} + x _ {2} + a x _ {3} = 0 \\end{array} \\right. \\quad \\left\\{ \\begin{array}{l} x _ {1} + b x _ {2} + c x _ {3} = 0 \\\\ 2 x _ {1} + b ^ {2} x _ {2} + (c + 1) x _ {3} = 0 \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":41,"id":"d5940937-0a44-4c44-b0ad-3b404140c154","page_size":[595,841],"block_position":"41-9"},{"bbox":[54,412,539,439],"type":"text","angle":0,"index":10,"merge_prev":false,"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 $B = CA$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"90bafcad-40ae-4f32-9d45-d542a12fe002","page_size":[595,841],"block_position":"41-10"},{"bbox":[53,439,475,470],"type":"text","angle":0,"index":11,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"5c91068c-96a1-40bc-a635-24cda754e9d7","page_size":[595,841],"block_position":"41-11"},{"bbox":[73,470,307,503],"type":"text","angle":0,"index":12,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"0b9465c1-63c4-4c76-b5be-f6e45ed80cb4","page_size":[595,841],"block_position":"41-12"},{"bbox":[57,519,174,534],"type":"title","angle":0,"index":13,"level":2,"text":"## 4.1.3 解个数判定定理","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"6230c238-b134-496e-9690-fef4de278cac","page_size":[595,841],"block_position":"41-13"},{"bbox":[73,547,190,560],"type":"title","angle":0,"index":14,"level":2,"text":"## 定理 4.3 (解个数判定定理)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"c58fb97b-9ba6-44ad-af3d-9ffdc6d18e20","page_size":[595,841],"block_position":"41-14"},{"bbox":[69,565,221,576],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"若 K 上 Ax = b 满足 $r(A) = r(\\overline{A})$ : ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"1de6282e-27cb-4c49-bfdc-6c9c51a50d3c","page_size":[595,841],"block_position":"41-15"},{"bbox":[83,581,244,592],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"若 $r(A) = n$ ：则 $Ax = b$ 有唯一解 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"58dd5738-8207-43e7-b516-dbbc244c5483","page_size":[595,841],"block_position":"41-16"},{"bbox":[83,596,244,608],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"若 $r(A) < n$ ：则 $Ax = b$ 有无穷解 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"cf15066b-4f65-41eb-ab55-49c06375f586","page_size":[595,841],"block_position":"41-17"},{"bbox":[57,645,201,661],"type":"title","angle":0,"index":18,"level":2,"text":"## 4.2 线性方程组的求解","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"acc5da59-dac1-49be-b0e5-74449037e35e","page_size":[595,841],"block_position":"41-18"},{"bbox":[57,679,155,692],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"4.2.1 $Ax = 0$ 求解 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"0d69a0ff-8334-4a37-bcde-2c9b75c8b41a","page_size":[595,841],"block_position":"41-19"},{"bbox":[73,706,170,719],"type":"title","angle":0,"index":20,"level":2,"text":"## 定义 4.1 (自由未知量)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"44bd0dac-99e1-4f7b-9f6d-1b7711438eba","page_size":[595,841],"block_position":"41-20"},{"bbox":[68,723,389,735],"type":"text","angle":0,"index":21,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":41,"id":"ffc6f7fb-9c9f-4e1b-838d-af6759d2e0f4","page_size":[595,841],"block_position":"41-21"}],[{"bbox":[439,47,539,58],"type":"header","angle":0,"index":-99998,"text":"4.2 线性方程组的求解 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":42,"is_discarded":true,"id":"d93e8b4e-c8d1-4bb0-a41d-dc8165ee0923","page_size":[595,841],"block_position":"42-0"},{"bbox":[35,248,47,264],"type":"image","angle":0,"index":0,"img_path":"/569bb7571fe1b96279850b2e831c71dd14f2853e8248c29c1c1dd0d70ae8dd6c.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":42,"id":"b395c1b0-1e26-4ea8-9f50-3d1e189cc908","page_size":[595,841],"block_position":"42-1"},{"bbox":[73,74,191,86],"type":"title","angle":0,"index":2,"level":2,"text":"## 定理 4.4 (自由变量的个数)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"050fa0dc-21a3-409d-89f3-947175da5eb6","page_size":[595,841],"block_position":"42-2"},{"bbox":[70,90,192,102],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"自由变量的个数为 $n - r(A)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"35d717fd-1d67-4f09-b599-8f66e4c02580","page_size":[595,841],"block_position":"42-3"},{"bbox":[73,123,160,136],"type":"title","angle":0,"index":4,"level":2,"text":"## 定义 4.2 (基础解系)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"f13db577-a3f5-45b8-94a8-ee0b5d750b58","page_size":[595,841],"block_position":"42-4"},{"bbox":[69,141,368,153],"type":"text","angle":0,"index":5,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"ed33a245-2795-4242-bb92-8ce2216db447","page_size":[595,841],"block_position":"42-5"},{"bbox":[225,158,358,233],"type":"interline_equation","angle":0,"index":6,"text":"$$\n\\left\\{ \\begin{array}{l} \\eta_ {1} = (*, \\dots , *, 1, 0, \\dots , 0) \\\\ \\eta_ {2} = (*, \\dots , *, 0, 1, \\dots , 0) \\\\ \\vdots \\\\ \\eta_ {n - r} = (*, \\dots , *, 0, 0, \\dots , 1) \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":42,"id":"ee7a68ce-950a-4a51-a509-c7692f0089d5","page_size":[595,841],"block_position":"42-6"},{"bbox":[52,252,470,264],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"39d077a5-3ba7-4252-a4a3-4edb534c74dc","page_size":[595,841],"block_position":"42-7"},{"bbox":[73,273,211,285],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"定理 4.5 (齐次线性方程组的解) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"687407b6-52dc-434e-8f28-1f240cfc9861","page_size":[595,841],"block_position":"42-8"},{"bbox":[69,290,328,302],"type":"text","angle":0,"index":9,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"a914441c-12e3-4015-b46a-88202cd9c491","page_size":[595,841],"block_position":"42-9"},{"bbox":[57,334,154,348],"type":"title","angle":0,"index":10,"level":2,"text":"## 4.2.2 $Ax = b$ 求解","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"f81424df-6edf-46d3-b99a-ea0263b0af94","page_size":[595,841],"block_position":"42-10"},{"bbox":[72,362,221,374],"type":"title","angle":0,"index":11,"level":2,"text":"## 定理 4.6 (非齐次线性方程组的解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"7a254c08-d848-4a18-87b7-25a3b34d1bbb","page_size":[595,841],"block_position":"42-11"},{"bbox":[67,377,536,412],"type":"text","angle":0,"index":12,"merge_prev":false,"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 $x = \\gamma_{0} + L\\{\\eta_{1}, \\cdots, \\eta_{n-r}\\}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"825d3f04-fa11-4bed-8c3b-91758e91c7d6","page_size":[595,841],"block_position":"42-12"},{"bbox":[35,439,343,452],"type":"text","angle":0,"index":13,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"befb10dc-7f8c-4b7d-a8e8-e4aa2c8465bc","page_size":[595,841],"block_position":"42-13"},{"bbox":[242,456,342,511],"type":"interline_equation","angle":0,"index":14,"text":"$$\n\\left\\{ \\begin{array}{l} x _ {1} + x _ {2} - x _ {3} = 1 \\\\ 2 x _ {1} + 3 x _ {2} + k x _ {3} = 3 \\\\ x _ {1} + k x _ {2} + 3 x _ {3} = 2 \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":42,"id":"3ad88d00-05e2-4284-832e-a42d72cc340d","page_size":[595,841],"block_position":"42-14"},{"bbox":[53,515,539,576],"type":"text","angle":0,"index":15,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"daebffbc-25d0-44de-b9c7-a956587bf337","page_size":[595,841],"block_position":"42-15"},{"bbox":[53,580,204,592],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"解(1)对增广矩阵做初等变换得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"12ff6df0-60e4-43dc-9c22-603bc708e36f","page_size":[595,841],"block_position":"42-16"},{"bbox":[223,591,370,640],"type":"interline_equation","angle":0,"index":18,"text":"$$\n\\left[ \\begin{array}{c c c c} 1 & 1 & - 1 & 1 \\\\ 0 & 1 & k + 2 & 1 \\\\ 0 & 0 & (k + 3) (k - 2) & k - 2 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":42,"id":"09bbfd08-7363-4e47-b9a4-e2dc6ce050d2","page_size":[595,841],"block_position":"42-17"},{"bbox":[54,647,539,675],"type":"text","angle":0,"index":19,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"9da016c0-8cd2-4070-a3b8-07ff12a04b3a","page_size":[595,841],"block_position":"42-18"},{"bbox":[201,673,390,724],"type":"interline_equation","angle":0,"index":20,"text":"$$\n\\gamma = \\left( \\begin{array}{c} 0 \\\\ 1 \\\\ 0 \\end{array} \\right), y = \\left( \\begin{array}{c} 5 \\\\ - 4 \\\\ 1 \\end{array} \\right) \\Rightarrow x = \\gamma + k _ {1} y\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":42,"id":"916d81d7-6e54-4d03-a36a-f72453fe8fc8","page_size":[595,841],"block_position":"42-19"},{"bbox":[54,730,191,741],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"唯一解情况直接解出结果即可。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":42,"id":"fa870b69-10fa-466a-913a-8e66f64ee6f3","page_size":[595,841],"block_position":"42-20"},{"bbox":[291,802,302,812],"type":"page_number","angle":0,"index":1000016,"text":"38 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":42,"is_discarded":true,"id":"7aeb9df4-6892-4c0c-b3f2-dddf0dc6bda5","page_size":[595,841],"block_position":"42-21"}],[{"bbox":[469,47,539,58],"type":"header","angle":0,"index":-99999,"text":"4.3 最小二乘解 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":43,"is_discarded":true,"id":"7ee5f43b-9c0f-40f2-acd1-a1214d92db57","page_size":[595,841],"block_position":"43-0"},{"bbox":[73,72,201,84],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"(2) 做增广矩阵初等变换得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"06fb0cec-bb78-497e-8224-c69519c294c5","page_size":[595,841],"block_position":"43-1"},{"bbox":[236,81,357,132],"type":"interline_equation","angle":0,"index":2,"text":"$$\n\\left[ \\begin{array}{c c c c} 1 & - 1 & - 1 & 1 \\\\ 0 & t + 4 & 5 & - 2 \\\\ 0 & 0 & t - 2 & 2 - t \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":43,"id":"399edb84-eec4-4dc4-8fdb-04b73bbdc3e4","page_size":[595,841],"block_position":"43-2"},{"bbox":[52,139,358,151],"type":"text","angle":0,"index":3,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"8945afd5-8b00-4117-92b9-964830a8975b","page_size":[595,841],"block_position":"43-3"},{"bbox":[55,169,223,184],"type":"title","angle":0,"index":4,"level":2,"text":"## 4.2.3 线性方程组公共解的计算","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"d37bd660-a68c-4151-97a0-0b26b7230872","page_size":[595,841],"block_position":"43-4"},{"bbox":[35,193,539,222],"type":"text","angle":0,"index":5,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"c75c0cd6-65fe-4348-9edc-54038b1cf71e","page_size":[595,841],"block_position":"43-5"},{"bbox":[241,225,343,263],"type":"interline_equation","angle":0,"index":6,"text":"$$\n\\left\\{ \\begin{array}{l} 7 x _ {1} - 6 x _ {2} + 3 x _ {3} = b \\\\ 8 x _ {1} - 9 x _ {2} + a x _ {4} = 7 \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":43,"id":"513bd57f-3966-414e-a3d7-31a4c89e9116","page_size":[595,841],"block_position":"43-6"},{"bbox":[52,266,539,293],"type":"text","angle":0,"index":7,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"f31a65fb-1dfa-415e-b438-a0ef983470d6","page_size":[595,841],"block_position":"43-7"},{"bbox":[167,293,417,331],"type":"interline_equation","angle":0,"index":8,"text":"$$\n\\left\\{ \\begin{array}{l} 4 k _ {1} - 4 k _ {2} = b - 1 \\\\ 8 k _ {1} + (a - 1 1) k _ {2} = 8 \\end{array} \\right. \\Rightarrow \\left\\{ \\begin{array}{l} 4 k _ {1} - 4 k _ {2} = b - 1 \\\\ 0 k _ {1} + (a - 3) k _ {2} = 1 0 - 2 b \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":43,"id":"b4f45949-8ccd-4dcf-988e-2cd2a9a4d971","page_size":[595,841],"block_position":"43-8"},{"bbox":[52,338,524,367],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"85db04dd-589e-4b6c-b198-94cffe4ec2f1","page_size":[595,841],"block_position":"43-9"},{"bbox":[55,389,158,405],"type":"title","angle":0,"index":10,"level":2,"text":"## 4.3 最小二乘解","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"f64ab296-4b71-43b0-8a87-8cd53f626baa","page_size":[595,841],"block_position":"43-10"},{"bbox":[55,421,154,436],"type":"title","angle":0,"index":11,"level":2,"text":"## 4.3.1 $A^T A$ 的研究","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"3a39f0e9-06a0-4500-b8cd-6d154af344b2","page_size":[595,841],"block_position":"43-11"},{"bbox":[72,449,188,463],"type":"title","angle":0,"index":12,"level":2,"text":"## 定理 4.7 ( $A^{T}A$ 与 A 的秩)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"9df432be-1985-4396-8194-c7aebcc7e631","page_size":[595,841],"block_position":"43-12"},{"bbox":[66,466,535,502],"type":"text","angle":0,"index":13,"merge_prev":false,"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 $r(A^{T}A) = r(A)$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"d8bc69e6-9fed-4ad2-af01-65654faf0910","page_size":[595,841],"block_position":"43-13"},{"bbox":[52,514,539,559],"type":"text","angle":0,"index":14,"merge_prev":false,"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 $r(A) = r(A^{T}A)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"e37e5c27-d6c6-48de-acdc-4ed9423bbf78","page_size":[595,841],"block_position":"43-14"},{"bbox":[52,560,495,590],"type":"text","angle":0,"index":15,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"d3540923-3684-457e-ba75-e47b3df0036d","page_size":[595,841],"block_position":"43-15"},{"bbox":[35,608,398,621],"type":"text","angle":0,"index":16,"merge_prev":false,"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： $r(AA^{T}A) = r(A)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"3dcb30b4-bfcb-4781-b900-b2b70fb9458b","page_size":[595,841],"block_position":"43-16"},{"bbox":[53,623,381,637],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"证明 (1) $r(AA^{T}A)\\leq r(A)$ ，而 $r(AA^{T}A)\\geq r(A^{T}AA^{T}A)=r(A^{T}A)=r(A)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"6e93aa4c-7dd6-4495-a272-086714dcbf3e","page_size":[595,841],"block_position":"43-17"},{"bbox":[56,655,151,668],"type":"title","angle":0,"index":18,"level":2,"text":"## 4.3.2 正规方程组","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"6e7a5f92-e99f-4392-b05f-c0e1d03c3b9e","page_size":[595,841],"block_position":"43-18"},{"bbox":[72,682,171,694],"type":"title","angle":0,"index":19,"level":2,"text":"## 定义 4.3 (最小二乘解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"2e5df389-2e8a-4e0a-a632-feae5eb5ce23","page_size":[595,841],"block_position":"43-19"},{"bbox":[68,698,434,711],"type":"text","angle":0,"index":20,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"1718209f-7c82-4d24-a3a5-1ea4a9fd68b8","page_size":[595,841],"block_position":"43-20"},{"bbox":[220,714,371,745],"type":"interline_equation","angle":0,"index":21,"text":"$$\nd = \\sum_ {i = 1} ^ {n} (a _ {i 1} x _ {1} + \\dots + a _ {i s} x _ {s} - b _ {i}) ^ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":43,"id":"fdc5046f-9aa1-4b3b-b539-ca8738e1e4d1","page_size":[595,841],"block_position":"43-21"},{"bbox":[69,748,215,761],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"最小，则称该向量为最小二乘解。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":43,"id":"95ba1526-7afb-4449-8e6f-f210429a74c9","page_size":[595,841],"block_position":"43-22"},{"bbox":[291,802,302,811],"type":"page_number","angle":0,"index":1000022,"text":"39 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":43,"is_discarded":true,"id":"bef6a4d4-6f5d-43fa-ae98-9efb91d3cd18","page_size":[595,841],"block_position":"43-23"}],[{"bbox":[469,47,539,58],"type":"header","angle":0,"index":-99998,"text":"4.3 最小二乘解 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":44,"is_discarded":true,"id":"28cfdbdb-e349-4a1e-9971-d751e7b16029","page_size":[595,841],"block_position":"44-0"},{"bbox":[35,320,47,335],"type":"image","angle":0,"index":0,"img_path":"/fc5927c4943f5711971e1159904e69cf3096da10f51513e01b79f3f32935c532.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":44,"id":"140c6b44-d72a-4931-89ec-0203833d0cd6","page_size":[595,841],"block_position":"44-1"},{"bbox":[73,74,171,85],"type":"title","angle":0,"index":2,"level":2,"text":"## 定理 4.8 (正规方程组)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":44,"id":"a17c1ed1-4b7c-4709-b731-dda96dda98a6","page_size":[595,841],"block_position":"44-2"},{"bbox":[69,90,480,103],"type":"text","angle":0,"index":3,"merge_prev":false,"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 $A^T A X = A^T b$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":44,"id":"445d1971-c952-4d21-a5e6-1e2bfa71b976","page_size":[595,841],"block_position":"44-3"},{"bbox":[54,119,540,148],"type":"text","angle":0,"index":4,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":44,"id":"9d5578ec-1254-465b-b3c8-ad631c40f46c","page_size":[595,841],"block_position":"44-4"},{"bbox":[179,154,414,168],"type":"interline_equation","angle":0,"index":5,"text":"$$\n\\alpha_ {i} ^ {T} (b - A X _ {0}) \\Rightarrow A ^ {T} (b - A X _ {0}) = 0 \\Rightarrow A ^ {T} A X _ {0} = A ^ {T} b\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":44,"id":"bd07f262-3a38-4655-8c46-eabf032e310f","page_size":[595,841],"block_position":"44-5"},{"bbox":[73,180,220,193],"type":"title","angle":0,"index":6,"level":2,"text":"## 定理 4.9 (正规方程组解的存在性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":44,"id":"f64af806-cd66-4a78-b74e-131b2be73c93","page_size":[595,841],"block_position":"44-6"},{"bbox":[69,198,213,210],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"方程组 $A^T A X = A^T b$ 一定有解。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":44,"id":"74dfb850-1b7e-431d-b962-fc17ab377465","page_size":[595,841],"block_position":"44-7"},{"bbox":[53,225,539,270],"type":"text","angle":0,"index":8,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":44,"id":"b6f7508a-8675-4bfd-b5b8-a00b9e1ce58c","page_size":[595,841],"block_position":"44-8"},{"bbox":[73,279,246,291],"type":"title","angle":0,"index":9,"level":2,"text":"## 定理 4.10 (凸闭集最佳逼近元的唯一性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":44,"id":"c4988838-4128-49d5-845d-7f8ef8f44bc0","page_size":[595,841],"block_position":"44-9"},{"bbox":[68,296,408,307],"type":"text","angle":0,"index":10,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":44,"id":"8d06c5f7-c8c1-4091-9567-37b87c4f872f","page_size":[595,841],"block_position":"44-10"},{"bbox":[52,324,273,337],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"笔记 一般而言最小二乘解不唯一，除非是凸闭集。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":44,"id":"1da1090a-3c10-4922-ae76-74567b6927cf","page_size":[595,841],"block_position":"44-11"},{"bbox":[291,803,303,811],"type":"page_number","angle":0,"index":1000011,"text":"40 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":44,"is_discarded":true,"id":"a03ea497-23ae-4daf-b049-4cbe7d6a56c0","page_size":[595,841],"block_position":"44-12"}],[{"bbox":[193,74,411,94],"type":"title","angle":0,"index":0,"level":1,"text":"# 第 5 章 线性空间与线性变换","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"723d18ae-66bc-4563-b291-d4cef53a92d7","page_size":[595,841],"block_position":"45-0"},{"bbox":[56,123,143,138],"type":"title","angle":0,"index":1,"level":2,"text":"## 5.1 线性空间","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"272f11de-ddb8-4483-99fb-1444b96c49cd","page_size":[595,841],"block_position":"45-1"},{"bbox":[55,155,198,169],"type":"title","angle":0,"index":2,"level":2,"text":"## 5.1.1 线性空间的基本概念","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"2709cf0c-2dc9-45da-865f-33d89ca3f1a7","page_size":[595,841],"block_position":"45-2"},{"bbox":[73,183,161,195],"type":"title","angle":0,"index":3,"level":2,"text":"## 定义 5.1 (线性空间)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"603be624-5f24-484b-86db-a6669cea6ee8","page_size":[595,841],"block_position":"45-3"},{"bbox":[67,199,437,211],"type":"text","angle":0,"index":4,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"f4825af0-46bc-4f1e-af76-6f579dec81f9","page_size":[595,841],"block_position":"45-4"},{"bbox":[83,216,290,227],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"二元运算：(1) 交换 (2) 结合 (3) 零元 (4) 负元 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"e2c91c66-9b09-427e-840e-afa26fa06634","page_size":[595,841],"block_position":"45-5"},{"bbox":[83,230,524,259],"type":"text","angle":0,"index":6,"merge_prev":false,"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 $c(\\alpha +\\beta) = c\\alpha +c\\beta$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"b09726d9-42ff-48e0-88e9-11c9c65b360d","page_size":[595,841],"block_position":"45-6"},{"bbox":[69,262,186,274],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"则称 $V$ 是 $\\mathbb{P}$ 上的线性空间 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"b1ddeaa3-1ebb-4b01-9de9-d047f023d565","page_size":[595,841],"block_position":"45-7"},{"bbox":[56,306,270,320],"type":"title","angle":0,"index":8,"level":2,"text":"## 5.1.2 线性相关、线性无关与向量组的秩","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"ec8644a8-c522-4387-aa31-a98613ca5493","page_size":[595,841],"block_position":"45-8"},{"bbox":[73,333,211,346],"type":"title","angle":0,"index":9,"level":2,"text":"## 定义 5.2 (线性相关和线性无关)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"2ddb8954-4530-4e29-bf38-1b614a1c38fc","page_size":[595,841],"block_position":"45-9"},{"bbox":[68,350,320,363],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"f3d123d9-6ffe-4133-83ff-129d9ae43e72","page_size":[595,841],"block_position":"45-10"},{"bbox":[229,370,364,383],"type":"interline_equation","angle":0,"index":11,"text":"$$\nk _ {1} \\alpha_ {1} + k _ {2} \\alpha_ {2} + \\dots + k _ {n} \\alpha_ {n} = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":45,"id":"c5c283a1-0513-4ecf-a88e-ed1910e9f556","page_size":[595,841],"block_position":"45-11"},{"bbox":[68,389,437,401],"type":"text","angle":0,"index":12,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"d2b8a6c1-fefa-47f5-a7e5-55c47802bf96","page_size":[595,841],"block_position":"45-12"},{"bbox":[35,433,539,495],"type":"text","angle":0,"index":13,"merge_prev":false,"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 $k_{0}, k_{1}, \\cdots, k_{n-1} = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"a64ad714-d4dd-4999-b620-8dc3d5c3e416","page_size":[595,841],"block_position":"45-13"},{"bbox":[73,515,221,528],"type":"title","angle":0,"index":14,"level":2,"text":"## 定理 5.1 (极大线性无关组的计算)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"cefb5ed0-672e-4ab3-b5e7-3729e6151cce","page_size":[595,841],"block_position":"45-14"},{"bbox":[67,533,525,576],"type":"text","angle":0,"index":15,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"c169b458-177e-422f-b3dc-d6e9ed791bde","page_size":[595,841],"block_position":"45-15"},{"bbox":[35,608,348,620],"type":"text","angle":0,"index":16,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":45,"id":"e417c58d-9bee-421c-a392-d3adce6a9f2c","page_size":[595,841],"block_position":"45-16"},{"bbox":[251,624,334,716],"type":"interline_equation","angle":0,"index":17,"text":"$$\n\\left\\{ \\begin{array}{l} \\alpha_ {1} = (1, - 1, 2, 4) \\\\ \\alpha_ {2} = (0, 3, 1, 2) \\\\ \\alpha_ {3} = (3, 0, 7, 1 4) \\\\ \\alpha_ {4} = (1, - 1, 2, 0) \\\\ \\alpha_ {5} = (2, 1, 5, 6) \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":45,"id":"fafbb8da-af6a-45b3-9cea-b73b3119b8cb","page_size":[595,841],"block_position":"45-17"}],[{"bbox":[480,47,539,58],"type":"header","angle":0,"index":-99999,"text":"5.1 线性空间 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":46,"is_discarded":true,"id":"72cb42c4-e6cb-4c66-a777-f707dbf25295","page_size":[595,841],"block_position":"46-0"},{"bbox":[54,72,214,84],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"解 拼成一个矩阵，并做初等行变换： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"7755f1e5-bfa3-41f9-982b-d3484684a25a","page_size":[595,841],"block_position":"46-1"},{"bbox":[187,86,406,149],"type":"interline_equation","angle":0,"index":2,"text":"$$\n\\left[\\begin{array}{c c c c c}1&0&3&1&2\\\\- 1&3&0&- 1&1\\\\2&1&7&2&5\\\\4&2&1 4&0&6\\end{array}\\right]\\rightarrow \\left[\\begin{array}{c c c c c}1&0&3&1&- 2\\\\0&1&1&0&1\\\\0&0&0&1&1\\\\0&0&0&0&0\\end{array}\\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":46,"id":"b1faac06-f3ef-4316-842d-b9ae693dd586","page_size":[595,841],"block_position":"46-2"},{"bbox":[55,153,201,165],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"因此极大线性无关组为 $\\alpha_{1},\\alpha_{2},\\alpha_{4}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"954d1ff1-8d2a-411d-a878-612e7332b13a","page_size":[595,841],"block_position":"46-3"},{"bbox":[74,188,170,200],"type":"title","angle":0,"index":4,"level":2,"text":"## 定义 5.3 (向量组的秩)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"1e11aebe-d41e-40f0-9fc3-e009d062a481","page_size":[595,841],"block_position":"46-4"},{"bbox":[68,205,374,217],"type":"text","angle":0,"index":5,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"a9f4b21b-dadd-4e6f-bd78-90d3bac500dd","page_size":[595,841],"block_position":"46-5"},{"bbox":[73,238,200,250],"type":"title","angle":0,"index":6,"level":2,"text":"## 定理 5.2 (源泉定理及其推论)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"e3230f68-de20-4645-b684-04a4cad20126","page_size":[595,841],"block_position":"46-6"},{"bbox":[68,255,398,268],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"ea87ce9e-cb0e-4e4c-97b1-eecd7da3435e","page_size":[595,841],"block_position":"46-7"},{"bbox":[57,300,163,314],"type":"title","angle":0,"index":8,"level":2,"text":"## 5.1.3 线性空间同构","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"f38eed22-8e68-4e5b-b7d2-6b8d897499c8","page_size":[595,841],"block_position":"46-8"},{"bbox":[73,327,141,339],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"定义 5.4 (同构) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"1c25979d-b99d-4343-9ad5-827438964641","page_size":[595,841],"block_position":"46-9"},{"bbox":[69,343,455,357],"type":"text","angle":0,"index":10,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"598dcd71-bc10-4f37-baed-6f9bf28b4a4a","page_size":[595,841],"block_position":"46-10"},{"bbox":[73,376,240,390],"type":"title","angle":0,"index":11,"level":2,"text":"## 定理 5.3 (有限维线性空间的同构映射)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"d699ba6c-c3fa-4083-b156-5bb70df8b858","page_size":[595,841],"block_position":"46-11"},{"bbox":[68,393,525,423],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"f07f14c4-c902-471c-b3c9-2f0579ae6cf4","page_size":[595,841],"block_position":"46-12"},{"bbox":[74,458,200,470],"type":"title","angle":0,"index":13,"level":2,"text":"## 定理 5.4 (用坐标研究向量组)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"c725c193-8e24-4167-99fb-9121e6ac09bc","page_size":[595,841],"block_position":"46-13"},{"bbox":[68,476,325,487],"type":"text","angle":0,"index":14,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"51833879-6e21-427f-9fab-ae779bf747c0","page_size":[595,841],"block_position":"46-14"},{"bbox":[232,494,361,508],"type":"interline_equation","angle":0,"index":15,"text":"$$\n(\\beta_ {1}, \\dots , \\beta_ {s}) = (\\alpha_ {1}, \\dots , \\alpha_ {n}) A\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":46,"id":"80b06260-2001-4e78-8a23-ff5a139ccc46","page_size":[595,841],"block_position":"46-15"},{"bbox":[69,513,184,526],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"则 $\\dim(\\beta_{1},\\cdots,\\beta_{s})=r(A)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"b4141386-1c4e-4f3f-804d-750092f89ed4","page_size":[595,841],"block_position":"46-16"},{"bbox":[54,543,536,557],"type":"text","angle":0,"index":17,"merge_prev":false,"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 $r(A) = \\dim(\\beta_{1}, \\cdots, \\beta_{s})$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"50b24d80-f099-4a1d-ac8f-75c19a969c2e","page_size":[595,841],"block_position":"46-17"},{"bbox":[73,565,180,578],"type":"title","angle":0,"index":18,"level":2,"text":"## 推论 5.1 (线性无关情况)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"ef19fa27-0e48-4df5-ae86-a02370420dba","page_size":[595,841],"block_position":"46-18"},{"bbox":[68,581,449,596],"type":"text","angle":0,"index":19,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"5bc70435-b388-49ca-809e-7553c5e73218","page_size":[595,841],"block_position":"46-19"},{"bbox":[36,628,499,640],"type":"text","angle":0,"index":20,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"bf6928a7-6b08-4afc-808d-8bbe29a72b0c","page_size":[595,841],"block_position":"46-20"},{"bbox":[73,645,488,656],"type":"text","angle":0,"index":21,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"55f4ac27-1cb0-4d60-896f-c3062d9a77af","page_size":[595,841],"block_position":"46-21"},{"bbox":[52,660,539,686],"type":"text","angle":0,"index":22,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"070f328a-ef77-49ed-aa79-67f26170cdb5","page_size":[595,841],"block_position":"46-22"},{"bbox":[54,690,298,702],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"证明 (1) 如下同构映射至坐标, 显然 $A$ 可逆, 因此等价 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":46,"id":"2b604b1b-cc46-45f7-89de-d5c03c87ea01","page_size":[595,841],"block_position":"46-23"},{"bbox":[173,706,420,755],"type":"interline_equation","angle":0,"index":25,"text":"$$\n(\\beta_ {1}, \\beta_ {2}, \\beta_ {3}) = (\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}) \\left[ \\begin{array}{c c c} 1 & 1 & 1 \\\\ 1 & & 1 \\\\ 0 & 1 \\end{array} \\right] := (\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}) A\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":46,"id":"0bd2d557-b2da-417a-b912-007a5b6d8eff","page_size":[595,841],"block_position":"46-24"},{"bbox":[291,803,302,811],"type":"page_number","angle":0,"index":1000022,"text":"42 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":46,"is_discarded":true,"id":"baa3efc2-b7b0-4627-8917-27134fe91b8c","page_size":[595,841],"block_position":"46-25"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99998,"text":"5.1 线性空间 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":47,"is_discarded":true,"id":"64ab3da6-935e-4684-bec4-04a70a5e66dd","page_size":[595,841],"block_position":"47-0"},{"bbox":[73,72,258,84],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"(2) 将后者设为 $\\beta_{1}, \\cdots, \\beta_{n}$ ，如下线性表出 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"ebf88c16-dc35-4c63-9546-e07c8b0013e7","page_size":[595,841],"block_position":"47-1"},{"bbox":[186,87,405,173],"type":"interline_equation","angle":0,"index":3,"text":"$$\n(\\beta_ {1}, \\dots , \\beta_ {n}) = (\\alpha_ {1}, \\dots , \\alpha_ {n}) \\left[ \\begin{array}{c c c c c} 1 & & & & 1 \\\\ 1 & 1 & & & \\\\ & 1 & \\ddots & & \\\\ & & \\ddots & 1 & \\\\ & & & 1 & 1 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":47,"id":"dbabe0f7-0478-4040-8c8c-2a7e7fcd2ae5","page_size":[595,841],"block_position":"47-2"},{"bbox":[54,177,303,190],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"用 Gauss 消元法可看出 $n$ 为奇数时矩阵满秩，因此等价。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"c7c5efd5-aea6-4055-8a48-447429ff8a7f","page_size":[595,841],"block_position":"47-3"},{"bbox":[72,193,378,206],"type":"text","angle":0,"index":5,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"c5c7d96b-8855-4c6b-8d18-65227b4b4250","page_size":[595,841],"block_position":"47-4"},{"bbox":[55,224,139,238],"type":"title","angle":0,"index":6,"level":2,"text":"## 5.1.4 基、坐标","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"e2c126ff-445e-4b8d-a18d-0b6b51162c25","page_size":[595,841],"block_position":"47-5"},{"bbox":[73,251,161,264],"type":"title","angle":0,"index":7,"level":2,"text":"## 定义 5.5 (基与坐标)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"abf19910-f339-4f74-8e31-990faf2fea39","page_size":[595,841],"block_position":"47-6"},{"bbox":[67,268,524,295],"type":"text","angle":0,"index":8,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"c32df3ab-2803-47fe-bc02-5ba850e72e58","page_size":[595,841],"block_position":"47-7"},{"bbox":[255,300,337,312],"type":"interline_equation","angle":0,"index":9,"text":"$$\n\\alpha = (\\epsilon_ {1}, \\dots , \\epsilon_ {n}) X\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":47,"id":"5759fec6-1499-4be8-86b2-96b2191d2d21","page_size":[595,841],"block_position":"47-8"},{"bbox":[68,322,235,335],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"则称 X 为 $\\alpha$ 在 $\\epsilon_{1},\\cdots,\\epsilon_{n}$ 基下的坐标 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"60d785a9-1e80-4f95-8e9e-4d222c31350c","page_size":[595,841],"block_position":"47-9"},{"bbox":[36,351,421,364],"type":"text","angle":0,"index":11,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"ce519fe7-473f-4fb1-b9a1-3f059e9260e7","page_size":[595,841],"block_position":"47-10"},{"bbox":[249,370,344,385],"type":"interline_equation","angle":0,"index":12,"text":"$$\nX = (\\epsilon_ {1}, \\dots , \\epsilon_ {n}) ^ {- 1} \\alpha\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":47,"id":"44ed3332-2554-44a5-aeca-ed2d9f5adba8","page_size":[595,841],"block_position":"47-11"},{"bbox":[35,406,530,433],"type":"text","angle":0,"index":13,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"a3740427-5473-4bfc-8dac-6ebda61f9b7b","page_size":[595,841],"block_position":"47-12"},{"bbox":[35,453,460,465],"type":"text","angle":0,"index":14,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"633f3333-4013-43df-8dbd-ab2e00072c45","page_size":[595,841],"block_position":"47-13"},{"bbox":[188,470,404,486],"type":"interline_equation","angle":0,"index":15,"text":"$$\n\\epsilon_ {1} = (1, 1, 1) ^ {T}, \\epsilon_ {2} = (1, 1, - 1) ^ {T}, \\epsilon_ {3} = (1, - 1, - 1) ^ {T}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":47,"id":"91fc045f-3c80-4d52-bb71-b76b0c34a770","page_size":[595,841],"block_position":"47-14"},{"bbox":[52,491,351,504],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"解(1)只需要解方程组 $\\alpha = k_{1}\\epsilon_{1} + k_{2}\\epsilon_{2} + k_{3}\\epsilon_{3}$ ，即 $[\\epsilon_1,\\epsilon_2,\\epsilon_3]X = \\alpha$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"aae75c96-c7c1-4c32-8f35-beb094b63cbd","page_size":[595,841],"block_position":"47-15"},{"bbox":[234,506,359,558],"type":"interline_equation","angle":0,"index":17,"text":"$$\n\\left[ \\begin{array}{c c c} 1 & 1 & 1 \\\\ 1 & 1 & - 1 \\\\ 1 & - 1 & - 1 \\end{array} \\right] X = \\left[ \\begin{array}{c} 1 \\\\ 2 \\\\ 1 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":47,"id":"b4dcea12-3448-45ae-9895-c194dedbf45b","page_size":[595,841],"block_position":"47-16"},{"bbox":[54,559,185,582],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"解得 $k_{1} = 1, k_{2} = \\frac{1}{2}, k_{3} = -\\frac{1}{2}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"635651df-852d-456c-8479-c763d92cc87e","page_size":[595,841],"block_position":"47-17"},{"bbox":[35,595,405,608],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"d57e766a-da35-4fee-976a-e24821820936","page_size":[595,841],"block_position":"47-18"},{"bbox":[195,613,389,686],"type":"interline_equation","angle":0,"index":20,"text":"$$\n\\left\\{ \\begin{array}{l} \\epsilon_ {1} = (1, 0, 0, 0) ^ {T} \\\\ \\epsilon_ {2} = (0, 1, 0, 0) ^ {T} \\\\ \\epsilon_ {3} = (0, 0, 1, 0) ^ {T} \\\\ \\epsilon_ {4} = (0, 0, 0, 1) ^ {T} \\end{array} \\right., \\quad \\left\\{ \\begin{array}{l} \\eta_ {1} = (2, 1, - 1, 1) ^ {T} \\\\ \\eta_ {2} = (0, 3, 1, 0) ^ {T} \\\\ \\eta_ {3} = (5, 3, 2, 1) ^ {T} \\\\ \\eta_ {4} = (6, 6, 1, 3) ^ {T} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":47,"id":"bf31f0d0-7f2d-4bf5-bf1b-ec5b0d08b8f2","page_size":[595,841],"block_position":"47-19"},{"bbox":[52,690,346,703],"type":"text","angle":0,"index":21,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":47,"id":"3eebf15e-7813-46e0-a9b8-a10166b02f78","page_size":[595,841],"block_position":"47-20"},{"bbox":[226,712,365,724],"type":"interline_equation","angle":0,"index":22,"text":"$$\n\\alpha = x _ {1} \\eta_ {1} + x _ {2} \\eta_ {2} + x _ {3} \\eta_ {3} + x _ {4} \\eta_ {4}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":47,"id":"8e7fba61-5b62-4c27-bd67-f92a67689019","page_size":[595,841],"block_position":"47-21"},{"bbox":[291,803,302,811],"type":"page_number","angle":0,"index":999999,"text":"43 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":47,"is_discarded":true,"id":"e95e28a5-8c09-4cd1-9dc3-0339eba88939","page_size":[595,841],"block_position":"47-22"}],[{"bbox":[480,47,539,58],"type":"header","angle":0,"index":-99999,"text":"5.1 线性空间 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":48,"is_discarded":true,"id":"5da0733a-d0c1-4cc8-bc69-46783fd93ced","page_size":[595,841],"block_position":"48-0"},{"bbox":[54,72,321,84],"type":"text","angle":0,"index":1,"merge_prev":false,"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): ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"a698c66f-d0b6-4705-8c77-35dd34f51cc6","page_size":[595,841],"block_position":"48-1"},{"bbox":[242,88,354,151],"type":"interline_equation","angle":0,"index":2,"text":"$$\n\\left[ \\begin{array}{r r r r} 1 & 0 & 5 & 6 \\\\ 1 & 2 & 3 & 6 \\\\ - 1 & 1 & 1 & 1 \\\\ 1 & 0 & 1 & 2 \\end{array} \\right] X = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":48,"id":"6abf9b6d-7eb4-4872-b2e9-3e4da02806a8","page_size":[595,841],"block_position":"48-2"},{"bbox":[54,154,153,166],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"解得 $\\alpha = k(1,1,1,-1)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"81c04745-dbb5-489b-8d9d-36419c423c11","page_size":[595,841],"block_position":"48-3"},{"bbox":[56,185,139,199],"type":"title","angle":0,"index":4,"level":2,"text":"## 5.1.5 坐标变换","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"6a6ca3dd-aebe-41a6-8e98-8ef0d4675f6d","page_size":[595,841],"block_position":"48-4"},{"bbox":[73,212,161,225],"type":"title","angle":0,"index":5,"level":2,"text":"## 定义 5.6 (过渡矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"fb85629a-0055-4941-bad9-a39f2e683394","page_size":[595,841],"block_position":"48-5"},{"bbox":[67,229,329,242],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"给定 $(\\epsilon_{1},\\cdots,\\epsilon_{n}),(\\eta_{1},\\cdots,\\eta_{n})$ 两组基，若存在矩阵T满足： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"76398cca-cd3c-44ab-a603-aad368c4bf39","page_size":[595,841],"block_position":"48-6"},{"bbox":[234,248,360,262],"type":"interline_equation","angle":0,"index":7,"text":"$$\n(\\epsilon_ {1}, \\dots , \\epsilon_ {n}) = (\\eta_ {1}, \\dots , \\eta_ {n}) T\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":48,"id":"d9d73da0-b382-41b9-8abc-9e5efae285b3","page_size":[595,841],"block_position":"48-7"},{"bbox":[68,268,318,280],"type":"text","angle":0,"index":8,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"f444a170-eecc-4ea6-96d9-6262a22d1349","page_size":[595,841],"block_position":"48-8"},{"bbox":[52,297,346,310],"type":"text","angle":0,"index":9,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"14a45d9c-cc94-44a6-93e0-2b8daf7d6e77","page_size":[595,841],"block_position":"48-9"},{"bbox":[227,317,364,330],"type":"interline_equation","angle":0,"index":10,"text":"$$\nT = (\\eta_ {1}, \\dots , \\eta_ {n}) ^ {- 1} (\\epsilon_ {1}, \\dots , \\epsilon_ {n})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":48,"id":"870c5b0f-e123-4184-b2be-1eff696b21b6","page_size":[595,841],"block_position":"48-10"},{"bbox":[54,336,245,349],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"用初等变换求 $A^{-1}B$ 的方式直接求 T 即可。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"83c6faf1-05b2-4794-95e7-4d05bada770e","page_size":[595,841],"block_position":"48-11"},{"bbox":[73,357,161,370],"type":"title","angle":0,"index":12,"level":2,"text":"## 定理 5.5 (坐标变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"60cd280d-a729-452b-9a39-76c9d847fdfe","page_size":[595,841],"block_position":"48-12"},{"bbox":[67,374,527,402],"type":"text","angle":0,"index":14,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"16a61028-885f-40da-8f1d-6bd0622c22bf","page_size":[595,841],"block_position":"48-13"},{"bbox":[276,405,317,417],"type":"interline_equation","angle":0,"index":15,"text":"$$\nX = T Y\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":48,"id":"354447a4-4791-4da8-ad77-d2c5a4d6a21a","page_size":[595,841],"block_position":"48-14"},{"bbox":[54,433,337,445],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"证明 $x=(\\epsilon_{1},\\cdots,\\epsilon_{n})X=(\\eta_{1},\\cdots,\\eta_{n})Y=(\\epsilon_{1},\\cdots,\\epsilon_{n})TY$ 即可。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"da5dd8bc-8ba4-4ee8-b99e-9dbd02661885","page_size":[595,841],"block_position":"48-15"},{"bbox":[53,448,540,476],"type":"text","angle":0,"index":17,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"ca1c9ff1-fd41-4ad8-9350-69b977b16ff6","page_size":[595,841],"block_position":"48-16"},{"bbox":[35,494,539,507],"type":"text","angle":0,"index":18,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"67282e4d-2aae-4f86-9863-84848eed3115","page_size":[595,841],"block_position":"48-17"},{"bbox":[124,513,467,541],"type":"interline_equation","angle":0,"index":19,"text":"$$\n\\begin{array}{l l l l} \\alpha_ {1} = (1, 1, 1, 1) & \\alpha_ {2} = (1, 1, - 1, - 1) & \\alpha_ {3} = (1, - 1, 1, - 1) & \\alpha_ {4} = (1, - 1, - 1, 1) \\\\ \\beta_ {1} = (1, 1, 0, 1) & \\beta_ {2} = (2, 1, 3, 1) & \\beta_ {3} = (1, 1, 0, 0) & \\beta_ {4} = (0, 1, - 1, - 1) \\end{array}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":48,"id":"b42d23ad-0f27-4c35-820c-3e195b998ba4","page_size":[595,841],"block_position":"48-18"},{"bbox":[53,547,204,559],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"解(1)先取标准基 $\\epsilon_{1}, \\epsilon_{2}, \\epsilon_{3}, \\epsilon_{4}$ ，设 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"479772fd-6c70-473e-8db0-d05f79c3e18e","page_size":[595,841],"block_position":"48-19"},{"bbox":[154,566,440,579],"type":"interline_equation","angle":0,"index":21,"text":"$$\n(\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}, \\alpha_ {4}) = (\\epsilon_ {1}, \\epsilon_ {2}, \\epsilon_ {3}, \\epsilon_ {4}) A, (\\beta_ {1}, \\beta_ {2}, \\beta_ {3}, \\beta_ {4}) = (\\epsilon_ {1}, \\epsilon_ {2}, \\epsilon_ {3}, \\epsilon_ {4}) B\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":48,"id":"665e92eb-8dc7-48c1-a933-7e7d36b50909","page_size":[595,841],"block_position":"48-20"},{"bbox":[55,585,67,597],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"1d8c39a1-409c-4427-b6dd-da568c4b8ca0","page_size":[595,841],"block_position":"48-21"},{"bbox":[182,595,409,658],"type":"interline_equation","angle":0,"index":23,"text":"$$\nA = \\left[ \\begin{array}{r r r r} 1 & 1 & 1 & 1 \\\\ 1 & 1 & - 1 & - 1 \\\\ 1 & - 1 & 1 & - 1 \\\\ 1 & - 1 & - 1 & 1 \\end{array} \\right], B = \\left[ \\begin{array}{r r r r} 1 & 2 & 1 & 0 \\\\ 1 & 1 & 1 & 1 \\\\ 0 & 3 & 0 & - 1 \\\\ 1 & 1 & 0 & - 1 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":48,"id":"5377e4a3-3208-4425-a20c-57b12569f8c0","page_size":[595,841],"block_position":"48-22"},{"bbox":[55,666,97,677],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"由此得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":48,"id":"4bf791f9-bf6f-4a85-810f-7eaba21c3e39","page_size":[595,841],"block_position":"48-23"},{"bbox":[174,682,420,695],"type":"interline_equation","angle":0,"index":25,"text":"$$\n(\\beta_ {1}, \\beta_ {2}, \\beta_ {3}, \\beta_ {4}) = (\\epsilon_ {1}, \\epsilon_ {2}, \\epsilon_ {3}, \\epsilon_ {4}) B = (\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}, \\alpha_ {4}) A ^ {- 1} B\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":48,"id":"2436769e-15d8-4cbd-a273-f1f58c61d0ef","page_size":[595,841],"block_position":"48-24"},{"bbox":[291,803,303,811],"type":"page_number","angle":0,"index":1000012,"text":"44 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":48,"is_discarded":true,"id":"316eccb5-1278-445b-b0a3-9897c2d2d338","page_size":[595,841],"block_position":"48-25"}],[{"bbox":[480,47,539,58],"type":"header","angle":0,"index":-99998,"text":"5.1 线性空间 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":49,"is_discarded":true,"id":"551eb595-5c05-446e-a9a3-436e0ed47961","page_size":[595,841],"block_position":"49-0"},{"bbox":[54,72,362,84],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"综上得到过渡矩阵为 (具体计算时用行初等变换: $(A|B) \\rightarrow (E|A^{-1}B)$ ) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"7031351a-b5ab-46b7-9f5f-3e6d72b196b4","page_size":[595,841],"block_position":"49-1"},{"bbox":[222,89,368,150],"type":"interline_equation","angle":0,"index":3,"text":"$$\nA ^ {- 1} B = \\frac {1}{4} \\left[ \\begin{array}{r r r r} 3 & 7 & 2 & - 1 \\\\ 1 & - 1 & 2 & 3 \\\\ - 1 & 3 & 0 & - 1 \\\\ 1 & - 1 & 0 & - 1 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":49,"id":"06f4ed6b-7542-4d08-8921-810d150a981e","page_size":[595,841],"block_position":"49-2"},{"bbox":[54,153,539,187],"type":"text","angle":0,"index":4,"merge_prev":false,"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 $(0, \\frac{1}{2}, \\frac{1}{2}, 0)^T$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"2d29c3b6-0a4a-49db-aa81-96b616999d06","page_size":[595,841],"block_position":"49-3"},{"bbox":[56,205,186,218],"type":"title","angle":0,"index":5,"level":2,"text":"## 5.1.6 和与交、维数定理","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"86e8c161-b0c3-461d-8a29-2ebf3971f641","page_size":[595,841],"block_position":"49-4"},{"bbox":[73,232,190,244],"type":"title","angle":0,"index":6,"level":2,"text":"## 定义 5.7 (线性空间和与交)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"537b7fe4-a980-4b26-a185-d276a75f902a","page_size":[595,841],"block_position":"49-5"},{"bbox":[69,249,169,260],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"$V_{1}, V_{2}$ 为线性空间，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"242d6241-eee4-4b33-81bd-69dc30cf7cb8","page_size":[595,841],"block_position":"49-6"},{"bbox":[129,268,463,281],"type":"interline_equation","angle":0,"index":8,"text":"$$\nV _ {1} \\cap V _ {2} := \\{\\alpha : \\alpha \\in V _ {1}, \\alpha \\in V _ {2} \\}, V _ {1} + V _ {2} := \\{\\alpha = \\alpha_ {1} + \\alpha_ {2}: \\alpha_ {1} \\in V _ {1}, \\alpha_ {2} \\in V _ {2} \\}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":49,"id":"71ad03e6-8b70-4ccb-b6ef-b4db4f02af4c","page_size":[595,841],"block_position":"49-7"},{"bbox":[69,287,230,300],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"且显然 $V_{1} \\cap V_{2}, V_{1} + V_{2}$ 都是子空间。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"aa72eaf1-71a3-4b0b-89c3-0ab3851054a0","page_size":[595,841],"block_position":"49-8"},{"bbox":[35,312,47,328],"type":"image","angle":0,"index":10,"img_path":"/6704a6fa582ccd0f7b0270cfe13b761ae03c428e9e7a8137480049d96a04f59a.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":49,"id":"e3790b1f-7561-4f89-bf26-8c0e9d36f970","page_size":[595,841],"block_position":"49-9"},{"bbox":[53,317,414,329],"type":"text","angle":0,"index":11,"merge_prev":false,"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 $V_{1} + V_{2}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"1afc9495-e208-4de1-9ce2-5879f103cc06","page_size":[595,841],"block_position":"49-10"},{"bbox":[35,348,382,360],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"6318a015-4c46-40ac-9c05-f1818200b144","page_size":[595,841],"block_position":"49-11"},{"bbox":[130,364,460,426],"type":"interline_equation","angle":0,"index":13,"text":"$$\n\\alpha_ {1} = \\left( \\begin{array}{c} 1 \\\\ 2 \\\\ 1 \\\\ 0 \\end{array} \\right), \\alpha_ {2} = \\left( \\begin{array}{c} - 1 \\\\ 1 \\\\ 1 \\\\ 1 \\end{array} \\right), \\alpha_ {3} = \\left( \\begin{array}{c} 0 \\\\ 3 \\\\ 2 \\\\ 1 \\end{array} \\right), \\beta_ {1} = \\left( \\begin{array}{c} 2 \\\\ - 1 \\\\ 0 \\\\ 1 \\end{array} \\right), \\beta_ {2} = \\left( \\begin{array}{c} 1 \\\\ - 1 \\\\ 3 \\\\ 7 \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":49,"id":"d3f4877e-5879-4d85-9f01-228dbe8e9ccf","page_size":[595,841],"block_position":"49-12"},{"bbox":[54,431,191,443],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"求 $V_{1} + V_{2}, V_{1} \\cap V_{2}$ 的基与维数 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"4897048f-9cea-44e8-97b6-df0ebc712b60","page_size":[595,841],"block_position":"49-13"},{"bbox":[73,446,430,459],"type":"text","angle":0,"index":15,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"96daff0c-c4ee-4fac-8237-d0324dd92c62","page_size":[595,841],"block_position":"49-14"},{"bbox":[120,462,471,524],"type":"interline_equation","angle":0,"index":16,"text":"$$\n\\alpha_ {1} = \\left( \\begin{array}{r} 1 \\\\ 2 \\\\ - 1 \\\\ - 3 \\end{array} \\right), \\alpha_ {2} = \\left( \\begin{array}{r} - 1 \\\\ - 1 \\\\ 2 \\\\ 1 \\end{array} \\right), \\alpha_ {3} = \\left( \\begin{array}{r} - 1 \\\\ - 3 \\\\ 0 \\\\ 5 \\end{array} \\right), \\beta_ {1} = \\left( \\begin{array}{r} - 1 \\\\ 0 \\\\ 4 \\\\ 2 \\end{array} \\right), \\beta_ {2} = \\left( \\begin{array}{r} 0 \\\\ 5 \\\\ 9 \\\\ - 1 4 \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":49,"id":"53cc57e0-d4d4-4e11-9c3d-f19b34ce3808","page_size":[595,841],"block_position":"49-15"},{"bbox":[52,528,366,541],"type":"text","angle":0,"index":17,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"1c12b8ea-ff17-43b7-8eb9-6fd4c2236dc3","page_size":[595,841],"block_position":"49-16"},{"bbox":[169,545,426,607],"type":"interline_equation","angle":0,"index":18,"text":"$$\n\\left( \\begin{array}{c c c c c} 1 & - 1 & 0 & 2 & 1 \\\\ 2 & 1 & 3 & - 1 & - 1 \\\\ 1 & 1 & 2 & 0 & 3 \\\\ 0 & 1 & 1 & 1 & 7 \\end{array} \\right) \\to \\left( \\begin{array}{c c c c c} 1 & 0 & 1 & 0 & - 1 \\\\ 0 & 1 & 1 & 0 & 4 \\\\ 0 & 0 & 0 & 1 & 3 \\\\ 0 & 0 & 0 & 0 & 0 \\end{array} \\right) := A\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":49,"id":"64fa5ebb-f8a4-44da-ac3d-ff520ea67090","page_size":[595,841],"block_position":"49-17"},{"bbox":[55,612,186,623],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"因此维数为3，基为 $\\alpha_{1},\\alpha_{2},\\beta_{1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"dea7cb9e-99eb-4c0a-ab8f-8d30c525888a","page_size":[595,841],"block_position":"49-18"},{"bbox":[73,627,468,639],"type":"text","angle":0,"index":20,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"982f8ed1-bd03-43ed-9035-ce0152b9877b","page_size":[595,841],"block_position":"49-19"},{"bbox":[210,647,384,659],"type":"interline_equation","angle":0,"index":21,"text":"$$\nx _ {1} \\alpha_ {1} + x _ {2} \\alpha_ {2} + x _ {3} \\alpha_ {3} - y _ {1} \\beta_ {1} - y _ {2} \\beta_ {2} = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":49,"id":"eec1ec2c-0c6d-49be-b086-7cd7a66008f3","page_size":[595,841],"block_position":"49-20"},{"bbox":[54,666,402,677],"type":"text","angle":0,"index":22,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":49,"id":"76afacd3-1e57-4aaf-bd61-6adac16e2d52","page_size":[595,841],"block_position":"49-21"},{"bbox":[229,682,362,761],"type":"interline_equation","angle":0,"index":23,"text":"$$\n\\eta_ {1} = \\left( \\begin{array}{c} - 1 \\\\ - 1 \\\\ 1 \\\\ 0 \\\\ 0 \\end{array} \\right), \\eta_ {2} = \\left( \\begin{array}{c} 1 \\\\ - 4 \\\\ 0 \\\\ - 3 \\\\ 1 \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":49,"id":"b6888d89-62f3-4416-b8b7-67464c361943","page_size":[595,841],"block_position":"49-22"},{"bbox":[291,803,302,811],"type":"page_number","angle":0,"index":999999,"text":"45 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":49,"is_discarded":true,"id":"25555b8a-2978-4178-827b-07d963dc518b","page_size":[595,841],"block_position":"49-23"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99999,"text":"5.1 线性空间 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":50,"is_discarded":true,"id":"d9d46055-1cba-4324-810d-6f6c44c727e9","page_size":[595,841],"block_position":"50-0"},{"bbox":[54,72,179,84],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"对应 $V_{1} \\cap V_{2}$ 的基础解系为： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"aced6e49-c38e-443c-ac7e-485d20624b48","page_size":[595,841],"block_position":"50-1"},{"bbox":[168,91,424,104],"type":"interline_equation","angle":0,"index":2,"text":"$$\n- \\alpha_ {1} - \\alpha_ {2} + \\alpha_ {3} = (0, 0, 0, 0) ^ {T}, \\alpha_ {1} - 4 \\alpha_ {2} = (5, - 2, - 3, - 4) ^ {T}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":50,"id":"e56ffdd3-608d-45b3-a674-6d61a2eb3c10","page_size":[595,841],"block_position":"50-2"},{"bbox":[54,111,352,123],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"只取非零的，故只能为 $k(5, -2, -3, -4)$ ，维数为1，基即 $\\alpha_{1} - 4\\alpha_{2}$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"810d6080-8efa-4508-aeec-b9bfa93b47e1","page_size":[595,841],"block_position":"50-3"},{"bbox":[35,141,539,169],"type":"text","angle":0,"index":4,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"26bc61d9-82af-4741-beb5-ea0e3ef21abc","page_size":[595,841],"block_position":"50-4"},{"bbox":[72,172,431,185],"type":"text","angle":0,"index":5,"merge_prev":false,"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 $V_{2} \\subseteq V_{1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"3b617a1e-b806-4e24-bb0c-f3e16c55e5ad","page_size":[595,841],"block_position":"50-5"},{"bbox":[52,188,537,216],"type":"text","angle":0,"index":6,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"6141f40f-8a27-478e-818f-44a4bd5d3c5c","page_size":[595,841],"block_position":"50-6"},{"bbox":[229,223,364,236],"type":"interline_equation","angle":0,"index":7,"text":"$$\n\\alpha \\notin V _ {1}, \\beta \\in V _ {1}, \\alpha \\in V _ {2}, \\beta \\notin V _ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":50,"id":"4a60958c-1226-427a-820d-cf729ae77e77","page_size":[595,841],"block_position":"50-7"},{"bbox":[54,243,179,255],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"因此 $\\gamma = \\alpha +\\beta \\notin V_1,\\gamma \\notin V_2$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"e9c81beb-fabd-4892-b6eb-970702e33e91","page_size":[595,841],"block_position":"50-8"},{"bbox":[73,258,132,270],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"(2) 用归纳法 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"0680113e-a84e-4755-b8b5-d333b5c5ab69","page_size":[595,841],"block_position":"50-9"},{"bbox":[73,273,456,285],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"5989d945-682c-4a1f-bf4f-fc7edaab58aa","page_size":[595,841],"block_position":"50-10"},{"bbox":[223,294,370,306],"type":"interline_equation","angle":0,"index":11,"text":"$$\n\\forall \\alpha_ {2} \\in V _ {2}, \\alpha = \\alpha_ {1} + \\alpha_ {2} \\in V _ {1} + V _ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":50,"id":"7890a440-524c-46a9-b657-60917e80f23f","page_size":[595,841],"block_position":"50-11"},{"bbox":[54,312,517,325],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"056af1e9-1087-45c5-a00e-1c8de8e627c2","page_size":[595,841],"block_position":"50-12"},{"bbox":[255,333,337,344],"type":"interline_equation","angle":0,"index":13,"text":"$$\n\\alpha_ {2} = \\alpha - \\alpha_ {1} \\in V _ {1}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":50,"id":"88916c0c-1303-45fe-8e01-cf9431a2c3a8","page_size":[595,841],"block_position":"50-13"},{"bbox":[54,351,133,364],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"综上得到 $V_{2} \\subseteq V_{1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"dbc2b4c7-b2ee-49b1-aa71-3a7d4f8dae4b","page_size":[595,841],"block_position":"50-14"},{"bbox":[73,386,161,398],"type":"title","angle":0,"index":16,"level":2,"text":"## 定理 5.6 (维数公式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"0a99cfbf-9820-4e20-9a2d-bf17b6273fe3","page_size":[595,841],"block_position":"50-15"},{"bbox":[69,402,285,415],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"设 $V_{1}, V_{2}$ 是有限维线性空间 $V$ 的两个子空间，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"7a3b0695-0007-45fc-bce6-a27ebf087156","page_size":[595,841],"block_position":"50-16"},{"bbox":[192,423,401,435],"type":"interline_equation","angle":0,"index":18,"text":"$$\n\\dim V _ {1} + \\dim V _ {2} = \\dim (V _ {1} + V _ {2}) + \\dim (V _ {1} \\cap V _ {2})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":50,"id":"b245be32-bd53-4414-a409-de1605133f93","page_size":[595,841],"block_position":"50-17"},{"bbox":[53,453,348,465],"type":"text","angle":0,"index":19,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"058cca42-b89a-4e07-b68a-c1a12b87e185","page_size":[595,841],"block_position":"50-18"},{"bbox":[232,469,350,505],"type":"interline_equation","angle":0,"index":20,"text":"$$\n\\left\\{ \\begin{array}{l} \\alpha_ {1}, \\dots , \\alpha_ {s}, \\beta_ {s + 1}, \\dots , \\beta_ {m} \\\\ \\alpha_ {1}, \\dots , \\alpha_ {s}, \\gamma_ {s + 1}, \\dots , \\gamma_ {n} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":50,"id":"dfe8f094-24c3-4b11-9bf0-cebb7a971628","page_size":[595,841],"block_position":"50-19"},{"bbox":[54,510,494,523],"type":"text","angle":0,"index":21,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"12eceaf9-5f73-48a2-9209-3be0d6f34894","page_size":[595,841],"block_position":"50-20"},{"bbox":[139,529,454,542],"type":"interline_equation","angle":0,"index":22,"text":"$$\nk _ {1} \\alpha_ {1} + \\dots + k _ {s} \\alpha_ {s} + k _ {s + 1} \\beta_ {s + 1} + \\dots + k _ {n} \\beta_ {n} + l _ {s + 1} \\gamma_ {s + 1} + \\dots + l _ {n} \\gamma_ {n} = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":50,"id":"41c3374e-042f-4717-914c-86e3f63f7fc4","page_size":[595,841],"block_position":"50-21"},{"bbox":[53,548,509,561],"type":"text","angle":0,"index":23,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"bcf16b67-53c3-4f7a-9b05-fccef757c46a","page_size":[595,841],"block_position":"50-22"},{"bbox":[66,568,527,580],"type":"interline_equation","angle":0,"index":24,"text":"$$\nk _ {1} \\alpha_ {1} + \\dots + k _ {s} \\alpha_ {s} + k _ {s + 1} \\beta_ {s + 1} + \\dots + k _ {n} \\beta_ {n} \\in V _ {1} \\cap V _ {2} \\Rightarrow k _ {s + 1} = k _ {s + 2} = \\dots = k _ {n} = 0 \\Rightarrow k _ {1} = \\dots = k _ {s} = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":50,"id":"d45dc904-c1ba-4eeb-9dc0-1572613e744e","page_size":[595,841],"block_position":"50-23"},{"bbox":[54,587,253,599],"type":"text","angle":0,"index":25,"merge_prev":false,"text":"因此左侧系数全为0，同理右侧系数也全为0 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"2a63fe0f-6a32-409b-a60d-7d243d1e7f00","page_size":[595,841],"block_position":"50-24"},{"bbox":[35,618,539,647],"type":"text","angle":0,"index":26,"merge_prev":false,"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 $V_{2} \\subseteq V_{1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"8c05e42a-dbe9-4739-a272-9dfd0523adf4","page_size":[595,841],"block_position":"50-25"},{"bbox":[54,650,176,661],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"证明 (1) 根据维数公式得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"bda956b2-5ab3-423d-a372-9dfddc99b42f","page_size":[595,841],"block_position":"50-26"},{"bbox":[138,669,455,682],"type":"interline_equation","angle":0,"index":28,"text":"$$\n\\dim (V _ {1}) + \\dim (V _ {2}) = \\dim (V _ {1} + V _ {2}) + \\dim (V _ {1} \\cap V _ {2}) = 2 \\dim (V _ {1} \\cap V _ {2}) + 1\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":50,"id":"d14130e0-f053-41e1-a007-df94221f47c2","page_size":[595,841],"block_position":"50-27"},{"bbox":[54,687,88,700],"type":"text","angle":0,"index":29,"merge_prev":false,"text":"等价于 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"513e133f-1d4c-44f8-8d20-0cdcd7c30104","page_size":[595,841],"block_position":"50-28"},{"bbox":[176,703,416,717],"type":"interline_equation","angle":0,"index":30,"text":"$$\n\\left(\\dim V _ {1} - \\dim (V _ {1} \\cap V _ {2})\\right) + \\left(\\dim V _ {2} - \\dim (V _ {1} \\cap V _ {2})\\right) = 1\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":50,"id":"e9d1a04e-76b3-4f88-9eff-40c18cac186f","page_size":[595,841],"block_position":"50-29"},{"bbox":[54,727,286,739],"type":"text","angle":0,"index":31,"merge_prev":false,"text":"由于都是整数，只能一个0，一个1，因此结论成立。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":50,"id":"d56540ee-bcf2-45f1-9b81-06f0b4a49994","page_size":[595,841],"block_position":"50-30"},{"bbox":[291,803,303,812],"type":"page_number","angle":0,"index":1000013,"text":"46 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":50,"is_discarded":true,"id":"0a9a0222-bc87-456f-b44d-60a3cd2a0509","page_size":[595,841],"block_position":"50-31"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99998,"text":"5.1 线性空间 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":51,"is_discarded":true,"id":"be31a0fb-69ac-4e47-bb65-1909d92c98d5","page_size":[595,841],"block_position":"51-0"},{"bbox":[35,191,47,207],"type":"image","angle":0,"index":0,"img_path":"/cee4c1078050101154e1beeff5719a1fa75de396fb1f2dfef978054d78e0b5d3.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":51,"id":"79fd4aad-484e-4128-a0df-c18f52e38ec0","page_size":[595,841],"block_position":"51-1"},{"bbox":[55,73,139,86],"type":"title","angle":0,"index":2,"level":2,"text":"## 5.1.7 直和理论","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"92a8af03-35e4-4566-8b15-9f6cf52e103f","page_size":[595,841],"block_position":"51-2"},{"bbox":[73,100,141,112],"type":"title","angle":0,"index":3,"level":2,"text":"## 定义 5.8 (直和)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"eae50577-bcd8-4ada-b5c9-8eab45b05172","page_size":[595,841],"block_position":"51-3"},{"bbox":[68,116,398,130],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"$V_{1}, V_{2}$ 是线性空间 $V$ 的子空间，若 $V_{1} \\cap V_{2} = \\{0\\}$ ，则记 $V_{1} \\oplus V_{2} := V_{1} + V_{2}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"f1732b91-d768-46bb-a752-aef782deebb7","page_size":[595,841],"block_position":"51-4"},{"bbox":[73,150,151,163],"type":"title","angle":0,"index":5,"level":2,"text":"## 定义 5.9 (补空间)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"5d035b91-61b0-41d9-ac18-065de70115d3","page_size":[595,841],"block_position":"51-5"},{"bbox":[69,167,253,179],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"若 $V = V_{1} \\oplus V_{2}$ ，则称 $V_{2}$ 是 $V_{1}$ 的补空间。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"65fab766-5b46-43a6-b734-d3f9f0212719","page_size":[595,841],"block_position":"51-6"},{"bbox":[47,195,408,209],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"4f9ba088-1c34-49e6-8f52-db800ff7ca6e","page_size":[595,841],"block_position":"51-7"},{"bbox":[72,218,181,230],"type":"title","angle":0,"index":8,"level":2,"text":"## 定理 5.7 (直和等价条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"629ddea5-7b3f-4b0f-b07d-802615df8c3e","page_size":[595,841],"block_position":"51-8"},{"bbox":[69,234,171,246],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"$W = V_{1} \\oplus V_{2}$ 当且仅当 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"656cfb7b-da54-4a42-b3ff-caf7a6de16f6","page_size":[595,841],"block_position":"51-9"},{"bbox":[83,250,333,263],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"b5324339-6d83-4d5c-8472-8a63ca551a07","page_size":[595,841],"block_position":"51-10"},{"bbox":[83,265,311,277],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"- $W$ 中零向量被 $V_{1}, V_{2}$ 中向量唯一表出: $0 = 0 + 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"8e02bdf3-254a-47a2-80d8-da5768264088","page_size":[595,841],"block_position":"51-11"},{"bbox":[83,281,204,294],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"- $\\dim W = \\dim V_1 + \\dim V_2$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"134b0ad6-38d0-428d-9786-8d80be166094","page_size":[595,841],"block_position":"51-12"},{"bbox":[35,325,540,354],"type":"text","angle":0,"index":13,"merge_prev":false,"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 $V = V_{1} \\oplus V_{2}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"9a115a83-a885-42a7-9d30-11c7919930ad","page_size":[595,841],"block_position":"51-13"},{"bbox":[53,356,499,369],"type":"text","angle":0,"index":14,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"d63f92d8-a7f1-4b7e-909e-3f54da8f3bb9","page_size":[595,841],"block_position":"51-14"},{"bbox":[241,371,351,396],"type":"interline_equation","angle":0,"index":15,"text":"$$\nA = \\frac {A + A ^ {T}}{2} + \\frac {A ^ {T} - A}{2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":51,"id":"31e21023-37ec-4543-af94-3508974737d9","page_size":[595,841],"block_position":"51-15"},{"bbox":[54,398,137,410],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"从而 $V = V_{1} + V_{2}$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"1bcbe092-dbdc-468c-9c73-31c3dd42186b","page_size":[595,841],"block_position":"51-16"},{"bbox":[54,412,539,442],"type":"text","angle":0,"index":17,"merge_prev":false,"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 $B = C = O$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"41963101-8e8a-44e4-97f0-09d25e2ff43a","page_size":[595,841],"block_position":"51-17"},{"bbox":[73,464,211,476],"type":"title","angle":0,"index":18,"level":2,"text":"## 定理 5.8 (多空间直和等价条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"68f95487-1dac-41cb-8d06-2698d50c2cc8","page_size":[595,841],"block_position":"51-18"},{"bbox":[68,480,476,493],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"77c4eb1a-aa4f-489a-ba4a-13afd9c5b55a","page_size":[595,841],"block_position":"51-19"},{"bbox":[83,496,190,507],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"- $W$ 中零向量分解唯一 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"21210ab6-3a8a-49d6-9cdf-98b389f9659b","page_size":[595,841],"block_position":"51-20"},{"bbox":[83,512,212,523],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"- $W$ 中任意向量 $\\alpha$ 分解唯一 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"cf19e8f2-c4e6-487c-914c-a1562308a91d","page_size":[595,841],"block_position":"51-21"},{"bbox":[93,525,171,550],"type":"interline_equation","angle":0,"index":22,"text":"$$\nV _ {i} \\cap \\sum_ {j \\neq i} V _ {j} = \\{0 \\}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":51,"id":"426329b6-9d55-4139-8065-f17f05d7a14d","page_size":[595,841],"block_position":"51-22"},{"bbox":[84,550,227,562],"type":"interline_equation","angle":0,"index":23,"text":"$$\n\\bullet \\dim W = \\dim V _ {1} + \\dots + \\dim V _ {s}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":51,"id":"eb312848-9e5f-4252-a34f-d8f546a8f06d","page_size":[595,841],"block_position":"51-23"},{"bbox":[53,577,245,590],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"证明 (3) 只证明第三条。第三条推直和：若 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"7f103075-cc6e-4606-ba53-8530e8b85b41","page_size":[595,841],"block_position":"51-24"},{"bbox":[163,599,429,611],"type":"interline_equation","angle":0,"index":25,"text":"$$\n0 = \\alpha_ {1} + \\dots + \\alpha_ {n} \\Rightarrow \\alpha_ {i} = \\alpha_ {1} + \\dots + \\alpha_ {i - 1} + \\alpha_ {i + 1} + \\dots + \\alpha_ {n}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":51,"id":"aea1f091-a1f4-4a30-baa6-7668d4d899b3","page_size":[595,841],"block_position":"51-25"},{"bbox":[54,615,348,640],"type":"text","angle":0,"index":26,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"c8b0f2fd-1988-4a07-a9e2-5267b2d9c750","page_size":[595,841],"block_position":"51-26"},{"bbox":[74,641,294,667],"type":"text","angle":0,"index":27,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"0ec46edc-9802-4c6a-83ad-ed495df34cb7","page_size":[595,841],"block_position":"51-27"},{"bbox":[104,670,489,682],"type":"interline_equation","angle":0,"index":29,"text":"$$\n\\alpha = \\alpha_ {1} + \\dots + \\alpha_ {i - 1} + \\alpha_ {i + 1} + \\dots + \\alpha_ {n} \\Rightarrow \\alpha_ {1} + \\dots + \\alpha_ {i - 1} + (- \\alpha) + \\alpha_ {i + 1} + \\dots + \\alpha_ {n} = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":51,"id":"979fbdf8-e873-4e0b-bc8f-073d57868c77","page_size":[595,841],"block_position":"51-28"},{"bbox":[55,685,289,716],"type":"text","angle":0,"index":30,"merge_prev":false,"text":"由于 $\\sum_{i=1}^{n} V_i$ 为直和，因此 $\\alpha = 0$ ，即第三条结论成立。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":51,"id":"0f04a8d1-63bc-45bc-a192-f4b48c1f015a","page_size":[595,841],"block_position":"51-29"},{"bbox":[291,802,302,811],"type":"page_number","angle":0,"index":1000027,"text":"47 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":51,"is_discarded":true,"id":"96a42999-ea1f-466c-a9c6-b615dc8096b0","page_size":[595,841],"block_position":"51-30"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99999,"text":"5.2 线性映射 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":52,"is_discarded":true,"id":"a0af8810-1096-490f-bbef-34aaa4fad64a","page_size":[595,841],"block_position":"52-0"},{"bbox":[73,73,181,85],"type":"title","angle":0,"index":1,"level":2,"text":"## 定理 5.9 (直和合并定理)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"8edcbdc0-4373-4bd7-bc40-7be754a4c70f","page_size":[595,841],"block_position":"52-1"},{"bbox":[69,89,374,103],"type":"text","angle":0,"index":2,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"fb2a2eeb-4f71-4e7f-bb88-9d2e1b1940ab","page_size":[595,841],"block_position":"52-2"},{"bbox":[219,109,373,121],"type":"interline_equation","angle":0,"index":3,"text":"$$\nV = V _ {1 1} \\oplus \\dots V _ {1 s} \\oplus V _ {2 1} \\oplus \\dots \\oplus V _ {2 t}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":52,"id":"c1a77a84-051e-4309-81a6-99253b569d05","page_size":[595,841],"block_position":"52-3"},{"bbox":[54,138,354,151],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"证明 只证明 $V = V_{1} \\oplus V_{2}$ ， $V_{1} = V_{11} \\oplus V_{12}$ 时， $V = V_{11} \\oplus V_{12} \\oplus V_{2}$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"ce5cb6ba-865d-43ad-b93f-bf99ccc591f1","page_size":[595,841],"block_position":"52-4"},{"bbox":[74,153,235,166],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"首先显然有 $V = V_{11} + V_{12} + V_2$ 。设 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"d82bff85-d1ea-4315-bb71-7adc6817aa5e","page_size":[595,841],"block_position":"52-5"},{"bbox":[209,174,383,186],"type":"interline_equation","angle":0,"index":6,"text":"$$\n0 = \\alpha_ {1 1} + \\alpha_ {1 2} + \\alpha_ {2} = (\\alpha_ {1 1} + \\alpha_ {1 2}) + \\alpha_ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":52,"id":"52064268-dfd1-4155-a2e6-45728cc63bff","page_size":[595,841],"block_position":"52-6"},{"bbox":[54,192,539,221],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"8e45f346-6d4b-4d61-bcb4-8cd714ad8ce2","page_size":[595,841],"block_position":"52-7"},{"bbox":[56,239,126,253],"type":"title","angle":0,"index":8,"level":2,"text":"## 5.1.8 商空间","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"c7d6bcc7-9df7-4fd3-9edc-47d594776090","page_size":[595,841],"block_position":"52-8"},{"bbox":[73,266,186,278],"type":"title","angle":0,"index":9,"level":2,"text":"## 定义 5.10 (商空间的元素)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"338b60b0-be54-40f9-9142-c6354af53f71","page_size":[595,841],"block_position":"52-9"},{"bbox":[69,283,253,295],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"W 是线性空间 V 的子空间， $\\alpha \\in V$ ，定义 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"c6aa58fd-295e-413f-9d6b-be258483877d","page_size":[595,841],"block_position":"52-10"},{"bbox":[235,302,357,315],"type":"interline_equation","angle":0,"index":11,"text":"$$\n\\alpha + W := \\{\\alpha + \\omega : \\omega \\in W \\}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":52,"id":"d99bb1a6-ad13-4b06-a841-0e60d20d5e39","page_size":[595,841],"block_position":"52-11"},{"bbox":[73,336,155,349],"type":"title","angle":0,"index":12,"level":2,"text":"## 定义 5.11 (商空间)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"465d9c54-9c77-4578-82dd-2a70edbdeed2","page_size":[595,841],"block_position":"52-12"},{"bbox":[68,352,465,366],"type":"text","angle":0,"index":14,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"99f4b689-87e5-46d6-b84a-79d6c2b6b45b","page_size":[595,841],"block_position":"52-13"},{"bbox":[73,386,166,399],"type":"title","angle":0,"index":15,"level":2,"text":"## 定义 5.12 (自然映射)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"96d8f814-4012-46d9-bbc8-5ece4fdf4093","page_size":[595,841],"block_position":"52-14"},{"bbox":[69,404,206,417],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"定义 $\\eta : V \\to V / W$ 为自然映射 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"f3e27a9c-e5da-4fcc-8937-5b55e9d73963","page_size":[595,841],"block_position":"52-15"},{"bbox":[57,454,143,470],"type":"title","angle":0,"index":17,"level":2,"text":"## 5.2 线性映射","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"f48bf37d-c8e1-4a57-8127-e725e86f7b92","page_size":[595,841],"block_position":"52-16"},{"bbox":[56,486,174,501],"type":"title","angle":0,"index":18,"level":2,"text":"## 5.2.1 线性映射的概念","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"ecc05bbd-8705-4b06-b297-87f4338de8c9","page_size":[595,841],"block_position":"52-17"},{"bbox":[73,513,167,526],"type":"title","angle":0,"index":19,"level":2,"text":"## 定义 5.13 (线性映射)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"07b71a7a-f3c8-4fe2-8a55-fef0d317a7fd","page_size":[595,841],"block_position":"52-18"},{"bbox":[69,530,273,543],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"$V, W$ 是 $\\mathbb{P}$ 上的线性空间， $\\varphi: V \\to W$ 若满足： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"cd527902-7f55-4b46-a1fe-1b098118974e","page_size":[595,841],"block_position":"52-19"},{"bbox":[83,547,202,559],"type":"interline_equation","angle":0,"index":21,"text":"$$\n\\bullet \\varphi (\\alpha + \\beta) = \\varphi (\\alpha) + \\varphi (\\beta)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":52,"id":"d3afc8c2-101e-4aad-bcea-1dc3e47a8398","page_size":[595,841],"block_position":"52-20"},{"bbox":[84,563,160,575],"type":"interline_equation","angle":0,"index":22,"text":"$$\n\\bullet \\varphi (c \\alpha) = c \\varphi (\\alpha)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":52,"id":"feef39a6-5e51-4de3-be98-3b2c18f9271b","page_size":[595,841],"block_position":"52-21"},{"bbox":[69,577,427,590],"type":"text","angle":0,"index":23,"merge_prev":false,"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 $\\mathrm{Hom}_{\\mathbb{P}}(V,W)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"7f09b6b7-ea1e-465f-ad43-4f8bc43712a3","page_size":[595,841],"block_position":"52-22"},{"bbox":[57,623,126,637],"type":"title","angle":0,"index":24,"level":2,"text":"## 5.2.2 核与像","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"68aee218-d1d8-4dc3-9064-ff36e423f706","page_size":[595,841],"block_position":"52-23"},{"bbox":[73,647,490,660],"type":"text","angle":0,"index":25,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"57549f6a-a1a3-4b2a-83f0-ff05998f250f","page_size":[595,841],"block_position":"52-24"},{"bbox":[73,667,156,680],"type":"title","angle":0,"index":26,"level":2,"text":"## 定义 5.14 (核与像)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"19a05ec2-c966-46fe-aa0f-4b22b6379223","page_size":[595,841],"block_position":"52-25"},{"bbox":[68,683,501,698],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"$U, V$ 是 $\\mathbb{P}$ 上的线性空间， $f: U \\to V$ ，定义 $\\operatorname{Ker}(f) = \\{\\alpha \\in U: f(\\alpha) = 0\\}$ ， $\\operatorname{Im}(f) = \\{f(\\alpha): \\alpha \\in U\\}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"a58dad78-0933-4536-b093-4f51b1a41a12","page_size":[595,841],"block_position":"52-26"},{"bbox":[53,714,469,726],"type":"text","angle":0,"index":28,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":52,"id":"187e6af8-b7af-4eae-a0c4-18d4acddb9ba","page_size":[595,841],"block_position":"52-27"},{"bbox":[291,803,302,811],"type":"page_number","angle":0,"index":1000012,"text":"48 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":52,"is_discarded":true,"id":"8cd0997d-373c-4af0-8f79-9b3e05cc41ec","page_size":[595,841],"block_position":"52-28"}],[{"bbox":[480,47,539,58],"type":"header","angle":0,"index":-99999,"text":"5.2 线性映射 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":53,"is_discarded":true,"id":"7a26ed67-7d66-4f21-a6d5-285a03b7dfbf","page_size":[595,841],"block_position":"53-0"},{"bbox":[73,73,186,85],"type":"title","angle":0,"index":1,"level":2,"text":"## 定理 5.10 (核像维数公式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"2879313e-b26b-487b-97da-fcf5e21dcb2c","page_size":[595,841],"block_position":"53-1"},{"bbox":[68,90,204,102],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"$f \\in \\text{Hom}(U, V)$ 其核与像满足： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"46901299-778a-4d4f-bbca-f86df4abc2e3","page_size":[595,841],"block_position":"53-2"},{"bbox":[218,110,374,122],"type":"interline_equation","angle":0,"index":3,"text":"$$\n\\dim \\operatorname{Ker} (f) + \\dim \\operatorname{Im} (f) = \\dim (U)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":53,"id":"f2433b65-0154-4b14-8708-99ccf94bf528","page_size":[595,841],"block_position":"53-3"},{"bbox":[54,138,539,184],"type":"text","angle":0,"index":4,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"139b81e1-567b-46ea-9b32-6f0706d8e2a5","page_size":[595,841],"block_position":"53-4"},{"bbox":[52,185,539,214],"type":"text","angle":0,"index":5,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"fa4f74de-2c83-4c56-beb3-7c64fad29285","page_size":[595,841],"block_position":"53-5"},{"bbox":[35,232,284,245],"type":"title","angle":0,"index":6,"level":2,"text":"## 练习 5.12 核像维数公式推广证明 证明以下核像命题","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"9ee1a9c0-c6c0-44d8-a6e6-12f6db356eae","page_size":[595,841],"block_position":"53-6"},{"bbox":[72,248,444,260],"type":"text","angle":0,"index":7,"merge_prev":false,"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: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"d58141e1-58ce-4381-878b-93285938f2c3","page_size":[595,841],"block_position":"53-7"},{"bbox":[207,268,385,280],"type":"interline_equation","angle":0,"index":8,"text":"$$\n\\dim (\\mathcal {A} W) + \\dim (\\operatorname{Ker} \\mathcal {A} \\cap W) = \\dim (W)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":53,"id":"fbdedaa3-af53-475a-8c88-4090d3ce362a","page_size":[595,841],"block_position":"53-8"},{"bbox":[73,287,376,300],"type":"text","angle":0,"index":9,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"926fdc3e-f6af-42a3-9008-f3460d50088a","page_size":[595,841],"block_position":"53-9"},{"bbox":[179,306,413,320],"type":"interline_equation","angle":0,"index":10,"text":"$$\n\\dim (\\operatorname{Ker} (\\sigma)) + \\dim (\\operatorname{Im} (\\sigma) \\cap \\operatorname{Ker} (\\tau)) = \\dim (\\operatorname{Ker} (\\tau \\sigma))\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":53,"id":"b6bd2322-af5f-4e33-8c1e-56509f8d17c8","page_size":[595,841],"block_position":"53-10"},{"bbox":[53,325,539,370],"type":"text","angle":0,"index":11,"merge_prev":false,"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 $k_{r+1}=\\cdots=k_{n}=0$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"3c4f37af-d891-4477-84c9-947010ace1ef","page_size":[595,841],"block_position":"53-11"},{"bbox":[52,372,537,401],"type":"text","angle":0,"index":12,"merge_prev":false,"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 $V_{1} \\subseteq \\mathrm{Im}(\\sigma) \\cap \\mathrm{Ker}(\\tau)$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"bac4780d-73a5-413f-9eb9-e2043d2edf7d","page_size":[595,841],"block_position":"53-12"},{"bbox":[52,403,539,433],"type":"text","angle":0,"index":13,"merge_prev":false,"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 $V_1 = \\mathrm{Im}(\\sigma) \\cap \\mathrm{Ker}(\\tau)$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"ba43d485-0302-44b6-8b11-3a2b5b85767e","page_size":[595,841],"block_position":"53-13"},{"bbox":[74,454,205,466],"type":"title","angle":0,"index":14,"level":2,"text":"## 定理 5.11 (核像与单射、满射)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"c7d83612-60f7-413e-bf31-1a27a8dd8d33","page_size":[595,841],"block_position":"53-14"},{"bbox":[68,470,204,482],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"$\\sigma$ 是 $U\\to W$ 的线性映射，则： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"15a7dcf2-490b-4e3d-a70a-725a3e56f943","page_size":[595,841],"block_position":"53-15"},{"bbox":[83,486,227,498],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"- $\\sigma$ 是满射当且仅当 $\\operatorname{Im}(\\sigma) = W$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"913bb823-3989-482e-a277-b88e5aa9d095","page_size":[595,841],"block_position":"53-16"},{"bbox":[83,502,236,513],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"- $\\sigma$ 是单射当且仅当 $\\operatorname{Ker}(\\sigma) = \\{\\mathbf{0}\\}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"13f0ea96-7c92-4636-bf21-d55be8e73385","page_size":[595,841],"block_position":"53-17"},{"bbox":[83,518,477,530],"type":"text","angle":0,"index":19,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"f86c60d9-4c3d-4200-ac87-4638b7533383","page_size":[595,841],"block_position":"53-18"},{"bbox":[74,550,186,563],"type":"title","angle":0,"index":20,"level":2,"text":"## 定理 5.12 (核、像任意性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"e7703ec2-49db-44d4-a8fb-8ddf4e10ec5c","page_size":[595,841],"block_position":"53-19"},{"bbox":[67,567,530,597],"type":"text","angle":0,"index":21,"merge_prev":false,"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 $\\operatorname{Ker} \\mathcal{A} = V_{1}, \\operatorname{Im} \\mathcal{A} = V_{2}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"85e7e57f-bc0a-4d48-b359-8c614318d325","page_size":[595,841],"block_position":"53-20"},{"bbox":[53,612,537,641],"type":"text","angle":0,"index":22,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"e2f87707-fb3e-4510-a21c-92768125623a","page_size":[595,841],"block_position":"53-21"},{"bbox":[35,659,290,671],"type":"text","angle":0,"index":23,"merge_prev":false,"text":"练习 5.13 计算核与像 (1) $A: P^{4} \\rightarrow P^{3}$ ，求 KerA, ImA ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"7581b2a4-bd56-4afb-9125-0c0ec39f9956","page_size":[595,841],"block_position":"53-22"},{"bbox":[198,676,393,735],"type":"interline_equation","angle":0,"index":24,"text":"$$\n\\mathcal {A} \\left( \\begin{array}{c} x _ {1} \\\\ x _ {2} \\\\ x _ {3} \\\\ x _ {4} \\end{array} \\right) = \\left( \\begin{array}{c} x _ {1} - 3 x _ {2} + x _ {3} - 2 x _ {4} \\\\ - x _ {1} - 1 1 x _ {2} + 2 x _ {3} - 5 x _ {4} \\\\ 3 x _ {1} + 5 x _ {2} + x _ {4} \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":53,"id":"83779251-5e7f-4ee3-9324-8fce217911df","page_size":[595,841],"block_position":"53-23"},{"bbox":[52,741,332,753],"type":"text","angle":0,"index":25,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":53,"id":"ac8325c2-e3a0-4bd2-8f12-48010b08f6a2","page_size":[595,841],"block_position":"53-24"},{"bbox":[291,803,302,811],"type":"page_number","angle":0,"index":1000017,"text":"49 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":53,"is_discarded":true,"id":"9b3910f2-f12d-47e3-9e7f-d7b0c285e81c","page_size":[595,841],"block_position":"53-25"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99999,"text":"5.3 线性变换 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":54,"is_discarded":true,"id":"e1aec219-2fdb-4422-ba0d-9f6512c5c4f0","page_size":[595,841],"block_position":"54-0"},{"bbox":[55,73,187,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 5.2.3 幂等变换的核与像","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"b78f32d3-34a4-4c14-9109-60bf0312459e","page_size":[595,841],"block_position":"54-1"},{"bbox":[54,96,539,124],"type":"text","angle":0,"index":2,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"77e18ac4-d90c-4369-8e13-5cb4ba31c190","page_size":[595,841],"block_position":"54-2"},{"bbox":[72,132,166,145],"type":"title","angle":0,"index":3,"level":2,"text":"## 定义 5.15 (幂等变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"b9edc7f8-6f74-4e4d-8cd9-d343010ba291","page_size":[595,841],"block_position":"54-3"},{"bbox":[69,149,285,163],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"若线性变换 $\\mathcal{A}$ 满足 $\\mathcal{A}^2 = \\mathcal{A}$ ，则称其为幂等变换。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"3993b3a0-dec0-45c5-bbbb-6d4a0d7f3a86","page_size":[595,841],"block_position":"54-4"},{"bbox":[72,183,226,195],"type":"title","angle":0,"index":5,"level":2,"text":"## 定理 5.13 (幂等变换核像空间分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"ea0708a9-8d11-4364-9a8c-e079f4be0be8","page_size":[595,841],"block_position":"54-5"},{"bbox":[68,200,185,211],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"$\\mathcal{A}$ 是 $V$ 上的幂等变换，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"dd80cf28-40e2-458d-91cc-060363153b61","page_size":[595,841],"block_position":"54-6"},{"bbox":[255,216,337,227],"type":"interline_equation","angle":0,"index":7,"text":"$$\nV = \\operatorname{Ker} \\mathcal {A} \\oplus \\operatorname{Im} \\mathcal {A}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":54,"id":"74553b55-9c26-4495-84ad-d554230582df","page_size":[595,841],"block_position":"54-7"},{"bbox":[54,243,538,271],"type":"text","angle":0,"index":8,"merge_prev":false,"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 $\\alpha - \\mathcal{A}\\alpha \\in \\operatorname{Ker}\\mathcal{A}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"79b6d005-70e9-4e6a-86fd-cf7a0314a632","page_size":[595,841],"block_position":"54-8"},{"bbox":[54,275,539,302],"type":"text","angle":0,"index":9,"merge_prev":false,"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 $\\alpha = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"3fdc07e3-71d9-4e09-9f69-7e7a29d1d528","page_size":[595,841],"block_position":"54-9"},{"bbox":[35,322,380,333],"type":"text","angle":0,"index":10,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"424147af-e5ca-4162-84d0-114d6ae705df","page_size":[595,841],"block_position":"54-10"},{"bbox":[73,337,280,349],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"(1) $\\mathcal{A},\\mathcal{B}$ 有相同的像当且仅当 $\\mathcal{AB} = \\mathcal{B},\\mathcal{BA} = \\mathcal{A}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"d25f5ea4-eaac-4f64-a0de-4a63b02d9909","page_size":[595,841],"block_position":"54-11"},{"bbox":[73,353,279,364],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"(2) $\\mathcal{A},\\mathcal{B}$ 有相同的核当且仅当 $\\mathcal{AB} = \\mathcal{A},\\mathcal{BA} = \\mathcal{B}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"538b01c5-10fb-42ea-b662-cdadc7b35d6e","page_size":[595,841],"block_position":"54-12"},{"bbox":[52,367,539,396],"type":"text","angle":0,"index":13,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"63569593-2ce9-4b43-a14b-22026581baf6","page_size":[595,841],"block_position":"54-13"},{"bbox":[241,399,352,412],"type":"interline_equation","angle":0,"index":14,"text":"$$\n\\mathcal {A} (\\mathcal {B} \\alpha) = \\mathcal {B} \\alpha \\Rightarrow \\mathcal {A B} = \\mathcal {B}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":54,"id":"e2ac9cb5-627a-445f-bf3d-f3888d6f9682","page_size":[595,841],"block_position":"54-14"},{"bbox":[54,423,137,434],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"同理得到 $\\mathcal{BA} = \\mathcal{A}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"436eef29-f059-4761-b6c9-1aa4ecbaa176","page_size":[595,841],"block_position":"54-15"},{"bbox":[73,438,502,451],"type":"text","angle":0,"index":16,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"cfe8ba27-cd45-4def-8a22-2e71f500fed4","page_size":[595,841],"block_position":"54-16"},{"bbox":[73,454,323,465],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"(2) 左推右：由于 $V = \\operatorname{Im}(\\mathcal{A}) \\oplus \\operatorname{Ker}(\\mathcal{A})$ ，因此 $\\forall \\alpha \\in V$ 有 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"f51e30bc-af42-45ac-b20f-a32fff5ad508","page_size":[595,841],"block_position":"54-17"},{"bbox":[214,473,379,486],"type":"interline_equation","angle":0,"index":18,"text":"$$\n\\alpha = \\alpha_ {1} + \\alpha_ {2}, \\alpha_ {1} \\in \\operatorname{Im} (\\mathcal {A}), \\alpha_ {2} \\in \\operatorname{Ker} \\mathcal {A}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":54,"id":"46419050-70e5-4a39-93b5-6a21aa8f12fa","page_size":[595,841],"block_position":"54-18"},{"bbox":[53,491,462,504],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"7cfed657-5411-494d-9eac-7cc6d6cdf72a","page_size":[595,841],"block_position":"54-19"},{"bbox":[251,512,342,524],"type":"interline_equation","angle":0,"index":20,"text":"$$\n(\\mathcal {B A}) \\alpha = \\mathcal {B} \\alpha_ {1} = \\mathcal {B} \\alpha\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":54,"id":"6e83be4d-781a-4ab0-8889-632e9b5cc8d8","page_size":[595,841],"block_position":"54-20"},{"bbox":[54,530,136,543],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"因此得出 $\\mathcal{BA} = \\mathcal{B}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"0481627f-3a18-43a6-8729-145a1f43bfde","page_size":[595,841],"block_position":"54-21"},{"bbox":[74,547,107,558],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"右推左 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"1f499adf-97d3-4d5a-96ca-9650495258fa","page_size":[595,841],"block_position":"54-22"},{"bbox":[56,577,223,592],"type":"title","angle":0,"index":23,"level":2,"text":"## 5.2.4 一般线性变换的核像分解","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"1559f5b6-baa3-49fc-8bee-07398aa93445","page_size":[595,841],"block_position":"54-23"},{"bbox":[73,605,226,618],"type":"title","angle":0,"index":24,"level":2,"text":"## 定理 5.14 (线性变换核像空间分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"a12a791a-bf42-4cee-90d9-f1483c8e4207","page_size":[595,841],"block_position":"54-24"},{"bbox":[69,621,389,634],"type":"text","angle":0,"index":25,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"50eb9dd4-8416-493f-9274-746f8d31d1b1","page_size":[595,841],"block_position":"54-25"},{"bbox":[243,641,349,654],"type":"interline_equation","angle":0,"index":26,"text":"$$\nV = \\operatorname{Ker} (\\mathcal {A} ^ {n}) \\oplus \\operatorname{Im} (\\mathcal {A} ^ {n})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":54,"id":"7fe897cc-96dc-425b-a108-bbb2eb47ad58","page_size":[595,841],"block_position":"54-26"},{"bbox":[35,666,47,682],"type":"image","angle":0,"index":28,"img_path":"/0c75a68015f2cdf632b969cf7c46095894a2cd05bcb7d257bbf03652b7bd42b8.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":54,"id":"eb3378ab-a223-411d-912f-fb9f93f93dba","page_size":[595,841],"block_position":"54-27"},{"bbox":[53,671,248,682],"type":"title","angle":0,"index":29,"level":2,"text":"## 笔记 该结论在后续证明空间分解法中会用到","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":54,"id":"c7439f81-5439-4fea-bc36-cfef6109295c","page_size":[595,841],"block_position":"54-28"},{"bbox":[291,803,303,811],"type":"page_number","angle":0,"index":1000026,"text":"50 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":54,"is_discarded":true,"id":"6a036983-85a1-49f9-86fe-f75977bd6482","page_size":[595,841],"block_position":"54-29"}],[{"bbox":[480,47,539,58],"type":"header","angle":0,"index":-99998,"text":"5.3 线性变换 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":55,"is_discarded":true,"id":"8e067ef7-9e3f-40e3-a16f-070a75a0ad38","page_size":[595,841],"block_position":"55-0"},{"bbox":[35,695,47,711],"type":"image","angle":0,"index":0,"img_path":"/b7f542caab526a8bf3277323ee7deae5a41b0c8eeb8714d2af1c4ee924c40700.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":55,"id":"4cb35821-4743-46a3-ab03-292f954b3d03","page_size":[595,841],"block_position":"55-1"},{"bbox":[55,74,143,90],"type":"title","angle":0,"index":2,"level":2,"text":"## 5.3 线性变换","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"b0d3d2d1-f561-4853-8533-8eea43b0097d","page_size":[595,841],"block_position":"55-2"},{"bbox":[56,107,186,121],"type":"title","angle":0,"index":3,"level":2,"text":"## 5.3.1 线性变换基本概念","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"79c6a5d1-1e9d-42d0-8386-34f0e902fbb1","page_size":[595,841],"block_position":"55-3"},{"bbox":[73,134,166,148],"type":"title","angle":0,"index":4,"level":2,"text":"## 定义 5.16 (线性变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"f2582488-4418-41de-bf92-ddf4e0615671","page_size":[595,841],"block_position":"55-4"},{"bbox":[67,151,525,180],"type":"text","angle":0,"index":5,"merge_prev":false,"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 $\\operatorname{End}(V)$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"3ab0f75b-450e-434d-b606-3b14202afecd","page_size":[595,841],"block_position":"55-5"},{"bbox":[36,212,539,239],"type":"text","angle":0,"index":6,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"e0f7efac-fd75-4110-900a-d092f602e433","page_size":[595,841],"block_position":"55-6"},{"bbox":[52,243,539,288],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"8e907b0d-54c0-4771-823c-eff3b333533e","page_size":[595,841],"block_position":"55-7"},{"bbox":[73,310,205,322],"type":"title","angle":0,"index":8,"level":2,"text":"## 定理 5.15 (线性变换的可逆性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"c4bd9d74-8385-46c3-b1b3-8219447ca924","page_size":[595,841],"block_position":"55-8"},{"bbox":[69,327,183,338],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"线性变换 T 可逆当且仅当 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"0d9bfb99-ba16-4395-b1c4-5e9794164287","page_size":[595,841],"block_position":"55-9"},{"bbox":[83,343,195,354],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"- $T\\epsilon_{1}, \\cdots, T\\epsilon_{n}$ 线性无关 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"b449f807-8fae-487e-92ac-29e7928c00b1","page_size":[595,841],"block_position":"55-10"},{"bbox":[83,358,137,370],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"- $T$ 是单射 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"d28ac7af-6a8a-403b-89e6-c92ea5a5536d","page_size":[595,841],"block_position":"55-11"},{"bbox":[83,373,136,385],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"- $T$ 是满射 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"4acf533c-047a-421c-9a7f-e4fc51fac0dc","page_size":[595,841],"block_position":"55-12"},{"bbox":[54,401,176,413],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"证明 (1) 从矩阵角度看即可 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"d3c598d0-eace-476f-a712-15525005c3ca","page_size":[595,841],"block_position":"55-13"},{"bbox":[73,417,333,429],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"(2) 左推右：若 $T\\alpha_{1} = T\\alpha_{2}$ ，则两侧作用 $T^{-1}$ 得到 $\\alpha_{1} = \\alpha_{2}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"06daaca3-b6a1-4b1e-a7ad-c34235c0eefe","page_size":[595,841],"block_position":"55-14"},{"bbox":[74,433,387,444],"type":"text","angle":0,"index":15,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"09249648-6571-4a52-922c-a8ab9bc72ded","page_size":[595,841],"block_position":"55-15"},{"bbox":[240,452,353,465],"type":"interline_equation","angle":0,"index":16,"text":"$$\nT (k _ {1} \\epsilon_ {1} + \\dots + k _ {n} \\epsilon_ {n}) = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":55,"id":"72d1c001-a459-4c9a-b8dc-ec3dbf6a1eb3","page_size":[595,841],"block_position":"55-16"},{"bbox":[55,470,449,483],"type":"text","angle":0,"index":17,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"5a47f2c4-6152-4d16-98c6-2cf45ef6c997","page_size":[595,841],"block_position":"55-17"},{"bbox":[73,486,426,498],"type":"text","angle":0,"index":18,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"5603bfbe-31d9-4948-a512-7598bc8065f4","page_size":[595,841],"block_position":"55-18"},{"bbox":[73,503,386,515],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"a7ee7a7c-e73c-4ea0-a0df-dcffc6217c36","page_size":[595,841],"block_position":"55-19"},{"bbox":[242,523,349,535],"type":"interline_equation","angle":0,"index":20,"text":"$$\nT \\epsilon_ {1} = \\eta_ {1}, \\dots , T \\epsilon_ {n} = \\eta_ {n}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":55,"id":"eab8d59b-231d-42b5-b929-56fc3ba66047","page_size":[595,841],"block_position":"55-20"},{"bbox":[54,541,318,554],"type":"text","angle":0,"index":21,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"df6e297a-3527-4db6-bc54-2d7cda43bb24","page_size":[595,841],"block_position":"55-21"},{"bbox":[196,560,398,573],"type":"interline_equation","angle":0,"index":22,"text":"$$\nT (k _ {1} \\epsilon_ {1} + \\dots + k _ {n} \\epsilon_ {n}) = k _ {1} \\eta_ {1} + \\dots + k _ {n} \\eta_ {n} = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":55,"id":"6458de1f-7549-4968-badb-449bf35960e7","page_size":[595,841],"block_position":"55-22"},{"bbox":[54,580,221,592],"type":"text","angle":0,"index":23,"merge_prev":false,"text":"这说明 $k_{1}=\\cdots=k_{n}=0$ ，因此可逆。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"e6631d91-25f5-4bee-b4ab-9896748ae3b2","page_size":[595,841],"block_position":"55-23"},{"bbox":[55,611,294,624],"type":"title","angle":0,"index":24,"level":2,"text":"## 5.3.2 线性变换在基下的矩阵、变换后的坐标","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"48ce403c-5db3-4578-8f5d-f6ee7a835356","page_size":[595,841],"block_position":"55-24"},{"bbox":[73,638,196,650],"type":"title","angle":0,"index":25,"level":2,"text":"## 定义 5.17 (线性变换的矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"8c853afe-5ac3-4c83-9c90-64786abec899","page_size":[595,841],"block_position":"55-25"},{"bbox":[67,655,527,686],"type":"text","angle":0,"index":26,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"68f5c6c9-9f7b-4910-824a-26c4ba0e5d18","page_size":[595,841],"block_position":"55-26"},{"bbox":[53,700,389,713],"type":"text","angle":0,"index":27,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"b9e11b96-332e-46e5-8e5a-7374b55ad5d2","page_size":[595,841],"block_position":"55-27"},{"bbox":[222,719,371,733],"type":"interline_equation","angle":0,"index":29,"text":"$$\nA = (\\epsilon_ {1}, \\dots , \\epsilon_ {n}) ^ {- 1} (\\mathcal {A} \\epsilon_ {1}, \\dots , \\mathcal {A} \\epsilon_ {n})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":55,"id":"8a5cf99f-e83b-4ca8-baf5-c21a485aeb17","page_size":[595,841],"block_position":"55-28"},{"bbox":[55,739,171,751],"type":"text","angle":0,"index":30,"merge_prev":false,"text":"可以用矩阵求逆方式求解。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":55,"id":"f6121e62-11b0-48f3-bb9a-fc05d36b6777","page_size":[595,841],"block_position":"55-29"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":1000027,"text":"51 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":55,"is_discarded":true,"id":"48e2e117-586c-46bd-94e8-3e1b68a7f82e","page_size":[595,841],"block_position":"55-30"}],[{"bbox":[479,47,539,58],"type":"header","angle":0,"index":-99999,"text":"5.3 线性变换 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":56,"is_discarded":true,"id":"4f92426f-f3b7-4663-9f4b-95376042db06","page_size":[595,841],"block_position":"56-0"},{"bbox":[35,72,466,84],"type":"text","angle":0,"index":1,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"4d65d684-cbdc-4429-9608-d978cd1f95a1","page_size":[595,841],"block_position":"56-1"},{"bbox":[73,87,189,100],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"(1) $\\mathcal{A}$ 在 $\\epsilon_3, \\epsilon_2, \\epsilon_1$ 下的矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"950ebfd3-98cc-4d68-b88c-55589d3e949f","page_size":[595,841],"block_position":"56-2"},{"bbox":[73,103,195,115],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"(2) $\\mathcal{A}$ 在 $\\epsilon_1, k\\epsilon_2, \\epsilon_3$ 下的矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"1cdabbf5-c967-4097-ba8c-06e942ade55e","page_size":[595,841],"block_position":"56-3"},{"bbox":[74,118,210,131],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"(3) $\\mathcal{A}$ 在 $\\epsilon_1 + \\epsilon_2, \\epsilon_2, \\epsilon_3$ 下的矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"f30b2397-f688-415f-9fc6-17e317d63027","page_size":[595,841],"block_position":"56-4"},{"bbox":[53,133,235,147],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"解 根据 $T(\\epsilon_1, \\epsilon_2, \\epsilon_3) = (\\epsilon_1, \\epsilon_2, \\epsilon_3)A$ ，得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"27f75163-2d87-4f66-9fd0-bfaf8f0917f2","page_size":[595,841],"block_position":"56-5"},{"bbox":[226,150,358,206],"type":"interline_equation","angle":0,"index":7,"text":"$$\n\\left\\{ \\begin{array}{l} T \\epsilon_ {1} = a _ {1 1} \\epsilon_ {1} + a _ {2 1} \\epsilon_ {2} + a _ {3 1} \\epsilon_ {3} \\\\ T \\epsilon_ {2} = a _ {1 2} \\epsilon_ {1} + a _ {2 2} \\epsilon_ {2} + a _ {3 2} \\epsilon_ {3} \\\\ T \\epsilon_ {3} = a _ {1 3} \\epsilon_ {1} + a _ {2 3} \\epsilon_ {2} + a _ {3 3} \\epsilon_ {3} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":56,"id":"72019dba-7405-4958-911b-c2bcb898c83d","page_size":[595,841],"block_position":"56-6"},{"bbox":[73,210,156,222],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"(1) 重新排列得到: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"e259fd2c-093f-4062-872a-ac6bc190029f","page_size":[595,841],"block_position":"56-7"},{"bbox":[226,222,358,278],"type":"interline_equation","angle":0,"index":9,"text":"$$\n\\left\\{ \\begin{array}{l} T \\epsilon_ {3} = a _ {3 3} \\epsilon_ {3} + a _ {2 3} \\epsilon_ {2} + a _ {1 3} \\epsilon_ {1} \\\\ T \\epsilon_ {2} = a _ {3 2} \\epsilon_ {3} + a _ {2 2} \\epsilon_ {2} + a _ {1 2} \\epsilon_ {1} \\\\ T \\epsilon_ {1} = a _ {3 1} \\epsilon_ {3} + a _ {2 1} \\epsilon_ {2} + a _ {1 1} \\epsilon_ {1} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":56,"id":"687fd44d-9bb1-40e2-a72c-67cb8de5f6de","page_size":[595,841],"block_position":"56-8"},{"bbox":[54,285,107,297],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"得到矩阵为 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"30bfbcd3-39f2-47bd-bf48-525291f86ea6","page_size":[595,841],"block_position":"56-9"},{"bbox":[255,295,338,346],"type":"interline_equation","angle":0,"index":11,"text":"$$\n\\left[ \\begin{array}{l l l} a _ {3 3} & a _ {3 2} & a _ {3 1} \\\\ a _ {2 3} & a _ {2 2} & a _ {2 1} \\\\ a _ {1 3} & a _ {1 2} & a _ {1 1} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":56,"id":"5c252249-df34-4d15-a64e-4feea978ec71","page_size":[595,841],"block_position":"56-10"},{"bbox":[73,352,163,364],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"(2)(3) 同理列开即可 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"d41b2098-1ae3-410c-bf2f-a9e0b28be53f","page_size":[595,841],"block_position":"56-11"},{"bbox":[35,382,539,410],"type":"text","angle":0,"index":13,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"71c2ffd3-33d3-4ee6-9012-ef39c093a26e","page_size":[595,841],"block_position":"56-12"},{"bbox":[249,408,343,477],"type":"interline_equation","angle":0,"index":14,"text":"$$\n\\left[ \\begin{array}{c c c c c} 0 & 0 & \\dots & 0 & 0 \\\\ 1 & 0 & \\dots & 0 & 0 \\\\ \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 0 & 0 & \\dots & 1 & 0 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":56,"id":"f62dca04-2af0-40db-b877-c8041a3fb918","page_size":[595,841],"block_position":"56-13"},{"bbox":[53,484,481,497],"type":"text","angle":0,"index":15,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"d61c1716-49c3-4452-a96c-4b7c2025adfe","page_size":[595,841],"block_position":"56-14"},{"bbox":[73,518,205,531],"type":"title","angle":0,"index":16,"level":2,"text":"## 定理 5.16 (线性变换后的坐标)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"7214af2b-ccd0-4b98-967d-0f72b022098d","page_size":[595,841],"block_position":"56-15"},{"bbox":[66,534,536,555],"type":"text","angle":0,"index":17,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"c38b9f24-6940-4bc5-9157-d96e2f07321c","page_size":[595,841],"block_position":"56-16"},{"bbox":[54,565,364,579],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"证明 $\\mathcal{A}\\alpha=\\mathcal{A}(x_{1}\\epsilon_{1}+\\cdots+x_{n}\\epsilon_{n})=\\mathcal{A}(\\epsilon_{1},\\cdots,\\epsilon_{n})X=(\\epsilon_{1},\\cdots,\\epsilon_{n})AX,$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"13fe199f-3cf8-4b42-b0b7-135905cf860d","page_size":[595,841],"block_position":"56-17"},{"bbox":[55,597,294,611],"type":"title","angle":0,"index":19,"level":2,"text":"## 5.3.3 线性变换在不同基下的矩阵：相似关系","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"e9bc5e11-0d38-43ba-ba47-1711dcef18a7","page_size":[595,841],"block_position":"56-18"},{"bbox":[71,624,246,637],"type":"title","angle":0,"index":20,"level":2,"text":"## 定理 5.17 (线性变换在不同基下的矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"107fe1f2-d20f-413b-b371-9a62008dcc52","page_size":[595,841],"block_position":"56-19"},{"bbox":[67,640,530,671],"type":"text","angle":0,"index":21,"merge_prev":false,"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 $B=T^{-1}AT$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"37abfe5f-032b-4900-8eab-cd250cd65ed1","page_size":[595,841],"block_position":"56-20"},{"bbox":[53,686,539,714],"type":"text","angle":0,"index":22,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"d8dc3b30-2909-4a89-98f6-af83f67a8142","page_size":[595,841],"block_position":"56-21"},{"bbox":[53,716,540,744],"type":"text","angle":0,"index":23,"merge_prev":false,"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 $B=T^{-1}AT$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":56,"id":"11c3aeef-d5a7-4906-a397-a036cd4df9b1","page_size":[595,841],"block_position":"56-22"},{"bbox":[291,803,303,811],"type":"page_number","angle":0,"index":1000005,"text":"52 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":56,"is_discarded":true,"id":"5e63cb2d-d1a7-4615-bccb-64db7d22b3a0","page_size":[595,841],"block_position":"56-23"}],[{"bbox":[429,47,539,58],"type":"header","angle":0,"index":-99998,"text":"5.4 特征值、相似对角化 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":57,"is_discarded":true,"id":"3ee9d3cc-7603-4773-8cd9-5a7130f55c0d","page_size":[595,841],"block_position":"57-0"},{"bbox":[35,71,417,84],"type":"text","angle":0,"index":2,"merge_prev":false,"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 $T \\epsilon_{i} = \\eta_{i}$ , ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"bdd93abf-40b9-4eea-9cf7-3310a5d20a05","page_size":[595,841],"block_position":"57-1"},{"bbox":[174,88,409,125],"type":"interline_equation","angle":0,"index":3,"text":"$$\n\\left\\{ \\begin{array}{l} \\epsilon_ {1} = (1, 0, 1) ^ {T}, \\epsilon_ {2} = (2, 1, 0) ^ {T}, \\epsilon_ {3} = (1, 1, 1) ^ {T} \\\\ \\eta_ {1} = (1, 2, - 1) ^ {T}, \\eta_ {2} = (2, 2, - 1) ^ {T}, \\eta_ {3} = (2, - 1, - 1) ^ {T} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":57,"id":"7c2e07de-2a0a-4260-8206-c53effaa2c91","page_size":[595,841],"block_position":"57-2"},{"bbox":[73,129,253,142],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"(1) 写出 $\\epsilon_{1}, \\epsilon_{2}, \\epsilon_{3}$ 到 $\\eta_{1}, \\eta_{2}, \\eta_{3}$ 的过渡矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"05402eca-b85e-44d4-a39f-b49a8e622d44","page_size":[595,841],"block_position":"57-3"},{"bbox":[73,144,214,157],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"(2) 写出 $T$ 在 $\\epsilon_1, \\epsilon_2, \\epsilon_3$ 下的矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"ad27e4da-57a7-4217-82d5-856505a74f5b","page_size":[595,841],"block_position":"57-4"},{"bbox":[73,159,215,173],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"(3) 写出 $T$ 在 $\\eta_1, \\eta_2, \\eta_3$ 下的矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"0dab3206-725b-4e30-8692-4f2efbd029a7","page_size":[595,841],"block_position":"57-5"},{"bbox":[54,175,170,188],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"解 (1) $e_{1}, e_{2}, e_{3}$ 下的矩阵为 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"d944b45f-41af-4a53-8236-e9df988ff142","page_size":[595,841],"block_position":"57-6"},{"bbox":[267,191,326,205],"type":"interline_equation","angle":0,"index":8,"text":"$$\n(\\eta_ {1}, \\eta_ {2}, \\eta_ {3}) =\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":57,"id":"873b5b74-f37b-42e4-b2e9-34a066c8e42d","page_size":[595,841],"block_position":"57-7"},{"bbox":[73,233,226,246],"type":"title","angle":0,"index":9,"level":2,"text":"## 定理 5.18 (线性变换构造的等价类)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"a26755c5-16ab-49a3-b4c2-4e102e183031","page_size":[595,841],"block_position":"57-8"},{"bbox":[68,250,370,263],"type":"text","angle":0,"index":10,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"36d71478-8e1d-402f-ab2d-18897d3fbf63","page_size":[595,841],"block_position":"57-9"},{"bbox":[56,295,174,310],"type":"title","angle":0,"index":11,"level":2,"text":"## 5.3.4 相似矩阵的性质","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"7b65e087-abf9-4889-a1ea-a6edf89183b6","page_size":[595,841],"block_position":"57-10"},{"bbox":[73,322,216,335],"type":"title","angle":0,"index":12,"level":2,"text":"## 定理 5.19 (相似矩阵的基本性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"4a59099f-1d1c-4de6-8659-a5060809d6ba","page_size":[595,841],"block_position":"57-11"},{"bbox":[68,339,276,352],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"$K$ 上矩阵 $A, B$ 相似，且满足 $B = P^{-1}AP$ ，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"b1b31d8b-64f8-4d53-85ef-0f8c42d8b9c4","page_size":[595,841],"block_position":"57-12"},{"bbox":[85,356,150,369],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"• $r(A) = r(B)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"5ed38758-4a10-4235-86c3-09f3b7fc1e24","page_size":[595,841],"block_position":"57-13"},{"bbox":[84,371,141,383],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"• $A^{m} \\sim B^{m}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"96a5a8d6-232a-4062-be77-65e136dff163","page_size":[595,841],"block_position":"57-14"},{"bbox":[83,386,238,400],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"- $f(x)$ 为多项式，则 $f(A) \\sim f(B)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"462a02e3-5997-403a-a8f7-0fa268b749da","page_size":[595,841],"block_position":"57-15"},{"bbox":[83,402,435,415],"type":"text","angle":0,"index":17,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"9504b9bc-8cef-43d7-a1da-422c6de5a9a2","page_size":[595,841],"block_position":"57-16"},{"bbox":[53,431,462,444],"type":"text","angle":0,"index":18,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"bebbf4ba-1ebb-4ed4-8578-4ba41b51d161","page_size":[595,841],"block_position":"57-17"},{"bbox":[55,467,215,483],"type":"title","angle":0,"index":19,"level":2,"text":"## 5.4 特征值、相似对角化","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"6234787d-c817-4cf4-91d9-aabcd1cb0e02","page_size":[595,841],"block_position":"57-18"},{"bbox":[56,500,186,514],"type":"title","angle":0,"index":20,"level":2,"text":"## 5.4.1 特征值与特征向量","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"1170c577-6a17-449e-88e7-5f158b4475ac","page_size":[595,841],"block_position":"57-19"},{"bbox":[73,527,206,539],"type":"title","angle":0,"index":21,"level":2,"text":"## 定义 5.18 (特征值与特征向量)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"0121bd74-41fd-40fe-9be4-08beb30ae3f0","page_size":[595,841],"block_position":"57-20"},{"bbox":[67,544,524,572],"type":"text","angle":0,"index":22,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"10d140e2-85d9-4641-9595-96cc268675fc","page_size":[595,841],"block_position":"57-21"},{"bbox":[73,593,176,606],"type":"title","angle":0,"index":23,"level":2,"text":"## 定义 5.19 (特征子空间)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"6776a32b-77c6-4fbb-8ded-11edf01fb9e6","page_size":[595,841],"block_position":"57-22"},{"bbox":[68,610,436,624],"type":"text","angle":0,"index":24,"merge_prev":false,"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 $V_{\\lambda}$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"4bef5565-b694-4127-822a-2bde49cfb1f5","page_size":[595,841],"block_position":"57-23"},{"bbox":[73,644,201,656],"type":"title","angle":0,"index":25,"level":2,"text":"## 引理 5.1 (特征子空间的性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"a5d9b0fe-b3fb-4f88-86a0-6438e93d6e39","page_size":[595,841],"block_position":"57-24"},{"bbox":[69,661,192,672],"type":"text","angle":0,"index":26,"merge_prev":false,"text":"特征子空间 $V_{\\lambda}$ 有如下性质： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"83d9708a-2b74-442e-8589-75304d499af9","page_size":[595,841],"block_position":"57-25"},{"bbox":[83,677,178,687],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"- $V_{\\lambda}$ 是 $V$ 的子空间。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"03bbfff4-cc2c-4cc0-b5ad-4c540f8b2870","page_size":[595,841],"block_position":"57-26"},{"bbox":[83,692,465,704],"type":"text","angle":0,"index":28,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"2251852f-949e-4210-9411-16015fad004e","page_size":[595,841],"block_position":"57-27"},{"bbox":[54,721,300,734],"type":"text","angle":0,"index":29,"merge_prev":false,"text":"证明 (1) 只需要证明 $V_{\\lambda}$ 对 $V$ 内的数乘和加法封闭即可。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"0948df5c-0fa8-41cf-94a6-210763ba7fdb","page_size":[595,841],"block_position":"57-28"},{"bbox":[52,737,539,766],"type":"text","angle":0,"index":30,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":57,"id":"09015091-4686-46eb-862f-e4007299ffd9","page_size":[595,841],"block_position":"57-29"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":999999,"text":"53 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":57,"is_discarded":true,"id":"be7751f1-f67b-4512-bce0-11cb3c732fbe","page_size":[595,841],"block_position":"57-30"}],[{"bbox":[429,47,539,58],"type":"header","angle":0,"index":-99999,"text":"5.4 特征值、相似对角化 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":58,"is_discarded":true,"id":"1e33e171-e2e5-4486-a160-3fe1c74995ff","page_size":[595,841],"block_position":"58-0"},{"bbox":[73,73,176,85],"type":"title","angle":0,"index":1,"level":2,"text":"## 定义 5.20 (特征多项式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"74ef1b43-5cd4-4699-8299-1e8a55832af8","page_size":[595,841],"block_position":"58-1"},{"bbox":[68,90,321,103],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"对于 P 上的矩阵 A，特征多项式定义为 $f(\\lambda) = |\\lambda E - A|$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"105f8b35-8c5f-4b27-8666-561621f06a27","page_size":[595,841],"block_position":"58-2"},{"bbox":[72,124,226,137],"type":"title","angle":0,"index":3,"level":2,"text":"## 定理 5.20 (相似矩阵的特征多项式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"44f6059b-fe4c-480d-8ecb-906503bff2db","page_size":[595,841],"block_position":"58-3"},{"bbox":[68,141,406,153],"type":"text","angle":0,"index":4,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"77c78411-6fb5-4eca-95f1-17a644274562","page_size":[595,841],"block_position":"58-4"},{"bbox":[54,169,433,183],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"证明 若 $B = T^{-1}AT$ ，则 $|\\lambda E - B| = |\\lambda E - T^{-1}AT| = |T^{-1}||\\lambda E - A||T| = |\\lambda E - A|$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"25a61e42-21ee-41c0-8809-13e093b56ed6","page_size":[595,841],"block_position":"58-5"},{"bbox":[73,192,196,205],"type":"title","angle":0,"index":6,"level":2,"text":"## 定理 5.21 (特征值基本性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"cafc1809-8988-4e3a-b0f3-447500c4dfcf","page_size":[595,841],"block_position":"58-6"},{"bbox":[68,210,346,226],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"$\\lambda_{1},\\cdots,\\lambda_{n}$ 是矩阵 A 的特征值。则 $\\sum\\lambda_{i}=\\mathrm{tr}(A),\\quad\\prod\\lambda_{i}=|A|$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"67c5126e-59fc-4f03-bc3e-16769a2926f0","page_size":[595,841],"block_position":"58-7"},{"bbox":[53,244,217,256],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"证明 用行列式展开与韦达定理证明。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"21a3c1a5-f58b-4fba-8520-ff71ec405764","page_size":[595,841],"block_position":"58-8"},{"bbox":[35,275,540,303],"type":"text","angle":0,"index":9,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"57c5238b-92a4-485b-b166-afabaa4e9c04","page_size":[595,841],"block_position":"58-9"},{"bbox":[52,306,375,335],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"d5e884f8-ba03-418d-82d4-75b7d780915a","page_size":[595,841],"block_position":"58-10"},{"bbox":[235,342,357,354],"type":"interline_equation","angle":0,"index":11,"text":"$$\nT (\\alpha_ {1} + \\alpha_ {2}) = \\lambda_ {1} \\alpha_ {1} + \\lambda_ {2} \\alpha_ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":58,"id":"d25ca171-670c-45af-8957-038b53dd05b6","page_size":[595,841],"block_position":"58-11"},{"bbox":[54,360,376,374],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"cfe63dfd-0856-419c-942e-7f07b4ae2616","page_size":[595,841],"block_position":"58-12"},{"bbox":[232,380,361,393],"type":"interline_equation","angle":0,"index":13,"text":"$$\n(\\lambda - \\lambda_ {1}) \\alpha_ {1} + (\\lambda - \\lambda_ {2}) \\alpha_ {2} = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":58,"id":"2688a8db-ea94-4565-9195-a563f842fd71","page_size":[595,841],"block_position":"58-13"},{"bbox":[54,400,242,412],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"因此 $\\lambda - \\lambda_{1} = \\lambda - \\lambda_{2} = 0$ ，这与假设矛盾。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"8b6f716b-2dc8-47dd-add2-f420498dacd9","page_size":[595,841],"block_position":"58-14"},{"bbox":[72,414,457,428],"type":"text","angle":0,"index":15,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"d4948123-713b-4eff-9f78-2adccfe6c012","page_size":[595,841],"block_position":"58-15"},{"bbox":[56,461,246,476],"type":"title","angle":0,"index":16,"level":2,"text":"## 5.4.2 特征值与特征向量进一步结论","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"6f0e2552-19dd-4b63-bba2-d849e2ef1d89","page_size":[595,841],"block_position":"58-16"},{"bbox":[72,489,213,501],"type":"title","angle":0,"index":17,"level":2,"text":"## 定理 5.22 (AB 与 BA 的特征值)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"400a89bf-0561-4564-806f-ef661ad854a4","page_size":[595,841],"block_position":"58-17"},{"bbox":[68,505,401,533],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"d4dbc376-0a06-47c5-9758-cacbe2567878","page_size":[595,841],"block_position":"58-18"},{"bbox":[83,536,401,550],"type":"text","angle":0,"index":20,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"b2f18310-6074-4b4c-80d1-9269d612b116","page_size":[595,841],"block_position":"58-19"},{"bbox":[54,565,417,579],"type":"text","angle":0,"index":21,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"6775b0f9-fabd-49a1-acdf-c1be99d9f177","page_size":[595,841],"block_position":"58-20"},{"bbox":[205,586,387,598],"type":"interline_equation","angle":0,"index":22,"text":"$$\n\\mathcal {B} (\\mathcal {A B}) \\xi = \\mathcal {B} (\\lambda_ {0} \\xi) \\Rightarrow (\\mathcal {B A}) (\\mathcal {B} \\xi) = \\lambda_ {0} (\\mathcal {B} \\xi)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":58,"id":"23e3bbd5-bef3-4832-b8e1-56aa31a191fb","page_size":[595,841],"block_position":"58-21"},{"bbox":[73,612,186,624],"type":"title","angle":0,"index":23,"level":2,"text":"## 定理 5.23 (Sylvester 公式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"e17b2261-b1eb-4c97-86f5-0db93f70cac3","page_size":[595,841],"block_position":"58-22"},{"bbox":[69,629,339,642],"type":"text","angle":0,"index":24,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"a593af67-f8d0-48d2-8641-dd6d69c32d5d","page_size":[595,841],"block_position":"58-23"},{"bbox":[244,648,348,662],"type":"interline_equation","angle":0,"index":25,"text":"$$\n\\lambda^ {n} f _ {A B} (\\lambda) = \\lambda^ {m} f _ {B A} (\\lambda)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":58,"id":"e715e322-12ba-48c6-b30b-5d9726db0aad","page_size":[595,841],"block_position":"58-24"},{"bbox":[54,678,179,690],"type":"text","angle":0,"index":26,"merge_prev":false,"text":"证明 证明：设 $\\lambda \\neq 0$ ，根据 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":58,"id":"399e0314-e108-4b51-9d8c-e48ee23c9b7e","page_size":[595,841],"block_position":"58-25"},{"bbox":[61,692,532,737],"type":"interline_equation","angle":0,"index":27,"text":"$$\n\\left[ \\begin{array}{c c} E _ {n} & 0 \\\\ - A & E _ {m} \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {n} & \\frac {1}{\\lambda} B \\\\ A & E _ {m} \\end{array} \\right] = \\left[ \\begin{array}{c c} E _ {n} & \\frac {1}{\\lambda} B \\\\ O & E _ {m} - \\frac {1}{\\lambda} A B \\end{array} \\right] \\Rightarrow | H | = \\left| \\begin{array}{c c} E _ {n} & \\frac {1}{\\lambda} B \\\\ O & E _ {m} - \\frac {1}{\\lambda} A B \\end{array} \\right| = \\left(\\frac {1}{\\lambda}\\right) ^ {m} | \\lambda E _ {m} - A B |\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":58,"id":"0ab8ab10-b766-4231-9687-4d797a9c2cf2","page_size":[595,841],"block_position":"58-26"},{"bbox":[291,803,303,811],"type":"page_number","angle":0,"index":1000017,"text":"54 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":58,"is_discarded":true,"id":"da77e047-ea69-449d-b3e9-c59efacd06d5","page_size":[595,841],"block_position":"58-27"}],[{"bbox":[429,47,539,58],"type":"header","angle":0,"index":-99999,"text":"5.4 特征值、相似对角化 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":59,"is_discarded":true,"id":"3ec0608d-ae15-4ef0-a9eb-59a295d87377","page_size":[595,841],"block_position":"59-0"},{"bbox":[55,72,98,84],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"同理根据 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"7d75135c-0ece-451a-bc49-bafe7c109463","page_size":[595,841],"block_position":"59-1"},{"bbox":[112,84,483,123],"type":"interline_equation","angle":0,"index":2,"text":"$$\n\\left[ \\begin{array}{c c} E _ {n} & \\frac {1}{\\lambda} B \\\\ A & E _ {m} \\end{array} \\right] \\cdot \\left[ \\begin{array}{c c} E _ {n} & 0 \\\\ - A & E _ {m} \\end{array} \\right] = \\left[ \\begin{array}{c c} E _ {n} - \\frac {1}{\\lambda} A B & \\frac {1}{\\lambda} B \\\\ 0 & E _ {m} \\end{array} \\right] \\Rightarrow | H | = \\left(\\frac {1}{\\lambda}\\right) ^ {n} | \\lambda E _ {n} - B A |\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":59,"id":"4f53c341-c826-4841-84c1-d9f2c231b386","page_size":[595,841],"block_position":"59-2"},{"bbox":[54,126,181,139],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"根据 $|H|$ 的值即可推出结论。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"f412d773-4c3a-4b15-8eab-fa7212e7e0b7","page_size":[595,841],"block_position":"59-3"},{"bbox":[35,158,467,170],"type":"text","angle":0,"index":4,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"1690c526-49c2-48ac-a747-7ef24eb0e395","page_size":[595,841],"block_position":"59-4"},{"bbox":[73,173,205,185],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"(2) 计算下面矩阵 $C$ 的特征值 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"6d2063ae-d7eb-4587-ab1d-62e1bea44ce7","page_size":[595,841],"block_position":"59-5"},{"bbox":[221,184,369,253],"type":"interline_equation","angle":0,"index":6,"text":"$$\nC = \\left[ \\begin{array}{c c c c} a _ {1} b _ {1} & a _ {1} b _ {2} & \\dots & a _ {1} b _ {n} \\\\ a _ {2} b _ {1} & a _ {2} b _ {2} & \\dots & a _ {2} b _ {n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a _ {n} b _ {1} & a _ {n} b _ {2} & \\dots & a _ {n} b _ {n} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":59,"id":"e2c204b6-b041-424b-96b3-f9f8b65af12e","page_size":[595,841],"block_position":"59-6"},{"bbox":[53,259,499,288],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"7bf93e35-6f19-4e1c-ba3b-cd5255884acc","page_size":[595,841],"block_position":"59-7"},{"bbox":[53,287,540,330],"type":"text","angle":0,"index":8,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"39603ca3-5119-452b-bf7d-e86731d482be","page_size":[595,841],"block_position":"59-8"},{"bbox":[255,332,338,364],"type":"interline_equation","angle":0,"index":9,"text":"$$\n\\lambda^ {n - 1} (\\lambda - \\sum_ {i = 1} ^ {n} a _ {i} b _ {i})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":59,"id":"22e366db-6f9b-4564-8397-ddc8c7c1f0d2","page_size":[595,841],"block_position":"59-9"},{"bbox":[73,365,148,378],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"(3) 即 (2) 的特例 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"2ebbd898-d39e-45f5-a563-214b4eef3eeb","page_size":[595,841],"block_position":"59-10"},{"bbox":[73,401,206,413],"type":"title","angle":0,"index":11,"level":2,"text":"## 定理 5.24 (可逆的特征值表达)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"255fb0dc-096d-41d7-aae3-37306078562d","page_size":[595,841],"block_position":"59-11"},{"bbox":[68,417,273,429],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"$n$ 阶方阵 $A$ 可逆当且仅当 $A$ 的特征值全不为0 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"c0a9bec9-ee7e-4e81-954a-7a1ba730b728","page_size":[595,841],"block_position":"59-12"},{"bbox":[54,445,179,459],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"证明 用 $\\left|A\\right|=\\lambda_{1}\\cdots\\lambda_{n}$ 即可 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"62f13ccd-6544-4e9e-81e7-fd407837d71e","page_size":[595,841],"block_position":"59-13"},{"bbox":[73,468,196,480],"type":"title","angle":0,"index":14,"level":2,"text":"## 定理 5.25 (逆矩阵的特征值)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"5f111373-9901-42e5-a757-73d4ef8e395c","page_size":[595,841],"block_position":"59-14"},{"bbox":[67,484,412,499],"type":"text","angle":0,"index":15,"merge_prev":false,"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 $\\lambda_{1}^{-1},\\cdots,\\lambda_{n}^{-1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"291ca283-0d78-4b07-bcd9-d5b099c1f8a9","page_size":[595,841],"block_position":"59-15"},{"bbox":[54,514,526,529],"type":"text","angle":0,"index":16,"merge_prev":false,"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 $\\lambda_{1}^{-1},\\cdots,\\lambda_{n}^{-1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"bf4fb6a7-06a2-4b63-aa42-a74d5af385d2","page_size":[595,841],"block_position":"59-16"},{"bbox":[56,546,186,560],"type":"title","angle":0,"index":17,"level":2,"text":"## 5.4.3 相似的进一步研究","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"146f4e37-b89c-49ff-9d5d-6a575e2960ed","page_size":[595,841],"block_position":"59-17"},{"bbox":[73,574,196,587],"type":"title","angle":0,"index":18,"level":2,"text":"## 定理 5.26 (相似的矩阵分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"704248c6-9aa5-41ba-9f1b-9bb33da447fa","page_size":[595,841],"block_position":"59-18"},{"bbox":[68,591,245,602],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"$A, B$ 相似当且仅当存在方阵 $P, Q$ ，使得 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"342d91a9-1598-4166-a38a-607fb5dd03c7","page_size":[595,841],"block_position":"59-19"},{"bbox":[257,610,337,623],"type":"interline_equation","angle":0,"index":20,"text":"$$\nA = P Q, B = Q P\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":59,"id":"85fd5f63-b04f-4f48-bf14-146decc644e0","page_size":[595,841],"block_position":"59-20"},{"bbox":[69,629,186,642],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"这里 $P, Q$ 至少有一个可逆 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"c7671d65-e387-47f4-9825-d7e4192feb3a","page_size":[595,841],"block_position":"59-21"},{"bbox":[54,659,487,671],"type":"text","angle":0,"index":22,"merge_prev":false,"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 $A = PQ, B = QP$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"a8041c94-25d3-4afc-be98-c7121a8181a6","page_size":[595,841],"block_position":"59-22"},{"bbox":[73,681,186,693],"type":"title","angle":0,"index":24,"level":2,"text":"## 定理 5.27 (一些相似结论)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"383e4f96-b68b-41e5-80ff-44ae8169e42c","page_size":[595,841],"block_position":"59-23"},{"bbox":[69,698,166,709],"type":"text","angle":0,"index":25,"merge_prev":false,"text":"$A, B$ 为 $n$ 阶方阵，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"603faad1-622a-4132-b3e8-3b2a49a96e1c","page_size":[595,841],"block_position":"59-24"},{"bbox":[83,714,307,724],"type":"text","angle":0,"index":26,"merge_prev":false,"text":"- $AB$ 与 $BA$ : 若方阵 $A$ 可逆, 则 $AB$ 与 $BA$ 相似 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"fca53da8-d330-4dce-abb2-7ce62ed446b1","page_size":[595,841],"block_position":"59-25"},{"bbox":[83,728,310,740],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"- $A^{-1}$ 与 $B^{-1}$ : 若 $A, B$ 可逆, 则 $A^{-1}$ 与 $B^{-1}$ 相似 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"1ecb5f13-c889-4b80-ae3a-c42afd467693","page_size":[595,841],"block_position":"59-26"},{"bbox":[83,744,302,756],"type":"text","angle":0,"index":28,"merge_prev":false,"text":"- 多项式：任一多项式 $f(x)$ ，有 $f(A), f(B)$ 相似 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":59,"id":"8960eb49-f5c2-45ef-852d-256d41e571fe","page_size":[595,841],"block_position":"59-27"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":1000022,"text":"55 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":59,"is_discarded":true,"id":"22c99a1f-9753-43f7-a1e5-dc84d5a3577e","page_size":[595,841],"block_position":"59-28"}],[{"bbox":[429,47,539,58],"type":"header","angle":0,"index":-99999,"text":"5.4 特征值、相似对角化 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":60,"is_discarded":true,"id":"a9a7ba6c-0295-4522-975f-8b5877d2f145","page_size":[595,841],"block_position":"60-0"},{"bbox":[83,81,286,93],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"- 伴随矩阵：若 $A, B$ 相似，则 $A^{*}$ 与 $B^{*}$ 相似 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"d65c2184-951b-4299-b5d7-faa4948c1752","page_size":[595,841],"block_position":"60-1"},{"bbox":[54,110,242,123],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"证明 (1) $BA = (A^{-1}A)(BA) = A^{-1}(AB)A$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"e1ac05f0-6f34-4bf9-a96d-5775c9fbdae3","page_size":[595,841],"block_position":"60-2"},{"bbox":[73,125,281,138],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"(2) $(P^{-1}AP)^{-1} = B^{-1}$ , 得到 $P^{-1}A^{-1}P = B^{-1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"59b8d3c0-24a7-4df1-949e-416bc3e39991","page_size":[595,841],"block_position":"60-3"},{"bbox":[74,142,112,153],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"(3) 显然 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"2da17f8b-4e4f-428a-982f-1adc6ba26804","page_size":[595,841],"block_position":"60-4"},{"bbox":[74,156,298,169],"type":"interline_equation","angle":0,"index":5,"text":"$$\n(4) B ^ {*} = (P ^ {- 1} A P) ^ {*} = P ^ {*} A ^ {*} (P ^ {- 1}) ^ {*} = P ^ {*} A ^ {*} (P ^ {*}) ^ {- 1}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":60,"id":"650a3713-ee55-4d0e-b25c-e4fb24152984","page_size":[595,841],"block_position":"60-5"},{"bbox":[74,191,200,205],"type":"title","angle":0,"index":6,"level":2,"text":"## 推论5.2(伴随矩阵的特征值)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"0eb6833d-2b18-4e33-9fb6-33064bf26bc9","page_size":[595,841],"block_position":"60-6"},{"bbox":[68,210,327,237],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"若矩阵 A 的特征值为 $\\lambda_{1},\\cdots,\\lambda_{m}$ ，则 $A^{*}$ 的特征值为 $\\prod_{i\\neq j}\\lambda_{i}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"c809fb08-b414-4f74-b76e-a96829c1f61a","page_size":[595,841],"block_position":"60-7"},{"bbox":[54,253,533,265],"type":"text","angle":0,"index":8,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"39b4a1e6-e74e-4062-98e9-3f8387740330","page_size":[595,841],"block_position":"60-8"},{"bbox":[56,285,151,298],"type":"title","angle":0,"index":9,"level":2,"text":"## 5.4.4 相似对角化","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"f266b033-002e-4400-96f5-6f9ed91a77a5","page_size":[595,841],"block_position":"60-9"},{"bbox":[73,312,176,324],"type":"title","angle":0,"index":10,"level":2,"text":"## 定义 5.21 (相似对角化)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"8cad3985-211c-4ceb-9c0a-bdedc94716a5","page_size":[595,841],"block_position":"60-10"},{"bbox":[68,327,498,341],"type":"text","angle":0,"index":11,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"b783baec-6760-4b8a-8abf-77a1a61d37ca","page_size":[595,841],"block_position":"60-11"},{"bbox":[73,361,215,374],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"定义 5.22 (代数重数、几何重数) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"b1743d7c-fecc-4a34-a02b-fbffcd5bfbaf","page_size":[595,841],"block_position":"60-12"},{"bbox":[69,378,468,391],"type":"text","angle":0,"index":13,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"387f3f84-944d-45ff-a219-e4e03b12e0e0","page_size":[595,841],"block_position":"60-13"},{"bbox":[74,412,230,425],"type":"title","angle":0,"index":14,"level":2,"text":"## 引理 5.2 (代数重数与几何重数关系)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"d7fb1355-c1d2-4639-91e1-64121309c85a","page_size":[595,841],"block_position":"60-14"},{"bbox":[69,429,287,441],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"对每个特征值 $\\lambda_{i}$ ，其代数重数大于等于几何重数。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"db781932-145b-4db6-98fb-8ef002528b2b","page_size":[595,841],"block_position":"60-15"},{"bbox":[73,462,221,475],"type":"title","angle":0,"index":17,"level":2,"text":"## 引理 5.3 (不同特征值的特征向量)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"df90820b-b1f3-4c7a-83ea-9246574c1c7c","page_size":[595,841],"block_position":"60-16"},{"bbox":[68,479,246,491],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"A 属于不同特征值的特征向量线性无关。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"dccfb689-10cf-4369-a72e-f4bf987cbde0","page_size":[595,841],"block_position":"60-17"},{"bbox":[54,507,540,534],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"4fcc0b52-f1a8-4a48-b990-616b982f68a9","page_size":[595,841],"block_position":"60-18"},{"bbox":[249,539,344,552],"type":"interline_equation","angle":0,"index":20,"text":"$$\nc _ {1} \\xi_ {1} + \\dots + c _ {k} \\xi_ {k} = 0\\tag{5.1}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":60,"id":"5936a050-0f72-421f-8fcd-faba2f2c2c17","page_size":[595,841],"block_position":"60-19"},{"bbox":[54,562,394,575],"type":"text","angle":0,"index":21,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"7b1207ae-40dc-4801-899b-c6740a395a6b","page_size":[595,841],"block_position":"60-20"},{"bbox":[193,582,401,594],"type":"interline_equation","angle":0,"index":22,"text":"$$\nc _ {1} \\left(\\lambda_ {1} - \\lambda_ {k}\\right) \\xi_ {1} + \\dots + c _ {k - 1} \\left(\\lambda_ {k - 1} - \\lambda_ {k}\\right) \\xi_ {k - 1} = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":60,"id":"2079abed-dd31-4d0b-9043-b03362484b1a","page_size":[595,841],"block_position":"60-21"},{"bbox":[55,601,383,613],"type":"text","angle":0,"index":23,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"8cf599ba-303c-4625-999d-a045588d1402","page_size":[595,841],"block_position":"60-22"},{"bbox":[74,623,186,635],"type":"title","angle":0,"index":24,"level":2,"text":"## 定理 5.28 (可对角化条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"c0921c1e-6439-47b8-ad5b-7772333aedb9","page_size":[595,841],"block_position":"60-23"},{"bbox":[69,640,373,651],"type":"text","angle":0,"index":25,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"a6d035da-f6ae-43d0-b4b4-3d4a692f2a2d","page_size":[595,841],"block_position":"60-24"},{"bbox":[83,655,280,667],"type":"text","angle":0,"index":26,"merge_prev":false,"text":"- 充要条件 1: $\\mathcal{A}$ 有 $n$ 个线性无关特征向量。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"d52a7e09-aee7-4424-8e02-8406cc346f0f","page_size":[595,841],"block_position":"60-25"},{"bbox":[83,671,323,682],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"- 充分条件：若 $\\mathcal{A}$ 有 $n$ 个不同特征值，则 $\\mathcal{A}$ 可对角化 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"d9f26f68-6da1-4514-b21c-97e8b311f7e5","page_size":[595,841],"block_position":"60-26"},{"bbox":[83,685,482,702],"type":"text","angle":0,"index":28,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"4b26d130-76d8-45d8-ab8e-11c946c9cf3b","page_size":[595,841],"block_position":"60-27"},{"bbox":[83,704,314,715],"type":"text","angle":0,"index":29,"merge_prev":false,"text":"- 充要条件3：所有特征值的代数重数等于几何重数 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"3291ebbb-3714-4aab-8556-4560461c2a04","page_size":[595,841],"block_position":"60-28"},{"bbox":[83,717,241,744],"type":"text","angle":0,"index":30,"merge_prev":false,"text":"- 充要条件4： $\\dim V = \\sum_{i=1}^{m} \\dim(V_i)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"5f8dd82c-fde4-4ba3-b3e0-c252e97b5b94","page_size":[595,841],"block_position":"60-29"},{"bbox":[83,744,340,756],"type":"text","angle":0,"index":31,"merge_prev":false,"text":"- 充要条件 5(最小多项式): 最小多项式都是一次幂的乘积 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":60,"id":"47f8c32c-0ee7-483e-9afc-9bbc0bfe5898","page_size":[595,841],"block_position":"60-30"},{"bbox":[291,803,303,812],"type":"page_number","angle":0,"index":1000015,"text":"56 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":60,"is_discarded":true,"id":"83774ea7-737c-411d-a15e-068645a28233","page_size":[595,841],"block_position":"60-31"}],[{"bbox":[470,47,539,58],"type":"header","angle":0,"index":-99999,"text":"5.5 最小多项式 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":61,"is_discarded":true,"id":"e5682526-9e1c-4162-9325-ba3e0603137e","page_size":[595,841],"block_position":"61-0"},{"bbox":[52,72,471,84],"type":"text","angle":0,"index":1,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"66a3bf96-2c73-4ce8-abc7-625cda4ec19b","page_size":[595,841],"block_position":"61-1"},{"bbox":[173,87,418,137],"type":"interline_equation","angle":0,"index":2,"text":"$$\n(A \\eta_ {1}, A \\eta_ {2}, \\dots , A \\eta_ {n}) = (\\eta_ {1}, \\eta_ {2}, \\dots , \\eta_ {n}) \\left[ \\begin{array}{c c c} \\lambda_ {1} & & \\\\ & \\lambda_ {2} & \\\\ & \\ddots & \\lambda_ {n} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":61,"id":"fd0d0172-a510-4c3d-aaf1-4f990558e3c7","page_size":[595,841],"block_position":"61-2"},{"bbox":[73,142,280,154],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"右推左：n 个线性无关特征向量当一组基即可。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"45254a30-eb05-407d-b675-9bb090da0f51","page_size":[595,841],"block_position":"61-3"},{"bbox":[73,158,165,169],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"(2) 根据引理5.3可知 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"74bf40d0-651b-4734-a44f-7ed6bb73c0c8","page_size":[595,841],"block_position":"61-4"},{"bbox":[74,173,217,185],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"后面三个都根据第一个可以推出 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"7c9e32d3-7b84-48ef-b794-bc5ae6e01e4c","page_size":[595,841],"block_position":"61-5"},{"bbox":[35,197,461,231],"type":"text","angle":0,"index":6,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"1dba6105-b9c2-44fa-9964-9e58500bf283","page_size":[595,841],"block_position":"61-6"},{"bbox":[73,243,236,255],"type":"title","angle":0,"index":7,"level":2,"text":"## 定理 5.29 (相关矩阵的可对角化性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"f98328d4-805c-4fde-b7fb-261fb8855c3a","page_size":[595,841],"block_position":"61-7"},{"bbox":[69,259,174,271],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"若矩阵 $A$ 可对角化，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"d737a598-93e3-4fbf-99e9-97eb07f8aa98","page_size":[595,841],"block_position":"61-8"},{"bbox":[81,275,152,287],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"1. $A^{*}$ 可对角化 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"635642b3-a81f-445f-8b93-dd6bc226217c","page_size":[595,841],"block_position":"61-9"},{"bbox":[80,290,244,303],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"2. $f(x)$ 为多项式，则 $f(A)$ 可对角化 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"c762ff9e-44ba-4bda-956e-a8cc9f22a9b8","page_size":[595,841],"block_position":"61-10"},{"bbox":[80,306,223,318],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"3. 若 $A$ 可逆，则 $A^{-1}$ 可对角化 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"d77d0842-7cfe-4e57-a45e-f2cc512b729f","page_size":[595,841],"block_position":"61-11"},{"bbox":[54,333,555,379],"type":"text","angle":0,"index":12,"merge_prev":false,"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 $A^{-1} = \\frac{1}{|A|}A^*$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"78ff3e1f-de35-42bb-93d3-5ef407631020","page_size":[595,841],"block_position":"61-12"},{"bbox":[73,382,119,395],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"(2,3) 显然 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"eaf9ec8d-faa7-48af-86d3-7a3910cda75b","page_size":[595,841],"block_position":"61-13"},{"bbox":[56,412,163,427],"type":"title","angle":0,"index":14,"level":2,"text":"## 5.4.5 相似上三角型","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"e3c89591-885a-4413-a599-4c36b2b899fd","page_size":[595,841],"block_position":"61-14"},{"bbox":[73,440,186,453],"type":"title","angle":0,"index":15,"level":2,"text":"## 定理 5.30 (相似上三角型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"59df6074-fd11-4e39-b402-226b788f89ea","page_size":[595,841],"block_position":"61-15"},{"bbox":[67,456,392,470],"type":"text","angle":0,"index":16,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"ebbbf140-133d-487c-b022-3ca2a1652e3c","page_size":[595,841],"block_position":"61-16"},{"bbox":[54,486,538,514],"type":"text","angle":0,"index":17,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"dddcd977-2ac8-4960-9653-672a6749df56","page_size":[595,841],"block_position":"61-17"},{"bbox":[82,516,510,585],"type":"interline_equation","angle":0,"index":18,"text":"$$\nA P _ {1} = (x _ {1}, \\dots , x _ {n}) \\left[ \\begin{array}{c c c c} \\lambda_ {1} & b _ {1 2} & \\dots & b _ {1 n} \\\\ 0 & b _ {2 2} & \\dots & b _ {2 n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ 0 & b _ {n 1} & \\dots & b _ {n n} \\end{array} \\right] \\Rightarrow P _ {1} ^ {- 1} A P _ {1} = \\left[ \\begin{array}{c c c c} \\lambda_ {1} & b _ {1 2} & \\dots & b _ {1 n} \\\\ 0 & b _ {2 2} & \\dots & b _ {2 n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ 0 & b _ {n 1} & \\dots & b _ {n n} \\end{array} \\right]. = \\left[ \\begin{array}{c c} \\lambda_ {1} & * \\\\ 0 & A _ {1} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":61,"id":"385defeb-f40f-4386-b841-33ef04bf2fdc","page_size":[595,841],"block_position":"61-18"},{"bbox":[53,588,539,634],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":61,"id":"c072c2ae-884e-4ba4-abff-c91b6a336ed9","page_size":[595,841],"block_position":"61-19"},{"bbox":[291,802,302,811],"type":"page_number","angle":0,"index":1000019,"text":"57 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":61,"is_discarded":true,"id":"df335464-6171-409d-9c05-bfd1b93bf64e","page_size":[595,841],"block_position":"61-20"}],[{"bbox":[469,47,539,58],"type":"header","angle":0,"index":-99999,"text":"5.5 最小多项式 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":62,"is_discarded":true,"id":"e5162d3a-d0b7-4115-8bd1-008e2c915439","page_size":[595,841],"block_position":"62-0"},{"bbox":[55,74,158,90],"type":"title","angle":0,"index":1,"level":2,"text":"## 5.5 最小多项式","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"1e3c4fdc-f419-4060-a4ed-39c2f4bc0538","page_size":[595,841],"block_position":"62-1"},{"bbox":[55,106,210,121],"type":"title","angle":0,"index":2,"level":2,"text":"## 5.5.1 最小多项式与基本性质","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"f8207014-905b-418b-b396-81b68bf23547","page_size":[595,841],"block_position":"62-2"},{"bbox":[73,134,236,147],"type":"title","angle":0,"index":3,"level":2,"text":"## 定义 5.23 (零化多项式与最小多项式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"d9b9c226-7c63-4ed7-b088-864db5fee7e3","page_size":[595,841],"block_position":"62-3"},{"bbox":[68,151,296,163],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"$A$ 是数域 $K$ 上的 $n$ 阶方阵， $g(x)$ 是 $K$ 上的多项式， ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"3c01e5e6-41d0-4a2d-8f4d-9d295b9bd3c0","page_size":[595,841],"block_position":"62-4"},{"bbox":[83,167,368,179],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"- 零化多项式：若 $g(A) = 0$ ，则称 $g(x)$ 是 $A$ 的一个零化多项式。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"41e6b620-5c7e-42b7-93bc-796e809908c6","page_size":[595,841],"block_position":"62-5"},{"bbox":[83,183,391,195],"type":"text","angle":0,"index":6,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"4ef4c05d-0a3e-4e6c-b135-87383ffc873c","page_size":[595,841],"block_position":"62-6"},{"bbox":[73,216,205,228],"type":"title","angle":0,"index":7,"level":2,"text":"## 定理 5.31 (最小多项式的性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"17e8d9d6-ce5e-4fdf-90cf-ec028de5b742","page_size":[595,841],"block_position":"62-7"},{"bbox":[69,232,176,244],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"最小多项式有以下性质： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"19263657-d928-45eb-84c7-cad9689bd10f","page_size":[595,841],"block_position":"62-8"},{"bbox":[83,248,227,259],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"- 唯一性：最小多项式是唯一的 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"2713d9dd-4fa8-412d-aee4-635c37d70a73","page_size":[595,841],"block_position":"62-9"},{"bbox":[83,264,277,275],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"- 整除性：最小多项式整除任意零化多项式 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"70ce0463-6a67-40a3-9225-ca9dc9ba0025","page_size":[595,841],"block_position":"62-10"},{"bbox":[83,279,446,292],"type":"text","angle":0,"index":11,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"37e00256-c782-4b70-9446-9ea31e5177c4","page_size":[595,841],"block_position":"62-11"},{"bbox":[83,295,247,307],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"相似：相似矩阵有相同最小多项式 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"37fee92d-c7a2-4363-9ea7-0627e9acc91b","page_size":[595,841],"block_position":"62-12"},{"bbox":[54,323,538,351],"type":"text","angle":0,"index":13,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"965d2bc8-7be1-4954-adfe-64bad96bb92a","page_size":[595,841],"block_position":"62-13"},{"bbox":[53,354,538,382],"type":"text","angle":0,"index":14,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"a39cc7cf-97fa-409c-ae78-ded728cb5cb6","page_size":[595,841],"block_position":"62-14"},{"bbox":[73,386,430,398],"type":"text","angle":0,"index":15,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"3bb680e3-1472-444b-b1fa-5a746e067175","page_size":[595,841],"block_position":"62-15"},{"bbox":[72,401,449,414],"type":"text","angle":0,"index":16,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"c5c98cac-7690-47a0-815a-3105eecc6bd5","page_size":[595,841],"block_position":"62-16"},{"bbox":[56,433,205,446],"type":"title","angle":0,"index":17,"level":2,"text":"## 5.5.2 Hamilton-Cayley 定理","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"d235ac77-14d2-408b-b488-394f5074ecd4","page_size":[595,841],"block_position":"62-17"},{"bbox":[72,460,221,472],"type":"title","angle":0,"index":18,"level":2,"text":"## 定理 5.32 (Hamilton-Cayley 定理)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"4ee07842-24e7-4066-a817-086c80215e1d","page_size":[595,841],"block_position":"62-18"},{"bbox":[67,476,464,490],"type":"text","angle":0,"index":19,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"3945f919-3d2f-431a-9497-6b9b9aed1ccd","page_size":[595,841],"block_position":"62-19"},{"bbox":[52,506,538,535],"type":"text","angle":0,"index":20,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"1f63b154-4107-44d7-99c2-5f5507434585","page_size":[595,841],"block_position":"62-20"},{"bbox":[123,541,470,555],"type":"interline_equation","angle":0,"index":21,"text":"$$\nB (\\lambda) (\\lambda E - A) = \\lambda^ {n} B _ {0} + \\lambda^ {n - 1} (B _ {1} - B _ {0} A) + \\dots + \\lambda (B _ {n - 1} - B _ {n - 2} A) - B _ {n - 1} A\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":62,"id":"2fae9e0f-0757-4c4f-add6-9ed661f77716","page_size":[595,841],"block_position":"62-21"},{"bbox":[54,560,435,574],"type":"text","angle":0,"index":22,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"8b74ed45-4617-4421-a499-a4f1aae97d7b","page_size":[595,841],"block_position":"62-22"},{"bbox":[154,577,429,673],"type":"interline_equation","angle":0,"index":23,"text":"$$\n\\left\\{ \\begin{array}{l} B _ {0} = E \\\\ B _ {1} - B _ {0} A = a _ {1} E \\\\ \\vdots \\\\ B _ {n - 1} - B _ {n - 2} A = a _ {n - 1} E \\\\ - B _ {n - 1} A = a _ {n} E \\end{array} \\right. \\Rightarrow \\left\\{ \\begin{array}{l} B _ {0} A ^ {n} = A ^ {n} \\\\ B _ {1} A ^ {n - 1} - B _ {0} A ^ {n} = a _ {1} A ^ {n - 1} \\\\ \\vdots \\\\ B _ {n - 1} A - B _ {n - 2} A ^ {2} = a _ {n - 2} A \\\\ - B _ {n - 1} A = a _ {n} E \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":62,"id":"e6edb518-fae4-4228-859c-e67e8e129da2","page_size":[595,841],"block_position":"62-23"},{"bbox":[54,677,235,689],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"两端分别相加，左侧为 0，因此右侧为 0 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"ba9b46e0-df7e-4d87-ab4f-ed730d122ea7","page_size":[595,841],"block_position":"62-24"},{"bbox":[52,692,538,720],"type":"text","angle":0,"index":25,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":62,"id":"38ec305c-d2bd-4944-853e-8ad1450f6b30","page_size":[595,841],"block_position":"62-25"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":1000025,"text":"58 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":62,"is_discarded":true,"id":"a0e24bb0-01fd-4e44-8013-e7ae689ca8c1","page_size":[595,841],"block_position":"62-26"}],[{"bbox":[470,47,539,58],"type":"header","angle":0,"index":-99999,"text":"5.5 最小多项式 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":63,"is_discarded":true,"id":"c9a933fc-8a13-49a9-98c4-dde57ad037db","page_size":[595,841],"block_position":"63-0"},{"bbox":[73,74,235,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 推论 5.3 (Hamilton-Cayley 定理推论)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":63,"id":"777184d9-0706-4c57-a480-e1526767301e","page_size":[595,841],"block_position":"63-1"},{"bbox":[68,91,217,102],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"Hamilton-Cayley 定理有以下推论: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":63,"id":"b822115e-42a9-447e-9074-85e1e7d6f673","page_size":[595,841],"block_position":"63-2"},{"bbox":[83,106,299,118],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"- 最小多项式与特征多项式同根(重数可以不同)。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":63,"id":"1bc5d2ad-504a-4bcd-98e0-4e6a736e6e85","page_size":[595,841],"block_position":"63-3"},{"bbox":[56,151,223,165],"type":"title","angle":0,"index":4,"level":2,"text":"## 5.5.3 最小多项式与相似对角化","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":63,"id":"81fb4b37-3a7e-4f19-bf0c-7df77eef3622","page_size":[595,841],"block_position":"63-4"},{"bbox":[72,178,236,190],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"定理 5.33 (最小多项式与相似对角化) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":63,"id":"d2eea14d-4d81-41dc-882b-3312b37cfea1","page_size":[595,841],"block_position":"63-5"},{"bbox":[68,195,461,207],"type":"text","angle":0,"index":6,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":63,"id":"4eb48ff0-e87d-4497-a65e-c8a1a3c925d4","page_size":[595,841],"block_position":"63-6"},{"bbox":[35,240,328,252],"type":"text","angle":0,"index":7,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":63,"id":"332993fb-cee3-46eb-bc17-13fd2163ae01","page_size":[595,841],"block_position":"63-7"},{"bbox":[73,255,230,267],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"(2) 对合矩阵 $A^2 = E$ 一定可对角化 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":63,"id":"33676024-9bba-4b54-bb86-8cf67f6a0d21","page_size":[595,841],"block_position":"63-8"},{"bbox":[73,270,238,282],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"(3) 幂零矩阵 $A^l = O$ 不一定可对角化 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":63,"id":"29829c52-e21a-46bf-94d6-68df3066ea66","page_size":[595,841],"block_position":"63-9"},{"bbox":[53,286,238,298],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"证明 (1) $A(A-E)=O$ ，一定为一次因子 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":63,"id":"47a090e9-4abd-465e-8246-4d409d5746bd","page_size":[595,841],"block_position":"63-10"},{"bbox":[73,301,251,314],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"(2) $(A + E)(A - E) = O$ ，一定可对角化 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":63,"id":"e14cdb8c-e97f-4771-9dde-462159d7748d","page_size":[595,841],"block_position":"63-11"},{"bbox":[73,317,154,329],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"(3) 例如 Jordan 块 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":63,"id":"91c924e5-190b-4065-8dc7-ef003860075a","page_size":[595,841],"block_position":"63-12"},{"bbox":[291,802,302,811],"type":"page_number","angle":0,"index":1000012,"text":"59 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":63,"is_discarded":true,"id":"42a5638b-ab55-4239-a004-fbee899b4f24","page_size":[595,841],"block_position":"63-13"}],[{"bbox":[217,74,387,94],"type":"title","angle":0,"index":0,"level":1,"text":"# 第 6 章 Jordan 标准型","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"7e160d3c-78b7-4851-8620-e7b30ce4673e","page_size":[595,841],"block_position":"64-0"},{"bbox":[55,123,263,139],"type":"title","angle":0,"index":1,"level":2,"text":"## 6.1 Jordan 标准形：空间分解法","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"2c4a529e-a1fa-4c82-9e37-439c4aff9776","page_size":[595,841],"block_position":"64-1"},{"bbox":[55,155,306,169],"type":"title","angle":0,"index":2,"level":2,"text":"## 6.1.1 空间第二分解定理：循环不变子空间分解","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"af99aa0d-56ff-4b1f-9f2c-515c67372fd6","page_size":[595,841],"block_position":"64-2"},{"bbox":[72,183,181,195],"type":"title","angle":0,"index":3,"level":2,"text":"## 定义 6.1 (幂零线性变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"a9e51999-0070-4d87-b77d-728418f9bc56","page_size":[595,841],"block_position":"64-3"},{"bbox":[67,200,368,212],"type":"text","angle":0,"index":4,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"588d5f0e-6be1-47e2-9167-7590f4726fe6","page_size":[595,841],"block_position":"64-4"},{"bbox":[73,232,151,245],"type":"title","angle":0,"index":5,"level":2,"text":"## 引理6.1(幂零基)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"b23fdf6d-ab2e-4954-bbe5-02af6ff206ef","page_size":[595,841],"block_position":"64-5"},{"bbox":[67,249,427,263],"type":"text","angle":0,"index":6,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"409ef9a0-7af5-4526-a9a4-edd0578e45e8","page_size":[595,841],"block_position":"64-6"},{"bbox":[54,279,537,308],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"8d8bd1ec-bce7-490e-b53e-8374dfab1456","page_size":[595,841],"block_position":"64-7"},{"bbox":[72,317,190,329],"type":"title","angle":0,"index":8,"level":2,"text":"## 定义 6.2 (循环不变子空间)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"4f3993b6-4874-4c97-95f8-547623ab1431","page_size":[595,841],"block_position":"64-8"},{"bbox":[68,333,525,364],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"b0d30736-a02a-496e-92ca-3635a80a5ff0","page_size":[595,841],"block_position":"64-9"},{"bbox":[73,381,210,394],"type":"title","angle":0,"index":10,"level":2,"text":"## 定理 6.1 (循环不变子空间分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"a75508d0-8541-4bec-b2d0-dca2d945c0e8","page_size":[595,841],"block_position":"64-10"},{"bbox":[68,398,535,431],"type":"text","angle":0,"index":11,"merge_prev":false,"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 $A^{m_{i}+1}v_{i}=0$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"38a4a94c-bdb0-48ce-b570-085e47d1ecfb","page_size":[595,841],"block_position":"64-11"},{"bbox":[52,444,539,505],"type":"text","angle":0,"index":12,"merge_prev":false,"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 $v_{j} = Au_{j}$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"b00d6808-4b0d-4273-8333-7af3f2f60a1e","page_size":[595,841],"block_position":"64-12"},{"bbox":[74,507,221,518],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"下面说明下面的向量组线性无关: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"02a4cfce-c613-437f-be0d-433a4d8f3b6c","page_size":[595,841],"block_position":"64-13"},{"bbox":[196,525,396,539],"type":"interline_equation","angle":0,"index":14,"text":"$$\n\\mathcal {A} ^ {m _ {1} + 1} u _ {1}, \\dots , \\mathcal {A} u _ {1}, u _ {1}, \\dots , \\mathcal {A} ^ {m _ {n} + 1} u _ {n}, \\dots , u _ {n}\\tag{6.1}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":64,"id":"e3760efe-cfa0-4600-8bcc-449e802422ff","page_size":[595,841],"block_position":"64-14"},{"bbox":[54,545,539,589],"type":"text","angle":0,"index":15,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"60f7a308-2916-4406-8594-a7a90e972033","page_size":[595,841],"block_position":"64-15"},{"bbox":[54,592,539,637],"type":"text","angle":0,"index":16,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"36e9bbd6-f162-4efa-8ffb-92c2ce0c116a","page_size":[595,841],"block_position":"64-16"},{"bbox":[55,654,198,668],"type":"title","angle":0,"index":17,"level":2,"text":"## 6.1.2 核像升链与根子空间","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"855e90fc-a9d7-4d31-a38e-56d9b515ed28","page_size":[595,841],"block_position":"64-17"},{"bbox":[73,677,530,691],"type":"text","angle":0,"index":18,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"79c5cd03-421c-4e6d-a7e9-06e56e5e5088","page_size":[595,841],"block_position":"64-18"},{"bbox":[73,700,181,713],"type":"title","angle":0,"index":19,"level":2,"text":"## 引理 6.2 (核像空间升链)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"46d79fa9-dab7-478b-983b-7c991ad72a38","page_size":[595,841],"block_position":"64-19"},{"bbox":[68,716,536,737],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"上述空间有： $\\{0\\}=M_{0}\\subseteq M_{1}\\subseteq M_{2}\\subseteq\\cdots,\\quad V=N_{0}\\supseteq N_{1}\\supseteq N_{2}\\supseteq\\cdots$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"a08d56c4-e15e-49a3-8f48-03167067d093","page_size":[595,841],"block_position":"64-20"},{"bbox":[54,746,399,760],"type":"text","angle":0,"index":21,"merge_prev":false,"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 $M_i \\subseteq M_{i+1}$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":64,"id":"0afedf16-805e-43cd-953e-792b40856eac","page_size":[595,841],"block_position":"64-21"}],[{"bbox":[400,47,539,58],"type":"header","angle":0,"index":-99998,"text":"6.1 Jordan 标准形：空间分解法 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":65,"is_discarded":true,"id":"89bd1a96-3072-4862-aa1b-8892edcaa1b9","page_size":[595,841],"block_position":"65-0"},{"bbox":[73,70,538,84],"type":"text","angle":0,"index":2,"merge_prev":false,"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 $N_i \\subseteq N_{i-1}$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"84d4b9de-c01d-4441-8ff8-e5888928ed2c","page_size":[595,841],"block_position":"65-1"},{"bbox":[74,94,170,106],"type":"title","angle":0,"index":3,"level":2,"text":"## 引理 6.3 (核空间增长)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"8101a596-ebc4-4af5-ba08-f0b4d384654e","page_size":[595,841],"block_position":"65-2"},{"bbox":[68,111,214,122],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"核空间 $M_{i}$ 的增长满足以下性质： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"2bd61b08-807f-42cc-8bba-e3bd50c8975f","page_size":[595,841],"block_position":"65-3"},{"bbox":[83,126,299,138],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"若 $M_{k} = M_{k + 1}$ ，则 $M_{k} = M_{k + 1} = M_{k + 2} = \\dots$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"3385d817-afeb-4897-ba83-749cae9a1a20","page_size":[595,841],"block_position":"65-4"},{"bbox":[83,142,299,154],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"若 dimV = n，则 $M_{n} = M_{n+1} = M_{n+2} = \\cdots$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"480e1d29-5359-452b-9994-21d3b27ba8ce","page_size":[595,841],"block_position":"65-5"},{"bbox":[54,170,539,214],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"645fe278-118a-4ac3-b53d-e98ce6754048","page_size":[595,841],"block_position":"65-6"},{"bbox":[54,218,538,247],"type":"text","angle":0,"index":8,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"3fed6bbd-ba41-49c7-8ebe-2cb477bbd03e","page_size":[595,841],"block_position":"65-7"},{"bbox":[73,255,200,268],"type":"title","angle":0,"index":9,"level":2,"text":"## 定理 6.2 (幂次变换核像分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"800da7cd-2d4f-401c-a74e-1784cebcb0f9","page_size":[595,841],"block_position":"65-8"},{"bbox":[69,272,215,285],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"$V = M_{n} \\oplus N_{n}$ , 其中 $n = dimV$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"2f78ce39-1b77-470a-b6f4-3ecda3ad2fa4","page_size":[595,841],"block_position":"65-9"},{"bbox":[54,301,538,329],"type":"text","angle":0,"index":11,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"bdfc7d5d-22ed-4f8b-9a23-5f00f43e1b90","page_size":[595,841],"block_position":"65-10"},{"bbox":[73,338,309,351],"type":"title","angle":0,"index":12,"level":2,"text":"## 定义 6.3 (广义特征向量与根子空间 (广义特征子空间))","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"37e674ce-a0a8-4f22-8ba1-88a52ce18f3e","page_size":[595,841],"block_position":"65-11"},{"bbox":[83,355,524,383],"type":"text","angle":0,"index":13,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"251b4953-caad-4890-a351-1b54255cc992","page_size":[595,841],"block_position":"65-12"},{"bbox":[83,386,524,415],"type":"text","angle":0,"index":14,"merge_prev":false,"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 $G(\\lambda, \\mathcal{A})$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"bd39cf2c-3082-43bd-ab95-8a700008e82f","page_size":[595,841],"block_position":"65-13"},{"bbox":[73,436,211,449],"type":"title","angle":0,"index":15,"level":2,"text":"## 定理 6.3 (根子空间的具体表达)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"17eaced3-2ac2-49f8-bb3f-5d6f7245484c","page_size":[595,841],"block_position":"65-14"},{"bbox":[68,454,301,466],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"根子空间 $G(\\lambda, \\mathcal{A})$ 满足 $G(\\lambda, \\mathcal{A}) = \\operatorname{Ker}(\\mathcal{A} - \\lambda E)^{dimV}$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"b4650db1-4e4a-4596-9f37-a1c10f992bed","page_size":[595,841],"block_position":"65-15"},{"bbox":[54,484,539,513],"type":"text","angle":0,"index":17,"merge_prev":false,"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 $v \\in Ker(\\mathcal{A} - \\lambda E)^{dimV}$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"3c85deb3-a43e-476a-ad17-f62d76e5eb35","page_size":[595,841],"block_position":"65-16"},{"bbox":[53,515,539,558],"type":"text","angle":0,"index":18,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"d208c747-55a3-4914-8be8-83d14eac5b67","page_size":[595,841],"block_position":"65-17"},{"bbox":[55,577,270,592],"type":"title","angle":0,"index":19,"level":2,"text":"## 6.1.3 空间第一分解定理：根子空间分解","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"d7de2270-575c-41da-b84a-f9c61c66264e","page_size":[595,841],"block_position":"65-18"},{"bbox":[72,604,201,617],"type":"title","angle":0,"index":20,"level":2,"text":"## 定理 6.4 (根子空间的不变性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"fe362fd0-8f23-4537-bcdc-aae522f5cdcc","page_size":[595,841],"block_position":"65-19"},{"bbox":[68,621,208,634],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"根子空间 $G(\\lambda, \\mathcal{A})$ 是 A 不变的。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"336cdec3-73c2-45ac-bb85-43e154b6b94f","page_size":[595,841],"block_position":"65-20"},{"bbox":[53,651,539,680],"type":"text","angle":0,"index":22,"merge_prev":false,"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 $\\mathcal{A} \\alpha \\in G(\\lambda, \\mathcal{A})$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"83774e15-0307-4521-9803-614f217e4145","page_size":[595,841],"block_position":"65-21"},{"bbox":[72,689,270,702],"type":"title","angle":0,"index":23,"level":2,"text":"## 定理 6.5 (空间第一分解定理：根子空间分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"1c9e5b12-f8bd-4210-a7ec-012c3559eef8","page_size":[595,841],"block_position":"65-22"},{"bbox":[67,706,497,719],"type":"text","angle":0,"index":24,"merge_prev":false,"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 $V=G(\\lambda_{1},\\mathcal{A})\\oplus\\cdots\\oplus G(\\lambda_{m},\\mathcal{A})$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"76a328a1-9516-4c45-8e1e-40865b4a4b38","page_size":[595,841],"block_position":"65-23"},{"bbox":[53,736,539,765],"type":"text","angle":0,"index":25,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":65,"id":"06aa906b-ad1e-4175-ad32-a7deb8996066","page_size":[595,841],"block_position":"65-24"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":999999,"text":"61 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":65,"is_discarded":true,"id":"8a5658a0-0abe-4d25-8ea0-237974c5cc73","page_size":[595,841],"block_position":"65-25"}],[{"bbox":[389,47,537,58],"type":"header","angle":0,"index":-99998,"text":"6.2 空间分解法：零化多项式角度 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":66,"is_discarded":true,"id":"18693c6d-eb65-4cc6-b7dd-acc76134c8d5","page_size":[595,841],"block_position":"66-0"},{"bbox":[54,71,539,100],"type":"text","angle":0,"index":2,"merge_prev":false,"lines_deleted":true,"text":"","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"06aa906b-ad1e-4175-ad32-a7deb8996066","page_size":[595,841],"block_position":"66-1"},{"bbox":[55,118,238,132],"type":"title","angle":0,"index":3,"level":2,"text":"## 6.1.4 Jordan 标准形：空间分解法","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"361dfb16-9c45-4df6-b021-6c5eb1c11cd2","page_size":[595,841],"block_position":"66-2"},{"bbox":[72,146,236,158],"type":"title","angle":0,"index":4,"level":2,"text":"## 定理 6.6 (幂零变换的 Jordan 标准型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"6af56ddd-1d6a-43c9-899a-8852a134b7cf","page_size":[595,841],"block_position":"66-3"},{"bbox":[69,163,440,175],"type":"text","angle":0,"index":5,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"947bb471-a8ad-4498-9208-63b0095138ae","page_size":[595,841],"block_position":"66-4"},{"bbox":[167,201,424,274],"type":"interline_equation","angle":0,"index":6,"text":"$$\nJ = \\left[ \\begin{array}{c c c c} J _ {1} & & & \\\\ & J _ {2} & & 0 \\\\ 0 & & \\ddots & \\\\ & & & J _ {s} \\end{array} \\right], \\quad J _ {i} = \\left[ \\begin{array}{c c c c} 0 & 1 & & 0 \\\\ & 0 & \\ddots & \\\\ & & \\ddots & 1 \\\\ 0 & & & 0 \\end{array} \\right] _ {n _ {i} \\times n _ {i}}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":66,"id":"8ce018b1-5a88-4221-a673-c9db69090594","page_size":[595,841],"block_position":"66-5"},{"bbox":[54,280,167,293],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"证明 代入计算显然可得。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"aa92aa3c-f834-4ce6-bebe-73910649d6cc","page_size":[595,841],"block_position":"66-6"},{"bbox":[73,301,220,314],"type":"title","angle":0,"index":8,"level":2,"text":"## 定理 6.7 (核限制变换是幂零变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"e2e526c4-e02e-452f-b13e-a471eb9cea49","page_size":[595,841],"block_position":"66-7"},{"bbox":[67,318,476,332],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"3c1cf44a-067a-46ac-bd98-75f52b13d375","page_size":[595,841],"block_position":"66-8"},{"bbox":[54,349,361,362],"type":"text","angle":0,"index":10,"merge_prev":false,"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 $(\\mathcal{B}|_{M_k})^k = 0$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"9fa7af86-12f2-4501-9387-dbb369e76011","page_size":[595,841],"block_position":"66-9"},{"bbox":[73,371,243,384],"type":"title","angle":0,"index":11,"level":2,"text":"## 定理 6.8 (Jordan 标准型：空间分解法)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"a81f7c8b-8164-4e19-9d99-8d8370b874ab","page_size":[595,841],"block_position":"66-10"},{"bbox":[67,388,526,417],"type":"text","angle":0,"index":12,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"5c74ab4b-e872-4224-837c-36a2ff3b52f6","page_size":[595,841],"block_position":"66-11"},{"bbox":[161,443,430,515],"type":"interline_equation","angle":0,"index":13,"text":"$$\nJ = \\left[ \\begin{array}{c c c c} J _ {1} & & & 0 \\\\ & J _ {2} & & \\\\ & & \\ddots & \\\\ 0 & & & J _ {s} \\end{array} \\right], \\quad J _ {i} = \\left[ \\begin{array}{c c c c} \\lambda_ {i} & 1 & & 0 \\\\ & \\lambda_ {i} & \\ddots & \\\\ & & \\ddots & 1 \\\\ 0 & & & \\lambda_ {i} \\end{array} \\right] _ {n _ {i} \\times n _ {i}}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":66,"id":"b9e948df-1225-4aff-9a49-9656d3c5146a","page_size":[595,841],"block_position":"66-12"},{"bbox":[53,522,539,550],"type":"text","angle":0,"index":14,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"c44359b7-e44a-4d4f-b27d-bab09c97746c","page_size":[595,841],"block_position":"66-13"},{"bbox":[55,573,272,590],"type":"title","angle":0,"index":15,"level":2,"text":"## 6.2 空间分解法：零化多项式角度","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"3a713b25-7e5f-4a9f-81c9-40f9993ca653","page_size":[595,841],"block_position":"66-14"},{"bbox":[55,606,222,620],"type":"title","angle":0,"index":16,"level":2,"text":"## 6.2.1 根子空间分解：矩阵角度","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"8fdaa6cd-9026-425e-aa3d-186320279906","page_size":[595,841],"block_position":"66-15"},{"bbox":[73,634,171,645],"type":"title","angle":0,"index":17,"level":2,"text":"## 定理 6.9 (解空间分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"af0ba201-ab5e-4869-8f62-f7ce2e843e4a","page_size":[595,841],"block_position":"66-16"},{"bbox":[67,650,525,679],"type":"text","angle":0,"index":18,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"7413402a-7251-4815-afc8-2158c44d2801","page_size":[595,841],"block_position":"66-17"},{"bbox":[267,685,325,698],"type":"interline_equation","angle":0,"index":19,"text":"$$\nV = V _ {1} \\oplus V _ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":66,"id":"3ee7149f-2fb0-45d0-a35a-66913cbe0f11","page_size":[595,841],"block_position":"66-18"},{"bbox":[53,714,434,726],"type":"text","angle":0,"index":20,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":66,"id":"3bd12ce0-43d7-4def-8e6e-5729c5c3741a","page_size":[595,841],"block_position":"66-19"},{"bbox":[201,734,391,746],"type":"interline_equation","angle":0,"index":21,"text":"$$\n\\alpha = u (A) f _ {1} (A) \\alpha + v (A) f _ {2} (A) \\alpha := \\alpha_ {1} + \\alpha_ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":66,"id":"3e1cddbf-a06f-4884-b928-c81a51d16d59","page_size":[595,841],"block_position":"66-20"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":999999,"text":"62 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":66,"is_discarded":true,"id":"9797b752-f30c-4160-ae6f-70055157a210","page_size":[595,841],"block_position":"66-21"}],[{"bbox":[389,47,537,58],"type":"header","angle":0,"index":-99998,"text":"6.2 空间分解法：零化多项式角度 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":67,"is_discarded":true,"id":"3bd8f32d-0889-4327-ade1-5c56a0f4caf3","page_size":[595,841],"block_position":"67-0"},{"bbox":[55,72,286,84],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"且满足 $f(A)\\alpha = f_{1}(A)f_{2}(A)\\alpha = 0$ 。故考察 $f_{2}(A)\\alpha_{1}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"628fccad-a2dd-4d60-9e68-d429707e16fe","page_size":[595,841],"block_position":"67-1"},{"bbox":[157,91,437,104],"type":"interline_equation","angle":0,"index":3,"text":"$$\nf _ {2} (A) \\alpha_ {1} = f _ {2} (A) u (A) f _ {1} (A) \\alpha = u (A) f _ {1} (A) f _ {2} (A) = 0 \\Rightarrow \\alpha_ {1} \\in V _ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":67,"id":"9286028c-4515-4898-9c9a-e9a278f11d46","page_size":[595,841],"block_position":"67-2"},{"bbox":[55,111,302,122],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"同理得到 $f_{1}(A)\\alpha_{2} = 0, \\alpha_{2} \\in V_{1}$ ，从而得到 $V = V_{1} + V_{2}$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"6971d7e9-af6f-4d1a-a761-8ac3e671dc84","page_size":[595,841],"block_position":"67-3"},{"bbox":[74,126,329,138],"type":"text","angle":0,"index":5,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"240b9a0c-d0bb-46f5-a6b5-3163e9ac7870","page_size":[595,841],"block_position":"67-4"},{"bbox":[217,146,376,158],"type":"interline_equation","angle":0,"index":6,"text":"$$\n\\alpha = u (A) f _ {1} (A) \\alpha + v (A) f _ {2} (A) \\alpha = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":67,"id":"9b58566c-3ce3-4c20-8eff-3ffc6d9348d0","page_size":[595,841],"block_position":"67-5"},{"bbox":[55,165,121,176],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"因此结论成立。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"c58d1d3a-89af-4f64-b029-b02895e417f5","page_size":[595,841],"block_position":"67-6"},{"bbox":[73,185,176,198],"type":"title","angle":0,"index":8,"level":2,"text":"## 定理 6.10 (多空间分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"228ad238-f9c9-4059-af64-fc9ba1dbdcb9","page_size":[595,841],"block_position":"67-7"},{"bbox":[67,201,524,230],"type":"text","angle":0,"index":9,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"bf791711-e158-4b36-9aaf-fffcf7b56894","page_size":[595,841],"block_position":"67-8"},{"bbox":[245,238,348,250],"type":"interline_equation","angle":0,"index":10,"text":"$$\nV = V _ {1} \\oplus V _ {2} \\oplus \\dots \\oplus V _ {s}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":67,"id":"b3a6ae2c-ec85-459f-8c6c-7c14bca1de62","page_size":[595,841],"block_position":"67-9"},{"bbox":[54,265,167,278],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"证明 用数学归纳法即可。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"2061283a-9842-460d-b553-d9c14443efa1","page_size":[595,841],"block_position":"67-10"},{"bbox":[73,301,215,313],"type":"title","angle":0,"index":12,"level":2,"text":"## 定理 6.11 (空间分解研究矩阵秩)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"ea3f6046-f654-4f9b-998f-814c02ab15d3","page_size":[595,841],"block_position":"67-11"},{"bbox":[69,317,213,331],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"考虑 $f(A)=f_{1}(A)\\cdots f_{s}(A)$ ，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"e3becc2e-0e77-4064-bca7-33494e850744","page_size":[595,841],"block_position":"67-12"},{"bbox":[190,338,402,350],"type":"interline_equation","angle":0,"index":14,"text":"$$\nn - \\mathrm{r} (f (A)) = n - \\mathrm{r} (f _ {1} (A)) + \\dots + n - \\mathrm{r} (f _ {s} (A))\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":67,"id":"4c6b12fb-93ba-4ffa-92b9-8d022e57243c","page_size":[595,841],"block_position":"67-13"},{"bbox":[52,367,539,396],"type":"text","angle":0,"index":15,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"6668cf0f-b589-4942-beff-b573d8c2a7db","page_size":[595,841],"block_position":"67-14"},{"bbox":[35,414,259,426],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"练习 6.1 空间分解研究矩阵的秩 证明以下命题 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"bb1c7d79-65d7-49a9-a6ca-1a6046be90bd","page_size":[595,841],"block_position":"67-15"},{"bbox":[73,429,258,441],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"(1) $A^{2}=A$ 当且仅当 $r(A)+r(A-E)=n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"672a57a4-36b1-4bf7-bbe1-995261675fb0","page_size":[595,841],"block_position":"67-16"},{"bbox":[73,444,279,457],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"(2) $A^2 = E$ 当且仅当 $r(A + E) + r(A - E) = n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"58af6345-77d7-4325-84a4-e1b628e16a0a","page_size":[595,841],"block_position":"67-17"},{"bbox":[73,460,303,472],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"(3) $A^{3}=E$ 当且仅当 $r(A-E)+r(A^{2}+A+E)=n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"d93837da-eb0b-4ad5-be2c-370a632468ae","page_size":[595,841],"block_position":"67-18"},{"bbox":[53,476,537,504],"type":"text","angle":0,"index":20,"merge_prev":false,"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 $r(A)+r(A-E)=n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"e660519e-8a98-4c12-be1a-1d56ef29d2f1","page_size":[595,841],"block_position":"67-19"},{"bbox":[56,538,246,552],"type":"title","angle":0,"index":21,"level":2,"text":"## 6.2.2 根子空间分解：线性变换角度","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"9ca180ab-28d1-4966-b869-ecec22fc6a96","page_size":[595,841],"block_position":"67-20"},{"bbox":[73,565,190,577],"type":"title","angle":0,"index":22,"level":2,"text":"## 引理 6.4 (零变换的核空间)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"26c72460-f22c-4d72-9687-67c97c8380be","page_size":[595,841],"block_position":"67-21"},{"bbox":[68,582,337,594],"type":"text","angle":0,"index":23,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"4d529da5-7989-41e2-a370-5ebd071d90b9","page_size":[595,841],"block_position":"67-22"},{"bbox":[268,602,325,614],"type":"interline_equation","angle":0,"index":24,"text":"$$\n\\operatorname{Ker} (\\mathcal {O}) = V\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":67,"id":"786cdc4d-0dcf-4020-a58f-9c61573116c7","page_size":[595,841],"block_position":"67-23"},{"bbox":[73,635,176,647],"type":"title","angle":0,"index":25,"level":2,"text":"## 定理 6.12 (核空间分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"3d063056-813f-4c66-9fb4-31536fef593b","page_size":[595,841],"block_position":"67-24"},{"bbox":[68,652,465,665],"type":"text","angle":0,"index":26,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"7a06e03b-1cde-4517-ac00-aaf33d444898","page_size":[595,841],"block_position":"67-25"},{"bbox":[210,671,382,686],"type":"interline_equation","angle":0,"index":27,"text":"$$\n\\operatorname{Ker} (f (\\mathcal {A})) = \\operatorname{Ker} (f _ {1} (\\mathcal {A})) \\oplus \\operatorname{Ker} (f _ {2} (\\mathcal {A}))\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":67,"id":"ed5169aa-fb6b-4ab7-8aea-d4ed853b4724","page_size":[595,841],"block_position":"67-26"},{"bbox":[54,702,422,714],"type":"text","angle":0,"index":28,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"0764cb58-1789-429f-aade-3c0117caee76","page_size":[595,841],"block_position":"67-27"},{"bbox":[232,721,360,734],"type":"interline_equation","angle":0,"index":29,"text":"$$\n\\mathcal {E} = u (\\mathcal {A}) f _ {1} (\\mathcal {A}) + v (\\mathcal {A}) f _ {2} (\\mathcal {A})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":67,"id":"ae3258a4-87df-4323-867e-449a1fb5db4d","page_size":[595,841],"block_position":"67-28"},{"bbox":[54,741,139,753],"type":"text","angle":0,"index":30,"merge_prev":false,"text":"对 $\\forall\\alpha\\in\\mathrm{Ker}(f(\\mathcal{A}))$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":67,"id":"2c8ab322-4b0a-48e2-85f1-ff29892c43f1","page_size":[595,841],"block_position":"67-29"},{"bbox":[201,756,392,769],"type":"interline_equation","angle":0,"index":31,"text":"$$\n\\alpha = \\alpha_ {1} + \\alpha_ {2} := u (\\mathcal {A}) f _ {1} (\\mathcal {A}) \\alpha + v (\\mathcal {A}) f _ {2} (\\mathcal {A}) \\alpha\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":67,"id":"3cfdbe37-6230-44fd-a54d-de2ef5cde372","page_size":[595,841],"block_position":"67-30"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":999999,"text":"63 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":67,"is_discarded":true,"id":"be6825c3-42ff-4fa0-a299-3eff347c52a1","page_size":[595,841],"block_position":"67-31"}],[{"bbox":[389,47,538,58],"type":"header","angle":0,"index":-99998,"text":"6.2 空间分解法：零化多项式角度 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":68,"is_discarded":true,"id":"e1bdd82c-4043-417b-9ada-1c380c2aab14","page_size":[595,841],"block_position":"68-0"},{"bbox":[54,71,465,84],"type":"text","angle":0,"index":2,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"5d801593-32e4-49c7-b299-a4e07a23a983","page_size":[595,841],"block_position":"68-1"},{"bbox":[210,91,383,104],"type":"interline_equation","angle":0,"index":3,"text":"$$\n\\operatorname{Ker} (f (\\mathcal {A})) = \\operatorname{Ker} (f _ {1} (\\mathcal {A})) + \\operatorname{Ker} (f _ {2} (\\mathcal {A}))\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":68,"id":"b2f3013d-6a96-46b9-8049-4b4d112de09a","page_size":[595,841],"block_position":"68-2"},{"bbox":[74,111,292,123],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"下证直和：对 $\\forall \\alpha \\in \\mathrm{Ker}(f_1(\\mathcal{A}))\\cap \\mathrm{Ker}(f_2(\\mathcal{A}))$ ，有 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"bd0dd9fc-2854-4c76-b83f-05d80a351534","page_size":[595,841],"block_position":"68-3"},{"bbox":[216,130,377,143],"type":"interline_equation","angle":0,"index":5,"text":"$$\n\\alpha = u (\\mathcal {A}) f _ {1} (\\mathcal {A}) \\alpha + v (\\mathcal {A}) f _ {2} (\\mathcal {A}) \\alpha = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":68,"id":"4e70081e-e062-457f-98d0-372538204dae","page_size":[595,841],"block_position":"68-4"},{"bbox":[55,149,107,161],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"因此为直和 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"6d37b12a-8c7e-4c2d-a1ff-d93f3c6c4150","page_size":[595,841],"block_position":"68-5"},{"bbox":[72,169,256,182],"type":"title","angle":0,"index":7,"level":2,"text":"## 定理 6.13 (全空间分解：零化多项式角度)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"c26a574f-1230-444e-b573-bf300184a2b9","page_size":[595,841],"block_position":"68-6"},{"bbox":[67,186,524,214],"type":"text","angle":0,"index":8,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"79c3e54e-97e4-49fe-a305-c25bb490a148","page_size":[595,841],"block_position":"68-7"},{"bbox":[217,218,375,232],"type":"interline_equation","angle":0,"index":9,"text":"$$\nV = \\operatorname{Ker} (f _ {1} (\\mathcal {A})) \\oplus \\dots \\oplus \\operatorname{Ker} (f _ {s} (\\mathcal {A}))\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":68,"id":"5265c14d-396f-49ff-9b64-6d11c8b37bb4","page_size":[595,841],"block_position":"68-8"},{"bbox":[54,248,361,260],"type":"text","angle":0,"index":10,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"88ed7121-3da0-4fd0-8adb-69f0691aa596","page_size":[595,841],"block_position":"68-9"},{"bbox":[73,270,193,282],"type":"title","angle":0,"index":11,"level":2,"text":"## 引理 6.5 (Hamilton-Cayley)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"a2fd6eea-1659-4e10-a016-372ec426ac95","page_size":[595,841],"block_position":"68-10"},{"bbox":[68,286,357,300],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"bc306c3e-d129-4b5d-a30f-d262cd8468ad","page_size":[595,841],"block_position":"68-11"},{"bbox":[54,317,251,329],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"证明 该定理在最小多项式一章已经证明过了 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"b4d4ba16-9696-48eb-a372-3d82bbdea3df","page_size":[595,841],"block_position":"68-12"},{"bbox":[72,338,266,350],"type":"title","angle":0,"index":14,"level":2,"text":"## 定理 6.14 (根子空间分解：零化多项式角度)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"a4b3ed2c-06e5-4150-90ee-64c4eef405c9","page_size":[595,841],"block_position":"68-13"},{"bbox":[67,354,525,384],"type":"text","angle":0,"index":15,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"629bf39f-0229-4e30-b232-43505fd7c31c","page_size":[595,841],"block_position":"68-14"},{"bbox":[105,389,487,403],"type":"interline_equation","angle":0,"index":16,"text":"$$\nV = \\operatorname{Ker} (\\mathcal {A} - \\lambda_ {1} \\mathcal {E}) ^ {k _ {1}} \\oplus \\dots \\oplus \\operatorname{Ker} (\\mathcal {A} - \\lambda_ {s} \\mathcal {E}) ^ {k _ {s}} = \\operatorname{Ker} (\\mathcal {A} - \\lambda_ {1} \\mathcal {E}) ^ {l _ {1}} \\oplus \\dots \\oplus \\operatorname{Ker} (\\mathcal {A} - \\lambda_ {s} \\mathcal {E}) ^ {l _ {s}}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":68,"id":"3692907a-1479-42a6-a5a1-5ec3e3ddd49d","page_size":[595,841],"block_position":"68-15"},{"bbox":[52,420,340,433],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"证明 根据 Hamilton-Cayley 定理以及零化多项式全空间分解可知。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"5425fabc-0f09-4815-88ce-c662ca5c6a8e","page_size":[595,841],"block_position":"68-16"},{"bbox":[35,451,539,480],"type":"text","angle":0,"index":18,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"4ad2b792-4ce2-4629-8253-3358eb2ac39c","page_size":[595,841],"block_position":"68-17"},{"bbox":[72,502,261,514],"type":"title","angle":0,"index":19,"level":2,"text":"## 定理 6.15 (根子空间分解与 Jordan 标准型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"a3462007-b723-41e3-8058-feacea0bff8d","page_size":[595,841],"block_position":"68-18"},{"bbox":[67,518,526,547],"type":"text","angle":0,"index":20,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"d86a990b-df2c-4a4c-bea5-006da45fe8a1","page_size":[595,841],"block_position":"68-19"},{"bbox":[55,578,238,592],"type":"title","angle":0,"index":21,"level":2,"text":"## 6.2.3 Jordan 阵与循环不变子空间","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"26bba02c-8c83-46ef-b12a-26b11aff0f58","page_size":[595,841],"block_position":"68-20"},{"bbox":[73,606,221,618],"type":"title","angle":0,"index":22,"level":2,"text":"## 引理 6.6 (核限制变换是幂零变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"83348ebc-90f0-44c1-a048-b94e66168c60","page_size":[595,841],"block_position":"68-21"},{"bbox":[67,622,406,640],"type":"text","angle":0,"index":23,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"7f094fec-427a-4db4-93f4-5469a953fe2c","page_size":[595,841],"block_position":"68-22"},{"bbox":[53,655,505,671],"type":"text","angle":0,"index":24,"merge_prev":false,"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 $(\\mathcal{A} - \\lambda_i\\mathcal{E})^k\\big|_{\\mathrm{Ker}(\\mathcal{A} - \\lambda_i\\mathcal{E})^k} = \\mathcal{O}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"132eabb8-57bb-4681-a337-8fcb110f8d65","page_size":[595,841],"block_position":"68-23"},{"bbox":[73,682,215,694],"type":"title","angle":0,"index":25,"level":2,"text":"## 定理 6.16 (循环不变子空间分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"0636cb0a-7327-40f4-a7ca-1077d248b13a","page_size":[595,841],"block_position":"68-24"},{"bbox":[66,698,532,731],"type":"text","angle":0,"index":26,"merge_prev":false,"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 $A^{m_{i}+1}v_{i}=0$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"0e3d7c77-f80e-4190-9e13-8e0dfa2aec84","page_size":[595,841],"block_position":"68-25"},{"bbox":[53,744,246,756],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"证明 在空间第二分解定理中已经证明过了。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":68,"id":"6e6efd01-2f13-46e8-b908-f7f4979da86f","page_size":[595,841],"block_position":"68-26"},{"bbox":[291,803,303,812],"type":"page_number","angle":0,"index":999999,"text":"64 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":68,"is_discarded":true,"id":"375caaa2-4f2d-4dc7-9067-f0e0ad17b9c5","page_size":[595,841],"block_position":"68-27"}],[{"bbox":[412,47,539,58],"type":"header","angle":0,"index":-99998,"text":"6.3 Jordan 标准形： $\\lambda$ 矩阵法 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":69,"is_discarded":true,"id":"52b0fdaf-8704-462b-a390-a76ea0198546","page_size":[595,841],"block_position":"69-0"},{"bbox":[73,73,248,86],"type":"title","angle":0,"index":2,"level":2,"text":"## 定理 6.17 (Jordan 阵与循环不变子空间)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"26af4c35-bab0-4316-a04d-d87011f40dda","page_size":[595,841],"block_position":"69-1"},{"bbox":[68,90,523,119],"type":"text","angle":0,"index":3,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"e8dea3b8-cd41-4985-bf59-ba3fa620d2f7","page_size":[595,841],"block_position":"69-2"},{"bbox":[218,123,373,206],"type":"interline_equation","angle":0,"index":4,"text":"$$\nJ _ {s} (\\lambda_ {i}) = \\left[ \\begin{array}{c c c c c} \\lambda_ {i} & 1 & & & \\\\ & \\lambda_ {i} & 1 & & \\\\ & & \\ddots & \\ddots & \\\\ & & & \\lambda_ {i} & 1 \\\\ & & & & \\lambda_ {i} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":69,"id":"f829808a-2bbf-4cde-8811-806581c299c5","page_size":[595,841],"block_position":"69-3"},{"bbox":[54,222,146,235],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"证明 直接验证即可。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"c67e58b0-1bbb-4c51-bda2-c0fea5d033ac","page_size":[595,841],"block_position":"69-4"},{"bbox":[73,243,219,256],"type":"title","angle":0,"index":6,"level":2,"text":"## 定理 6.18 (Jordan 阵与几何重数)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"815d52f3-d869-47c6-9809-ff6df3684e15","page_size":[595,841],"block_position":"69-5"},{"bbox":[68,260,532,290],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"57699b50-970f-498e-8000-a75ac18f3a5e","page_size":[595,841],"block_position":"69-6"},{"bbox":[54,305,540,333],"type":"text","angle":0,"index":8,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"b63f2fa9-317b-48a8-87a8-a3fa899cf64e","page_size":[595,841],"block_position":"69-7"},{"bbox":[53,336,396,349],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"e72f701b-68b6-4849-a4b8-35cd278d40b1","page_size":[595,841],"block_position":"69-8"},{"bbox":[55,371,245,388],"type":"title","angle":0,"index":10,"level":2,"text":"## 6.3 Jordan 标准形： $\\lambda$ 矩阵法","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"ee78ee22-4a18-47e4-94b9-c9f85d9e5de4","page_size":[595,841],"block_position":"69-9"},{"bbox":[56,404,160,418],"type":"title","angle":0,"index":11,"level":2,"text":"## 6.3.1 $\\lambda$ 矩阵与相抵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"205a1671-4e64-4247-a304-e1fc229f3a85","page_size":[595,841],"block_position":"69-10"},{"bbox":[73,431,149,444],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"定义 6.4 ( $\\lambda$ 矩阵) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"7e777079-d68d-411b-85e1-c7483b0f87e3","page_size":[595,841],"block_position":"69-11"},{"bbox":[68,449,317,461],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"$A(\\lambda)$ 是元素为 $\\lambda$ 多项式的矩阵，则将 $A(\\lambda)$ 称为 $\\lambda$ 矩阵。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"53ae064a-badc-4df1-b34b-7bd56a7dd2a7","page_size":[595,841],"block_position":"69-12"},{"bbox":[73,482,189,495],"type":"title","angle":0,"index":14,"level":2,"text":"## 定义 6.5 ( $\\lambda$ 矩阵初等变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"67d58b15-b583-419b-9c69-7c0056f3379c","page_size":[595,841],"block_position":"69-13"},{"bbox":[69,500,174,512],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"$\\lambda$ 矩阵有三种初等变换： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"323d09e9-d69f-4c0f-9554-f1bba8793d18","page_size":[595,841],"block_position":"69-14"},{"bbox":[83,516,149,527],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"- 互换两行/列 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"c2ac99d5-b826-433d-9d6f-7b9fe5ef211d","page_size":[595,841],"block_position":"69-15"},{"bbox":[83,530,269,543],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"- 用常数 $c$ 乘一行/列 (不能乘 $\\lambda$ 多项式！) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"75d302d5-722b-42c7-a871-4d420fbcde5f","page_size":[595,841],"block_position":"69-16"},{"bbox":[83,545,292,558],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"- 将一行/列乘以一个 $\\lambda$ 多项式加到另一行/列上 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"5ebeb375-6f3b-4384-90af-9920633dec71","page_size":[595,841],"block_position":"69-17"},{"bbox":[73,579,141,592],"type":"title","angle":0,"index":19,"level":2,"text":"## 定义 6.6 (相抵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"5fa4ae9b-7488-4d33-80fe-f725a9c4229f","page_size":[595,841],"block_position":"69-18"},{"bbox":[69,596,408,608],"type":"text","angle":0,"index":20,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"ffd2b361-86bd-4664-ad78-ca44b235cc04","page_size":[595,841],"block_position":"69-19"},{"bbox":[73,629,205,642],"type":"title","angle":0,"index":21,"level":2,"text":"## 定理 6.19 (相抵标准形：法式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"dff3963c-d57b-4ccf-9d89-eaa525670182","page_size":[595,841],"block_position":"69-20"},{"bbox":[67,645,525,676],"type":"text","angle":0,"index":22,"merge_prev":false,"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 $d_{i}(\\lambda)|d_{i+1}(\\lambda)$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"01f4144a-2b31-4caf-bcfe-ac46f251154f","page_size":[595,841],"block_position":"69-21"},{"bbox":[74,695,170,708],"type":"title","angle":0,"index":23,"level":2,"text":"## 引理 6.7 (对角整除化)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"bf52b42e-5887-4181-bc75-cb230af5584a","page_size":[595,841],"block_position":"69-22"},{"bbox":[68,712,512,726],"type":"text","angle":0,"index":24,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"e12b0009-11fe-4d3d-9ac7-9b19c0a0fb27","page_size":[595,841],"block_position":"69-23"},{"bbox":[54,742,287,756],"type":"text","angle":0,"index":25,"merge_prev":false,"text":"证明 先将 $A(\\lambda)$ 中次数最低的元素放置于对角 $a_{11}(\\lambda)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":69,"id":"24b91342-2683-42f5-a303-a0a966284a7d","page_size":[595,841],"block_position":"69-24"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":999999,"text":"65 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":69,"is_discarded":true,"id":"6ebd8e06-c5bf-41ec-bd3f-6f1c0a08ffc8","page_size":[595,841],"block_position":"69-25"}],[{"bbox":[412,47,539,58],"type":"header","angle":0,"index":-99997,"text":"6.3 Jordan 标准形： $\\lambda$ 矩阵法 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":70,"is_discarded":true,"id":"108cd5d7-e2bd-43b3-951d-307855063ff0","page_size":[595,841],"block_position":"70-0"},{"bbox":[35,539,47,555],"type":"image","angle":0,"index":0,"img_path":"/f82ca568e8c477de45bbe1b6a8fb8599b6d29b00dd1fab12ae185a1b2156e3f9.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":70,"id":"579ed5e2-d338-4849-a22a-3eb433e2b568","page_size":[595,841],"block_position":"70-1"},{"bbox":[52,72,538,100],"type":"text","angle":0,"index":3,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"3ed969c6-a2b3-4c1c-92fa-73fbe2b507cf","page_size":[595,841],"block_position":"70-2"},{"bbox":[53,102,539,131],"type":"text","angle":0,"index":4,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"63d58d3f-e680-4aca-ba08-4cf1dff8b465","page_size":[595,841],"block_position":"70-3"},{"bbox":[73,139,236,152],"type":"title","angle":0,"index":5,"level":2,"text":"## 定理 6.20 (法式的计算：初等变换法)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"c32929b6-7285-4e96-a686-711c4722cce2","page_size":[595,841],"block_position":"70-4"},{"bbox":[68,156,342,168],"type":"text","angle":0,"index":6,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"13037705-3690-40f6-84db-72baa65d7c6f","page_size":[595,841],"block_position":"70-5"},{"bbox":[83,172,406,184],"type":"text","angle":0,"index":7,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"6a1516bf-f454-4f7a-b870-1da8cf732177","page_size":[595,841],"block_position":"70-6"},{"bbox":[83,187,267,199],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"- 将矩阵对角整除化：具体见引理的过程 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"bbf46fbc-22ab-4e57-b664-d3e494c0a6ee","page_size":[595,841],"block_position":"70-7"},{"bbox":[83,202,307,215],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"- 消去一列，再消去一行，重复对角整除化和消去 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"44c8fbfd-e9de-4ff6-b4e9-6c5a78c0a92a","page_size":[595,841],"block_position":"70-8"},{"bbox":[73,235,176,248],"type":"title","angle":0,"index":10,"level":2,"text":"## 定理 6.21 (相抵与相似)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"d3d37611-ae32-4011-b6f0-95f8638dc642","page_size":[595,841],"block_position":"70-9"},{"bbox":[69,253,276,264],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"A 与 B 相似当且仅当 $\\lambda E - A$ 与 $\\lambda E - B$ 相抵。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"ef085739-579d-4a47-9d82-096bee98af4b","page_size":[595,841],"block_position":"70-10"},{"bbox":[53,280,466,293],"type":"text","angle":0,"index":12,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"7aaa07a0-cf72-4a86-a807-c2961847801f","page_size":[595,841],"block_position":"70-11"},{"bbox":[54,296,538,340],"type":"text","angle":0,"index":13,"merge_prev":false,"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 $R_m WB^m + \\cdots + R_1 WB + R_0 W = E$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"58e7d55f-46f3-40ef-bf88-a33aed01fee6","page_size":[595,841],"block_position":"70-12"},{"bbox":[54,343,539,371],"type":"text","angle":0,"index":14,"merge_prev":false,"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 $B$ )。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"c0ed7dcb-cd7b-4ee0-851f-693de480a7f0","page_size":[595,841],"block_position":"70-13"},{"bbox":[73,374,459,386],"type":"text","angle":0,"index":15,"merge_prev":false,"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 $B=W^{-1}AW$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"1ab55d30-db64-4bd5-ba7a-1709f0d2f4d2","page_size":[595,841],"block_position":"70-14"},{"bbox":[56,405,210,419],"type":"title","angle":0,"index":16,"level":2,"text":"## 6.3.2 行列式因子、不变因子","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"77bcf405-c4b4-4748-85ec-0829a3aca9f6","page_size":[595,841],"block_position":"70-15"},{"bbox":[73,432,192,444],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"定义 6.7 (行列式 k 阶子式) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"08231e43-a090-448b-af6d-616d9bb33ad3","page_size":[595,841],"block_position":"70-16"},{"bbox":[67,449,320,461],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"$|A|$ 的 k 阶子式是其任选 k 行、k 列组成的矩阵的行列式。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"f71b885c-f8e8-4d1e-9410-a5a053ebeb19","page_size":[595,841],"block_position":"70-17"},{"bbox":[73,483,170,496],"type":"title","angle":0,"index":19,"level":2,"text":"## 定义 6.8 (行列式因子)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"0b8b0131-b30a-40a9-987a-3da43105eaa7","page_size":[595,841],"block_position":"70-18"},{"bbox":[67,500,524,528],"type":"text","angle":0,"index":20,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"0e36a840-2539-4a26-aaf3-e93d197563a4","page_size":[595,841],"block_position":"70-19"},{"bbox":[52,544,539,571],"type":"text","angle":0,"index":21,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"5c3aad82-ef72-4ada-8d02-731223c1ab9f","page_size":[595,841],"block_position":"70-20"},{"bbox":[73,581,211,592],"type":"title","angle":0,"index":22,"level":2,"text":"## 引理 6.8 (行列式因子的相除性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"d1850c93-6f3c-403a-a395-f7d4648b8032","page_size":[595,841],"block_position":"70-21"},{"bbox":[69,597,357,610],"type":"text","angle":0,"index":23,"merge_prev":false,"text":"设 $D_{1}(\\lambda),\\cdots,D_{r}(\\lambda)$ 是 $A(\\lambda)$ 非零行列式因子，则 $D_{i}(\\lambda)|D_{i+1}(\\lambda)$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"676b011f-f0fb-4d47-a444-b105e9dd10ba","page_size":[595,841],"block_position":"70-22"},{"bbox":[53,627,539,671],"type":"text","angle":0,"index":24,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"c8607aaa-6a8e-4f02-9e86-3396bacc51b5","page_size":[595,841],"block_position":"70-23"},{"bbox":[73,679,160,692],"type":"title","angle":0,"index":25,"level":2,"text":"## 定义 6.9 (不变因子)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"c971e81e-8d04-496a-a9f0-de84a7f0d50e","page_size":[595,841],"block_position":"70-24"},{"bbox":[68,695,368,722],"type":"text","angle":0,"index":26,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":70,"id":"e8e9c401-1ee3-495f-852f-f22aed770486","page_size":[595,841],"block_position":"70-25"},{"bbox":[291,803,303,812],"type":"page_number","angle":0,"index":1000000,"text":"66 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":70,"is_discarded":true,"id":"88b17312-d4c0-484d-865b-a2dffa58fe4f","page_size":[595,841],"block_position":"70-26"}],[{"bbox":[431,47,539,58],"type":"header","angle":0,"index":-99999,"text":"6.4 $\\lambda$ 矩阵的进一步研究 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":71,"is_discarded":true,"id":"e3cc3cbe-971f-4be6-b770-4d9e49873af9","page_size":[595,841],"block_position":"71-0"},{"bbox":[73,73,255,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 定理 6.22 (行列式因子、不变因子的性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"9666f654-da58-4d5f-b979-5f8f0e7575dd","page_size":[595,841],"block_position":"71-1"},{"bbox":[83,89,524,117],"type":"text","angle":0,"index":2,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"669b3574-ce71-4676-a482-d572e06a106f","page_size":[595,841],"block_position":"71-2"},{"bbox":[83,121,405,133],"type":"text","angle":0,"index":3,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"094f5c10-4987-4a17-bdda-2e2824726fe3","page_size":[595,841],"block_position":"71-3"},{"bbox":[54,149,460,163],"type":"text","angle":0,"index":4,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"2d2f6820-2dd9-4069-9aea-3b6621f2f41d","page_size":[595,841],"block_position":"71-4"},{"bbox":[54,165,538,193],"type":"text","angle":0,"index":5,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"ea667c38-b175-4103-9493-9632a18ce07f","page_size":[595,841],"block_position":"71-5"},{"bbox":[73,200,226,213],"type":"title","angle":0,"index":6,"level":2,"text":"## 定理 6.23 (不变因子与最小多项式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"66e7379c-0190-44eb-bf98-838598a1546b","page_size":[595,841],"block_position":"71-6"},{"bbox":[68,218,279,231],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"$A$ 的最小多项式即 $\\lambda E - A$ 的最后一个不变因子 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"186f6af7-dae2-476c-8400-53cd81d2b067","page_size":[595,841],"block_position":"71-7"},{"bbox":[53,247,539,275],"type":"text","angle":0,"index":8,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"157c9651-f65d-4fd6-82fe-af5bd10526a9","page_size":[595,841],"block_position":"71-8"},{"bbox":[56,294,229,308],"type":"title","angle":0,"index":9,"level":2,"text":"## 6.3.3 初等因子与 Jordan 标准型","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"f86c6772-0230-4923-a18c-4dad18b594ee","page_size":[595,841],"block_position":"71-9"},{"bbox":[73,321,166,333],"type":"title","angle":0,"index":10,"level":2,"text":"## 定义 6.10 (初等因子)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"3ff28ca9-afab-4110-8e15-6f39f1136d3b","page_size":[595,841],"block_position":"71-10"},{"bbox":[68,338,415,351],"type":"text","angle":0,"index":11,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"3451826f-e848-4d66-9387-c3b1bb393dd8","page_size":[595,841],"block_position":"71-11"},{"bbox":[73,371,215,384],"type":"title","angle":0,"index":12,"level":2,"text":"## 定理 6.24 (对角化计算初等因子)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"7ca7398f-53f1-4ab5-a8b5-ecf223edb2bc","page_size":[595,841],"block_position":"71-12"},{"bbox":[68,387,286,400],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"$\\lambda E-A$ 经过初等变化变为对角阵 (不一定要法式) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"d9a60667-2b6b-4e6c-87a0-4291001c1eef","page_size":[595,841],"block_position":"71-13"},{"bbox":[227,403,365,470],"type":"interline_equation","angle":0,"index":14,"text":"$$\n\\left( \\begin{array}{c c c c} f _ {1} (\\lambda) & & & \\\\ & f _ {2} (\\lambda) & & \\\\ & & \\ddots & \\\\ & & & f _ {n} (\\lambda) \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":71,"id":"9be800a8-d87b-4fba-8bc3-8c03954c6326","page_size":[595,841],"block_position":"71-14"},{"bbox":[69,473,481,486],"type":"text","angle":0,"index":15,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"e3258692-edc8-47a4-92f9-d3002c9ff148","page_size":[595,841],"block_position":"71-15"},{"bbox":[74,507,266,520],"type":"title","angle":0,"index":16,"level":2,"text":"## 引理 6.9 (Jordan 块、Jordan 阵的初等因子)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"171f559f-eb58-4908-8bf7-5e0d9b97759d","page_size":[595,841],"block_position":"71-16"},{"bbox":[68,524,524,554],"type":"text","angle":0,"index":17,"merge_prev":false,"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 $(\\lambda - \\lambda_{1})^{r_{1}}, \\cdots, (\\lambda - \\lambda_{k})^{r_{k}}$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"e3ccf32b-2c5c-4e5c-bac1-8bb34678c723","page_size":[595,841],"block_position":"71-17"},{"bbox":[54,570,469,582],"type":"text","angle":0,"index":18,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"05f97d36-5b0d-43eb-9af5-f8aa8e73fd3b","page_size":[595,841],"block_position":"71-18"},{"bbox":[73,586,490,598],"type":"text","angle":0,"index":19,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"0add75c2-f062-4727-abc8-7aa47be0d51b","page_size":[595,841],"block_position":"71-19"},{"bbox":[73,608,271,619],"type":"title","angle":0,"index":20,"level":2,"text":"## 定理 6.25 (使用初等因子计算 Jordan 标准型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"498bf1cf-27e4-40ba-8171-7831e9143d20","page_size":[595,841],"block_position":"71-20"},{"bbox":[68,623,527,652],"type":"text","angle":0,"index":21,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"b334a6b2-fa46-4f8f-9cac-fd8d997cb852","page_size":[595,841],"block_position":"71-21"},{"bbox":[73,672,249,685],"type":"title","angle":0,"index":22,"level":2,"text":"## 定理 6.26 (Jordan 标准型存在充要条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"8d24c2e0-3067-42e2-9a55-016bf2813390","page_size":[595,841],"block_position":"71-22"},{"bbox":[68,689,391,702],"type":"text","angle":0,"index":23,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":71,"id":"e47fb419-1d8b-4ab9-8461-5e0c006106e7","page_size":[595,841],"block_position":"71-23"},{"bbox":[291,802,302,812],"type":"page_number","angle":0,"index":1000023,"text":"67 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":71,"is_discarded":true,"id":"74139880-fa4f-40ca-bc4c-00e658bbba61","page_size":[595,841],"block_position":"71-24"}],[{"bbox":[430,47,539,58],"type":"header","angle":0,"index":-99999,"text":"6.4 $\\lambda$ 矩阵的进一步研究 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":72,"is_discarded":true,"id":"e54ec659-e086-4d98-8ccf-b6c40fedf2fc","page_size":[595,841],"block_position":"72-0"},{"bbox":[55,74,214,91],"type":"title","angle":0,"index":1,"level":2,"text":"## 6.4 $\\lambda$ 矩阵的进一步研究","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":72,"id":"59c9fd11-a923-468f-b78c-23d87cb8bfd8","page_size":[595,841],"block_position":"72-1"},{"bbox":[55,107,227,121],"type":"title","angle":0,"index":2,"level":2,"text":"## 6.4.1 Jordan 标准型的直接计算","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":72,"id":"95f8e9c0-9dc3-4d6f-aaa7-a386c51b4b9e","page_size":[595,841],"block_position":"72-2"},{"bbox":[52,131,539,159],"type":"text","angle":0,"index":3,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":72,"id":"a2d5b26e-e6b7-4beb-a308-a4cc32e70103","page_size":[595,841],"block_position":"72-3"},{"bbox":[35,178,309,190],"type":"text","angle":0,"index":4,"merge_prev":false,"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: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":72,"id":"64129cb7-b372-43cd-95b7-bf08578033bb","page_size":[595,841],"block_position":"72-4"},{"bbox":[247,193,343,242],"type":"interline_equation","angle":0,"index":5,"text":"$$\nA = \\left[ \\begin{array}{c c c} - 1 & 1 & 3 \\\\ 3 & 0 & - 4 \\\\ - 2 & 1 & 4 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":72,"id":"06fbd73c-8c68-4d8d-87e7-6aee5a6bc7a5","page_size":[595,841],"block_position":"72-5"},{"bbox":[52,245,443,258],"type":"text","angle":0,"index":6,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":72,"id":"55c199dc-0d34-4942-ab43-bfd8088355f6","page_size":[595,841],"block_position":"72-6"},{"bbox":[55,260,595,310],"type":"interline_equation","angle":0,"index":7,"text":"$$\n\\left[\\begin{array}{c c c}1&0&0\\\\0&\\lambda - 1&- \\lambda + 1\\\\0&\\lambda^ {2} + \\lambda - 3&- 3 \\lambda + 4\\end{array}\\right]\\rightarrow \\left[\\begin{array}{c c c}1&0&0\\\\0&\\lambda - 1&- \\lambda + 1\\\\0&(\\lambda^ {2} + \\lambda - 3) - (\\lambda - 1) (\\lambda + 2)&(- 3 \\lambda + 4) - (\\lambda + 2) (- \\lambda + 1)\\end{array}\\right] = \\left[\\begin{array}{c c c}1&0&\\\\0&\\lambda - 1&- \\lambda\\\\0&- 1&\\lambda^ {2} - \\lambda^ {2}\\end{array}\\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":72,"id":"90bd5fe4-f481-4af1-b297-730aa0b7f9a5","page_size":[595,841],"block_position":"72-7"},{"bbox":[55,312,152,324],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"因此最终可以化简为： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":72,"id":"23431cad-538e-48ce-8585-5cfdc8e56559","page_size":[595,841],"block_position":"72-8"},{"bbox":[252,322,341,374],"type":"interline_equation","angle":0,"index":9,"text":"$$\n\\left[ \\begin{array}{c c c c} 1 & & & \\\\ & 1 & & \\\\ & & (\\lambda - 1) ^ {3} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":72,"id":"de88c971-1dad-4cd4-8c2a-05b28a30dec3","page_size":[595,841],"block_position":"72-9"},{"bbox":[35,380,542,534],"type":"text","angle":0,"index":10,"merge_prev":false,"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： $J = \\left[ \\begin{array}{ccccc}0 &  &  &  \\\\  & 0 &  &  \\\\  &  & 2 \\end{array} \\right]$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":72,"id":"260cf220-fe24-4d52-addc-53a11171ef85","page_size":[595,841],"block_position":"72-10"},{"bbox":[52,542,539,571],"type":"text","angle":0,"index":11,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":72,"id":"3d6d3ef6-5303-4657-baf7-81cd0318905b","page_size":[595,841],"block_position":"72-11"},{"bbox":[35,588,357,601],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":72,"id":"4ed4dc6e-352d-4e2f-9e88-044f7b7d0f59","page_size":[595,841],"block_position":"72-12"},{"bbox":[236,605,355,687],"type":"interline_equation","angle":0,"index":13,"text":"$$\nA = \\left[ \\begin{array}{c c c c c} a & b & \\dots & b & b \\\\ & a & \\dots & b & b \\\\ & & \\ddots & \\vdots & \\vdots \\\\ & & & a & b \\\\ & & & & a \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":72,"id":"cc772c37-91f7-4b48-b60e-bd13f6881c7f","page_size":[595,841],"block_position":"72-13"},{"bbox":[291,803,303,812],"type":"page_number","angle":0,"index":1000013,"text":"68 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":72,"is_discarded":true,"id":"b11080e5-1563-4e46-a8a4-a1a4d3313915","page_size":[595,841],"block_position":"72-14"}],[{"bbox":[431,47,539,58],"type":"header","angle":0,"index":-99998,"text":"6.4 $\\lambda$ 矩阵的进一步研究 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":73,"is_discarded":true,"id":"8735d35a-866b-4cbb-ad0a-7cabc7b650a1","page_size":[595,841],"block_position":"73-0"},{"bbox":[53,72,337,84],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"解 若 b=0，则 Jordan 标准型即 aE。若 $b\\neq0$ ，则考虑 $\\lambda E-A$ ： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":73,"id":"82dbbbb9-63cc-473e-a2c2-e9b9efbcd1d9","page_size":[595,841],"block_position":"73-1"},{"bbox":[186,88,404,169],"type":"interline_equation","angle":0,"index":3,"text":"$$\n\\lambda E - A = \\left[ \\begin{array}{c c c c c} \\lambda - a & - b & \\dots & - b & - b \\\\ & \\lambda - a & \\dots & - b & - b \\\\ & & \\ddots & \\vdots & \\vdots \\\\ & & & \\lambda - a & - b \\\\ & & & & \\lambda - a \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":73,"id":"91d5cf3e-1a6c-45a5-86cf-88f4ff2ed95b","page_size":[595,841],"block_position":"73-2"},{"bbox":[53,173,540,202],"type":"text","angle":0,"index":4,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":73,"id":"51455c34-9151-42cc-b496-7ec6ec723c55","page_size":[595,841],"block_position":"73-3"},{"bbox":[35,214,575,277],"type":"text","angle":0,"index":5,"merge_prev":false,"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) $A=\\left[\\begin{array}{cccc} \\lambda & -1 & & \\\\ & \\lambda & -1 & \\\\ & & \\lambda & -1 \\\\ 5 & 4 & 3 & \\lambda+2 \\end{array}\\right]$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":73,"id":"c241f7fd-f3d5-4632-a977-fdc713d818ad","page_size":[595,841],"block_position":"73-4"},{"bbox":[52,284,325,296],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"解(1)左上行列式因子为-1，因此不变因子为 $1,1,1,(\\lambda+2)^{4}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":73,"id":"6655d372-d6b4-407e-a7ce-886e9b5564fd","page_size":[595,841],"block_position":"73-5"},{"bbox":[73,300,301,312],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"(2) 右上行列式因子为 -1，因此不变因子 1,1,1, $|A|$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":73,"id":"4f4d3ece-277d-421a-9e36-93c4219e1687","page_size":[595,841],"block_position":"73-6"},{"bbox":[73,334,177,347],"type":"title","angle":0,"index":8,"level":2,"text":"## 定理 6.27 (重要 $\\lambda$ 矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":73,"id":"3a615281-ff18-4596-9791-5a0f8ada6ee2","page_size":[595,841],"block_position":"73-7"},{"bbox":[70,352,419,364],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"下面 $\\lambda$ 矩阵 A 的不变因子为 $1,1,\\cdots,\\lambda^{n}+a_{1}\\lambda^{n-1}+a_{2}\\lambda^{n-2}+\\cdots+a_{n-1}\\lambda+a_{n}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":73,"id":"d9563f23-b3a9-4b03-94cc-09be0f2dd4b7","page_size":[595,841],"block_position":"73-8"},{"bbox":[203,369,387,467],"type":"interline_equation","angle":0,"index":10,"text":"$$\nA = \\left[ \\begin{array}{c c c c c c} \\lambda & & & & & a _ {n} \\\\ - 1 & \\lambda & & & & a _ {n - 1} \\\\ & - 1 & \\lambda & & & a _ {n - 2} \\\\ & & \\ddots & \\ddots & & \\vdots \\\\ & & & - 1 & \\lambda & a _ {2} \\\\ & & & & - 1 & \\lambda + a _ {1} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":73,"id":"55816726-e552-4edf-9218-58a6939f8848","page_size":[595,841],"block_position":"73-9"},{"bbox":[54,483,539,512],"type":"text","angle":0,"index":11,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":73,"id":"1104e2ef-ecc4-440e-a00b-07e336dd2794","page_size":[595,841],"block_position":"73-10"},{"bbox":[73,520,112,533],"type":"title","angle":0,"index":12,"level":2,"text":"## 推论6.1","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":73,"id":"4e507c87-4070-4e26-9eeb-46899fbecd25","page_size":[595,841],"block_position":"73-11"},{"bbox":[69,535,477,550],"type":"text","angle":0,"index":13,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":73,"id":"5e3f064b-e12d-4dc0-9a97-1592b9a447ca","page_size":[595,841],"block_position":"73-12"},{"bbox":[206,553,384,651],"type":"interline_equation","angle":0,"index":14,"text":"$$\nA = \\left[ \\begin{array}{c c c c c c} 0 & & & & & a _ {n} \\\\ - 1 & 0 & & & & a _ {n - 1} \\\\ & - 1 & 0 & & & a _ {n - 2} \\\\ & & \\ddots & \\ddots & & \\vdots \\\\ & & & - 1 & 0 & a _ {2} \\\\ & & & & - 1 & a _ {1} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":73,"id":"40071f11-ce3a-4bbe-9bc4-8dc9b4e00480","page_size":[595,841],"block_position":"73-13"},{"bbox":[35,677,563,746],"type":"text","angle":0,"index":15,"merge_prev":false,"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) $A = \\left[ \\begin{array}{cccc}a & & & \\\\ a & a & & \\\\ \\vdots & \\vdots & \\ddots & \\\\ a & a & \\cdots & a \\end{array} \\right]$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":73,"id":"e018e23a-20ae-47d8-b8df-dbbde3968f64","page_size":[595,841],"block_position":"73-14"},{"bbox":[291,803,302,811],"type":"page_number","angle":0,"index":999999,"text":"69 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":73,"is_discarded":true,"id":"7adda60a-0281-4790-8772-24d71b9b0b3c","page_size":[595,841],"block_position":"73-15"}],[{"bbox":[430,47,539,58],"type":"header","angle":0,"index":-99999,"text":"6.4 $\\lambda$ 矩阵的进一步研究 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":74,"is_discarded":true,"id":"a3881b8d-796a-4095-9757-e5bf3f629e2e","page_size":[595,841],"block_position":"74-0"},{"bbox":[52,73,458,149],"type":"interline_equation","angle":0,"index":1,"text":"$$\n(3) A = \\left[ \\begin{array}{c c c c} a & & & \\\\ b _ {1} & a & & \\\\ & \\ddots & \\ddots & \\\\ & & b _ {n - 1} & a \\end{array} \\right] (4) A = \\left[ \\begin{array}{c c c c c} a & a _ {1 2} & a _ {1 3} & \\dots & a _ {1 n} \\\\ & a & a _ {2 3} & \\dots & a _ {2 n} \\\\ & & \\ddots & & \\vdots \\\\ & & & a & a _ {n - 1, n} \\\\ & & & & a \\end{array} \\right], a _ {1 2} a _ {2 3} \\dots a _ {n - 1, n} \\neq 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":74,"id":"fadb697e-b0b5-47ce-ad6d-36f650bb02df","page_size":[595,841],"block_position":"74-1"},{"bbox":[52,153,473,166],"type":"text","angle":0,"index":2,"merge_prev":false,"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 $1, 1, 1, |\\lambda E - A|$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"06fd68bf-df1d-4e82-9918-7d1b7b643a0e","page_size":[595,841],"block_position":"74-2"},{"bbox":[72,169,459,181],"type":"text","angle":0,"index":3,"merge_prev":false,"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 $1, 1, \\cdots, 1, |\\lambda E - A|$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"e1eb2ff3-ad6b-482f-871e-292366e4d3bb","page_size":[595,841],"block_position":"74-3"},{"bbox":[73,184,124,196],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"(3)(4) 显然 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"ea3bc69c-e8c7-4dba-9cbd-c6a22de841c9","page_size":[595,841],"block_position":"74-4"},{"bbox":[33,198,309,212],"type":"text","angle":0,"index":5,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"7cf1c074-fbd6-4f95-98d1-02099711dea4","page_size":[595,841],"block_position":"74-5"},{"bbox":[55,230,203,245],"type":"title","angle":0,"index":6,"level":2,"text":"## 6.4.2 $k$ 阶Jordan块的个数","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"79ee5411-3b6b-44d2-bb94-10bd76c0ece1","page_size":[595,841],"block_position":"74-6"},{"bbox":[72,258,210,270],"type":"title","angle":0,"index":7,"level":2,"text":"## 定理 6.28 (k 阶 Jordan 块个数)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"2766073e-4519-4ea2-ae85-4294e7460eee","page_size":[595,841],"block_position":"74-7"},{"bbox":[67,275,524,303],"type":"text","angle":0,"index":8,"merge_prev":false,"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 $r((A - \\lambda E_n)^0) = n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"059e5310-95fd-4e60-888c-5c78617b5d5f","page_size":[595,841],"block_position":"74-8"},{"bbox":[174,305,418,320],"type":"interline_equation","angle":0,"index":9,"text":"$$\nr ((A - \\lambda E _ {n}) ^ {k - 1}) - 2 r ((A - \\lambda E _ {n}) ^ {k}) + r ((A - \\lambda E _ {n}) ^ {k + 1})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":74,"id":"7d472f4e-8ea1-4d7b-a473-80cd57b7dfd0","page_size":[595,841],"block_position":"74-9"},{"bbox":[54,336,208,348],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"证明 考虑零对角 Jordan 块的幂次: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"f1aff54c-6311-4955-bf47-5fad1b5c0ab7","page_size":[595,841],"block_position":"74-10"},{"bbox":[124,351,467,388],"type":"interline_equation","angle":0,"index":11,"text":"$$\nr (J _ {r} (0) ^ {k}) = \\left\\{ \\begin{array}{l l} r - k, & 0 \\leq k <   r \\\\ 0, & k \\geq r \\end{array} \\right. \\quad r (J _ {r} (0) ^ {k}) - r (J _ {r} (0) ^ {k + 1}) = \\left\\{ \\begin{array}{l l} 1, & 0 \\leq k <   r \\\\ 0, & k \\geq r \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":74,"id":"0db25cc3-28aa-4164-ab4c-3ed5c4f18387","page_size":[595,841],"block_position":"74-11"},{"bbox":[54,392,326,406],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"而非零对角 Jordan 块的幂次满足 $r(J_{r}(a)^{k}) - r(J_{r}(a)^{k+1}) = 0$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"81254fc0-8245-4827-92ff-708eb43b6718","page_size":[595,841],"block_position":"74-12"},{"bbox":[53,407,538,451],"type":"text","angle":0,"index":13,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"5b3dadd3-4fb3-4722-ad5a-8ecd28b50041","page_size":[595,841],"block_position":"74-13"},{"bbox":[131,454,461,486],"type":"interline_equation","angle":0,"index":14,"text":"$$\nr ((A - \\lambda E) ^ {k}) - r ((A - \\lambda E) ^ {k + 1}) = \\sum_ {i = 1} ^ {s} [ r (J _ {r _ {i}} (\\lambda_ {i} - \\lambda) ^ {k}) - r (J _ {r _ {i}} (\\lambda_ {i} - \\lambda) ^ {k + 1}) ]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":74,"id":"ba355cc9-d69c-4e58-b500-cce90fa59645","page_size":[595,841],"block_position":"74-14"},{"bbox":[52,488,539,518],"type":"text","angle":0,"index":15,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"341e3e86-2b49-443c-8ba5-5d4686c6d957","page_size":[595,841],"block_position":"74-15"},{"bbox":[35,535,273,548],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"练习 6.8 上述定理的应用 计算 A 的 Jordan 标准型 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"e932392b-7fa3-46b4-bc3d-5cd6ead8859a","page_size":[595,841],"block_position":"74-16"},{"bbox":[236,550,355,642],"type":"interline_equation","angle":0,"index":17,"text":"$$\nA = \\left[ \\begin{array}{c c c c c} a & 0 & 1 & & \\\\ & a & 0 & \\ddots & \\\\ & & a & \\ddots & 1 \\\\ & & & \\ddots & 0 \\\\ & & & & a \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":74,"id":"4f3e0243-d9d3-467e-9b88-a6e3ce90272b","page_size":[595,841],"block_position":"74-17"},{"bbox":[35,651,539,680],"type":"text","angle":0,"index":18,"merge_prev":false,"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 $r((A - \\lambda E)^{k}) = r((B - \\lambda E)^{k})$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"646f6ddb-5f02-430b-a7cd-6d2f7e4e08c2","page_size":[595,841],"block_position":"74-18"},{"bbox":[53,682,152,695],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"证明 直接用结论即可 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":74,"id":"89a74382-5e4e-4f8b-8e4d-07b58dbb1a1c","page_size":[595,841],"block_position":"74-19"},{"bbox":[291,802,303,812],"type":"page_number","angle":0,"index":1000018,"text":"70 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":74,"is_discarded":true,"id":"c6e89714-0339-4696-9c99-3f364beba768","page_size":[595,841],"block_position":"74-20"}],[{"bbox":[431,47,539,58],"type":"header","angle":0,"index":-99999,"text":"6.4 $\\lambda$ 矩阵的进一步研究 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":75,"is_discarded":true,"id":"7cf6c275-d5f4-4bc0-998d-ce8ecb3d8ee1","page_size":[595,841],"block_position":"75-0"},{"bbox":[55,73,205,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 6.4.3 分块矩阵的 Jordan 阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"abcfa88c-e2b2-4ede-b620-1bec9d348709","page_size":[595,841],"block_position":"75-1"},{"bbox":[72,100,242,112],"type":"title","angle":0,"index":2,"level":2,"text":"## 定理 6.29 (分块矩阵的 Jordan 标准型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"4005aad7-5086-4018-84dd-279f88db63cc","page_size":[595,841],"block_position":"75-2"},{"bbox":[68,116,523,130],"type":"text","angle":0,"index":3,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"80584eb4-e7d1-4fad-8173-14de8d3c9902","page_size":[595,841],"block_position":"75-3"},{"bbox":[240,132,354,165],"type":"interline_equation","angle":0,"index":4,"text":"$$\n\\left[ \\begin{array}{c c} A & C \\\\ O & B \\end{array} \\right] \\sim \\operatorname{diag} \\{J _ {1}, J _ {2} \\}\\tag{#}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":75,"id":"1488714b-4c91-4d20-9600-417fc0155496","page_size":[595,841],"block_position":"75-4"},{"bbox":[53,183,539,211],"type":"text","angle":0,"index":5,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"1d69944e-4289-43ee-9459-318f8f84333f","page_size":[595,841],"block_position":"75-5"},{"bbox":[110,212,482,245],"type":"interline_equation","angle":0,"index":6,"text":"$$\n\\left[ \\begin{array}{c c} E _ {m} & X _ {0} \\\\ O & E _ {n} \\end{array} \\right] \\left[ \\begin{array}{c c} A & C \\\\ O & B \\end{array} \\right] \\left[ \\begin{array}{c c} E _ {m} & - X _ {0} \\\\ O & E _ {n} \\end{array} \\right] = \\left[ \\begin{array}{c c} A & C - (A X _ {0} - X _ {0} B) \\\\ O & B \\end{array} \\right] = \\left[ \\begin{array}{c c} A & O \\\\ O & B \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":75,"id":"0f4a6d12-eed0-4097-85da-dbf38ada6cbd","page_size":[595,841],"block_position":"75-6"},{"bbox":[54,248,101,260],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"因此得到： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"45114040-bd02-431c-9e52-d085e51e2f25","page_size":[595,841],"block_position":"75-7"},{"bbox":[209,258,383,292],"type":"interline_equation","angle":0,"index":8,"text":"$$\n\\left[ \\begin{array}{c c} A & C \\\\ O & B \\end{array} \\right] \\sim \\left[ \\begin{array}{c c} A & O \\\\ O & B \\end{array} \\right] \\sim \\left[ \\begin{array}{c c} J _ {1} & O \\\\ O & J _ {2} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":75,"id":"4a810254-603a-4e85-8a86-49af94a3ba4d","page_size":[595,841],"block_position":"75-8"},{"bbox":[52,298,451,311],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"0395f1f7-ebe3-4f45-ab43-128ad3ab54a5","page_size":[595,841],"block_position":"75-9"},{"bbox":[73,333,113,346],"type":"title","angle":0,"index":10,"level":2,"text":"## 推论6.2","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"6cca06ac-8059-418f-9b8f-c6039806e5a6","page_size":[595,841],"block_position":"75-10"},{"bbox":[68,349,524,378],"type":"text","angle":0,"index":11,"merge_prev":false,"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 $\\mathrm{diag}\\{J_1, \\cdots, J_s\\}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"41a28bc5-821e-4646-a4a6-ba11bd70c3d3","page_size":[595,841],"block_position":"75-11"},{"bbox":[228,377,362,445],"type":"interline_equation","angle":0,"index":12,"text":"$$\nA = \\left[ \\begin{array}{c c c c} A _ {1 1} & A _ {1 2} & \\dots & A _ {1 s} \\\\ & A _ {2 2} & \\dots & A _ {2 s} \\\\ & & \\ddots & \\vdots \\\\ & & & A _ {s s} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":75,"id":"ed8d84b2-b762-49c6-a2da-c07488289625","page_size":[595,841],"block_position":"75-12"},{"bbox":[35,478,262,490],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"练习 6.10 分块矩阵应用 求 A 的 Jordan 标准型: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"2d6b9f19-cbc9-4c06-ba23-1db0305b76c0","page_size":[595,841],"block_position":"75-13"},{"bbox":[240,493,350,555],"type":"interline_equation","angle":0,"index":14,"text":"$$\nA = \\left[ \\begin{array}{c c c c} 1 & 5 & - 1 & 2 \\\\ 0 & 2 & 4 & 3 \\\\ 0 & 0 & - 3 & 0 \\\\ 0 & 0 & 4 & - 3 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":75,"id":"b870876e-23b1-4fb8-a1d0-3bc3255a10c0","page_size":[595,841],"block_position":"75-14"},{"bbox":[54,560,145,572],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"解 $\\mathrm{diag}\\{1,2,J_2(-3)\\}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"b00a38ec-80a8-4c34-a6db-0a8362c776b1","page_size":[595,841],"block_position":"75-15"},{"bbox":[55,590,174,604],"type":"title","angle":0,"index":16,"level":2,"text":"## 6.4.4 过渡矩阵的计算","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"54896fa8-8051-45ba-a47d-db24552860e2","page_size":[595,841],"block_position":"75-16"},{"bbox":[73,618,205,629],"type":"title","angle":0,"index":17,"level":2,"text":"## 定理 6.30 (过渡矩阵计算总结)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"96006aef-c78d-4e16-be43-fd92c6e3326d","page_size":[595,841],"block_position":"75-17"},{"bbox":[69,634,437,646],"type":"text","angle":0,"index":18,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"25f44260-5aa9-4c5c-b252-1b4880037186","page_size":[595,841],"block_position":"75-18"},{"bbox":[80,650,448,662],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"e446882f-127a-435d-be59-80ae5873ea73","page_size":[595,841],"block_position":"75-19"},{"bbox":[185,671,433,682],"type":"interline_equation","angle":0,"index":20,"text":"$$\nA X _ {1} = \\lambda X _ {1}, A X _ {2} = \\lambda X _ {2} + X _ {1}, \\dots , A X _ {k} = \\lambda X _ {k} + X _ {k - 1}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":75,"id":"425519c7-4fbe-46ef-870d-7aedbe000178","page_size":[595,841],"block_position":"75-20"},{"bbox":[93,688,524,718],"type":"text","angle":0,"index":21,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"824622f1-1f8c-45af-b6a8-738b6f16b39d","page_size":[595,841],"block_position":"75-21"},{"bbox":[80,719,524,748],"type":"text","angle":0,"index":23,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":75,"id":"248c383f-5e76-4d5b-b9ad-e2f1dea6a90a","page_size":[595,841],"block_position":"75-22"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":1000021,"text":"71 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":75,"is_discarded":true,"id":"17e7495d-9243-488e-89c4-80afeeb73a74","page_size":[595,841],"block_position":"75-23"}],[{"bbox":[431,47,539,58],"type":"header","angle":0,"index":-99999,"text":"6.4 $\\lambda$ 矩阵的进一步研究 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":76,"is_discarded":true,"id":"a949de94-085f-4896-abba-e34879f1bd48","page_size":[595,841],"block_position":"76-0"},{"bbox":[35,71,391,84],"type":"text","angle":0,"index":1,"merge_prev":false,"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 $P^{-1}AP = J$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":76,"id":"6efe1fb7-ea12-453d-848a-96710213b2da","page_size":[595,841],"block_position":"76-1"},{"bbox":[196,85,394,136],"type":"interline_equation","angle":0,"index":2,"text":"$$\nA = \\left[ \\begin{array}{c c c} 2 & 3 & 2 \\\\ 1 & 8 & 2 \\\\ - 2 & - 1 4 & - 3 \\end{array} \\right] \\quad J = \\left[ \\begin{array}{c c c} 1 & 0 & 0 \\\\ 0 & 3 & 1 \\\\ 0 & 0 & 3 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":76,"id":"64a42612-ef1b-4cfa-b2c2-3bc989d3a48d","page_size":[595,841],"block_position":"76-2"},{"bbox":[53,138,539,182],"type":"text","angle":0,"index":3,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":76,"id":"dff9f342-2ee8-4a66-a7c1-2ee8a554a931","page_size":[595,841],"block_position":"76-3"},{"bbox":[193,182,398,234],"type":"interline_equation","angle":0,"index":4,"text":"$$\n[ A - 3 E | b ] \\rightarrow \\left[\\begin{array}{c c c c}{- 1}&{3}&{2}&{b _ {1}}\\\\{0}&{8}&{4}&{b _ {1} + b _ {2}}\\\\{0}&{8}&{4}&{\\frac {1}{2} b _ {1} + \\frac {5}{2} b _ {2} + b _ {3}}\\end{array}\\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":76,"id":"0af8eb95-929b-4ee1-be3d-1a18f639a81b","page_size":[595,841],"block_position":"76-4"},{"bbox":[54,238,311,263],"type":"text","angle":0,"index":5,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":76,"id":"ecf6f3d3-ff15-470d-8647-a7253140bc0e","page_size":[595,841],"block_position":"76-5"},{"bbox":[246,265,349,317],"type":"interline_equation","angle":0,"index":6,"text":"$$\n\\left[ \\begin{array}{c c c} - 1 & 3 & 2 \\\\ 0 & 8 & 4 \\\\ \\frac {1}{2} & \\frac {5}{2} & 1 \\end{array} \\right] X _ {2} = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":76,"id":"1f482404-c250-4f0c-a89f-2ef58f005758","page_size":[595,841],"block_position":"76-6"},{"bbox":[54,321,474,334],"type":"text","angle":0,"index":7,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":76,"id":"a82f70e3-1dec-4516-85f0-a17ad63d16dc","page_size":[595,841],"block_position":"76-7"},{"bbox":[236,337,358,386],"type":"interline_equation","angle":0,"index":8,"text":"$$\n\\left[ \\begin{array}{c c c} - 1 & 3 & 2 \\\\ 0 & 8 & 4 \\\\ 0 & 0 & 0 \\end{array} \\right] X _ {3} = \\left[ \\begin{array}{c} 1 \\\\ 0 \\\\ 0 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":76,"id":"7e3dbef9-5738-45ee-9a4c-e651e3798530","page_size":[595,841],"block_position":"76-8"},{"bbox":[54,390,433,403],"type":"text","angle":0,"index":9,"merge_prev":false,"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 $X_{1} = [-2,0,1]^{T}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":76,"id":"42754bf5-59c4-4937-978f-36bbb83348bb","page_size":[595,841],"block_position":"76-9"},{"bbox":[35,421,390,433],"type":"text","angle":0,"index":10,"merge_prev":false,"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 $P^{-1}AP = J$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":76,"id":"cbca375d-c661-4173-95f6-b04b7358df28","page_size":[595,841],"block_position":"76-10"},{"bbox":[209,436,381,486],"type":"interline_equation","angle":0,"index":11,"text":"$$\nA = \\left[ \\begin{array}{r r r} 0 & 1 & 0 \\\\ - 4 & 4 & 0 \\\\ - 2 & 1 & 2 \\end{array} \\right], J = \\left[ \\begin{array}{c c c} 2 & & \\\\ & 2 & 1 \\\\ & & 2 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":76,"id":"67556a56-0403-48b0-8882-512d68a7247b","page_size":[595,841],"block_position":"76-11"},{"bbox":[52,488,539,516],"type":"text","angle":0,"index":12,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":76,"id":"1cc6ee06-3cb3-455c-8475-4870bcee4659","page_size":[595,841],"block_position":"76-12"},{"bbox":[220,519,370,569],"type":"interline_equation","angle":0,"index":13,"text":"$$\n[ A | b ] \\to \\left[ \\begin{array}{c c c c} - 2 & 1 & 0 & b _ {1} \\\\ 0 & 0 & 0 & b _ {2} - 2 b _ {1} \\\\ 0 & 0 & 0 & b _ {3} - b _ {1} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":76,"id":"511b0771-5ff2-4e31-ab9b-74a4a0241f64","page_size":[595,841],"block_position":"76-13"},{"bbox":[54,575,539,607],"type":"text","angle":0,"index":14,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":76,"id":"6b1ff061-494e-4231-a280-7519b98ed685","page_size":[595,841],"block_position":"76-14"},{"bbox":[291,802,302,811],"type":"page_number","angle":0,"index":1000014,"text":"72 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":76,"is_discarded":true,"id":"0c724984-5ee4-4933-8bc4-e8cd2c7185cc","page_size":[595,841],"block_position":"76-15"}],[{"bbox":[193,74,411,94],"type":"title","angle":0,"index":0,"level":1,"text":"# 第 7 章 双线性函数与二次型","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"a41e450f-31e4-4321-a278-2986f16bdf43","page_size":[595,841],"block_position":"77-0"},{"bbox":[56,122,158,138],"type":"title","angle":0,"index":1,"level":2,"text":"## 7.1 双线性函数","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"8dd9d52b-89be-4dea-af1e-f9fffd7d80ec","page_size":[595,841],"block_position":"77-1"},{"bbox":[56,156,174,169],"type":"title","angle":0,"index":2,"level":2,"text":"## 7.1.1 双线性函数概念","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"7e6964e3-b643-4ee1-bb1d-9ed0e7a44f8e","page_size":[595,841],"block_position":"77-2"},{"bbox":[73,183,161,195],"type":"title","angle":0,"index":3,"level":2,"text":"## 定义 7.1 (线性函数)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"97535614-d45d-4eb6-9eb2-b2d47f37170c","page_size":[595,841],"block_position":"77-3"},{"bbox":[67,199,525,227],"type":"text","angle":0,"index":4,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"556ce505-d731-4426-9e2c-758c034b41c8","page_size":[595,841],"block_position":"77-4"},{"bbox":[73,248,171,260],"type":"title","angle":0,"index":5,"level":2,"text":"## 定义 7.2 (双线性函数)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"e4d56a52-07c5-4607-9061-216040fc013b","page_size":[595,841],"block_position":"77-5"},{"bbox":[67,264,525,293],"type":"text","angle":0,"index":6,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"a406b60c-ddf6-4712-944a-86313b57a15d","page_size":[595,841],"block_position":"77-6"},{"bbox":[57,325,198,339],"type":"title","angle":0,"index":7,"level":2,"text":"## 7.1.2 双线性函数度量矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"a6059e53-dcae-4228-8dd4-47a74c96fa69","page_size":[595,841],"block_position":"77-7"},{"bbox":[73,352,160,364],"type":"title","angle":0,"index":8,"level":2,"text":"## 定义 7.3 (度量矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"a95f454b-5251-440e-b8b3-511985c7965f","page_size":[595,841],"block_position":"77-8"},{"bbox":[67,370,524,396],"type":"text","angle":0,"index":9,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"7fc08623-8f25-4a6d-944b-584886985266","page_size":[595,841],"block_position":"77-9"},{"bbox":[196,396,393,464],"type":"interline_equation","angle":0,"index":10,"text":"$$\nG = \\left[ \\begin{array}{c c c c} f (\\epsilon_ {1}, \\epsilon_ {1}) & f (\\epsilon_ {1}, \\epsilon_ {2}) & \\dots & f (\\epsilon_ {1}, \\epsilon_ {n}) \\\\ f (\\epsilon_ {2}, \\epsilon_ {1}) & f (\\epsilon_ {2}, \\epsilon_ {2}) & \\dots & f (\\epsilon_ {2}, \\epsilon_ {n}) \\\\ \\vdots & \\vdots & & \\vdots \\\\ f (\\epsilon_ {n}, \\epsilon_ {1}) & f (\\epsilon_ {n}, \\epsilon_ {2}) & \\dots & f (\\epsilon_ {n}, \\epsilon_ {n}) \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":77,"id":"55594a1e-f16a-42ac-bb8f-959f90843064","page_size":[595,841],"block_position":"77-10"},{"bbox":[73,484,251,497],"type":"title","angle":0,"index":11,"level":2,"text":"## 定理 7.1 (双线性函数度量矩阵唯一确定)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"e37763c6-3157-4ca0-8f00-e658bb273422","page_size":[595,841],"block_position":"77-11"},{"bbox":[68,502,350,515],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"若 $x, y \\in V$ 在 $\\epsilon_{1}, \\cdots, \\epsilon_{n}$ 下的坐标为 X, Y，则 $f(\\alpha, \\beta) = X^{T}GY$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"f6f5e928-07bc-42f0-9879-ca6b0fc37c45","page_size":[595,841],"block_position":"77-12"},{"bbox":[57,548,270,562],"type":"title","angle":0,"index":13,"level":2,"text":"## 7.1.3 双线性函数不同基下的矩阵：合同","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"36f24ea9-ddec-4a7d-b1f8-344830a8f277","page_size":[595,841],"block_position":"77-13"},{"bbox":[73,575,141,587],"type":"title","angle":0,"index":14,"level":2,"text":"## 定义 7.4 (合同)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"cfd30741-1a32-418b-9be4-2922c1371fea","page_size":[595,841],"block_position":"77-14"},{"bbox":[68,592,477,605],"type":"text","angle":0,"index":15,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"c63fe07f-e115-4456-a154-b0ec90ded460","page_size":[595,841],"block_position":"77-15"},{"bbox":[52,621,382,634],"type":"text","angle":0,"index":16,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"35d5797a-8f1a-449c-bc9a-972f751ad1fc","page_size":[595,841],"block_position":"77-16"},{"bbox":[73,643,221,655],"type":"title","angle":0,"index":17,"level":2,"text":"## 定理 7.2 (不同基下度量矩阵关系)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"f394c6ca-acfd-44fc-ad9e-85984252837c","page_size":[595,841],"block_position":"77-17"},{"bbox":[69,660,330,672],"type":"text","angle":0,"index":18,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"0bf2f1d4-191e-41a1-b133-481eb3151f78","page_size":[595,841],"block_position":"77-18"},{"bbox":[73,693,180,706],"type":"title","angle":0,"index":19,"level":2,"text":"## 定理 7.3 (分块矩阵合同)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"d41e1aec-0d4c-4869-82f5-20e35ae54e49","page_size":[595,841],"block_position":"77-19"},{"bbox":[69,710,391,723],"type":"text","angle":0,"index":20,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":77,"id":"01d0bcfc-49e5-443a-a0b0-101a6b14ec25","page_size":[595,841],"block_position":"77-20"}],[{"bbox":[489,47,539,58],"type":"header","angle":0,"index":-99999,"text":"7.2 二次型 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":78,"is_discarded":true,"id":"e2eec35c-a6f6-43eb-8400-d694c5f47c14","page_size":[595,841],"block_position":"78-0"},{"bbox":[83,80,479,114],"type":"text","angle":0,"index":1,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"836d6545-7106-489a-a0aa-ad9d1f243d83","page_size":[595,841],"block_position":"78-1"},{"bbox":[224,117,393,149],"type":"interline_equation","angle":0,"index":2,"text":"$$\nP ^ {T} A P = \\left( \\begin{array}{c c} E _ {n - 1} & 0 \\\\ 0 & a _ {n n} - \\alpha^ {T} A _ {1} ^ {- 1} \\alpha \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":78,"id":"cd926ae9-6da8-40f6-8f11-00e8a8de60df","page_size":[595,841],"block_position":"78-2"},{"bbox":[83,153,387,190],"type":"text","angle":0,"index":3,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"b66e1d91-1748-4f64-8672-6ef9d66b0310","page_size":[595,841],"block_position":"78-3"},{"bbox":[225,195,391,234],"type":"interline_equation","angle":0,"index":4,"text":"$$\nP ^ {T} A P = \\left( \\begin{array}{c c} a _ {1 1} & 0 \\\\ 0 & A _ {n - 1} - \\frac {1}{a _ {1 1}} \\alpha \\alpha^ {T} \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":78,"id":"43e990ea-27b9-4038-a0d5-941167296d15","page_size":[595,841],"block_position":"78-4"},{"bbox":[83,235,525,295],"type":"text","angle":0,"index":5,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"765aee02-4271-4910-8116-9f9fc81173e1","page_size":[595,841],"block_position":"78-5"},{"bbox":[202,298,412,339],"type":"interline_equation","angle":0,"index":6,"text":"$$\nQ ^ {T} A Q = \\left( \\begin{array}{c c} a _ {1 1} & 0 \\\\ 0 & P _ {1} ^ {T} \\left(A _ {n - 1} - \\frac {1}{a _ {1 1}} \\alpha \\alpha^ {T}\\right) P _ {1} \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":78,"id":"297904f5-6791-4bd6-9cfa-e16f6a9f070b","page_size":[595,841],"block_position":"78-6"},{"bbox":[56,375,129,391],"type":"title","angle":0,"index":7,"level":2,"text":"## 7.2 二次型","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"b5c16b28-0c28-4476-8292-1f86d5c99441","page_size":[595,841],"block_position":"78-7"},{"bbox":[56,408,198,423],"type":"title","angle":0,"index":8,"level":2,"text":"## 7.2.1 二次型概念及其矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"ac6c69f4-6090-4e32-93b3-acd98ffa6fb2","page_size":[595,841],"block_position":"78-8"},{"bbox":[73,435,151,449],"type":"title","angle":0,"index":9,"level":2,"text":"## 定义 7.5 (二次型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"0bddc3f4-1e23-4b8d-b257-4b4d5a7f471c","page_size":[595,841],"block_position":"78-9"},{"bbox":[68,452,415,486],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"639a6756-b093-4f08-af1f-1ed5b738ab13","page_size":[595,841],"block_position":"78-10"},{"bbox":[73,507,180,520],"type":"title","angle":0,"index":11,"level":2,"text":"## 定义 7.6 (二次型的矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"fa65134f-2906-4990-a84d-fb1529fa2a07","page_size":[595,841],"block_position":"78-11"},{"bbox":[68,523,525,553],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"c36ec2a6-b8d0-4390-8d98-66d497292569","page_size":[595,841],"block_position":"78-12"},{"bbox":[52,568,539,596],"type":"text","angle":0,"index":13,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"c899e2e6-4f25-4b28-9bae-6340e6ccafdb","page_size":[595,841],"block_position":"78-13"},{"bbox":[35,615,481,629],"type":"text","angle":0,"index":14,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"79c2fb0e-656d-4688-a12b-3761ace1a476","page_size":[595,841],"block_position":"78-14"},{"bbox":[73,631,260,644],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"(2) 给定二次型 $f(X)$ ，计算二次型的矩阵： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"61e27891-346e-4e69-bf10-7e00e77ab17b","page_size":[595,841],"block_position":"78-15"},{"bbox":[199,647,391,679],"type":"interline_equation","angle":0,"index":16,"text":"$$\nf (x _ {1}, x _ {2}) = \\left( \\begin{array}{c c} x _ {1} & x _ {2} \\end{array} \\right) \\left( \\begin{array}{c c} 2 & 1 \\\\ 3 & 1 \\end{array} \\right) \\binom{x _ {1}}{x _ {2}}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":78,"id":"1453b41e-31f8-4b44-9454-f7189ffc1682","page_size":[595,841],"block_position":"78-16"},{"bbox":[73,682,348,714],"type":"text","angle":0,"index":17,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"846221e7-c755-4539-8281-48d60cb24caa","page_size":[595,841],"block_position":"78-17"},{"bbox":[53,714,178,761],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"解(1) $A = \\left( \\begin{array}{ccc}0 & 1 & 1\\\\ 1 & 0 & -3\\\\ 1 & -3 & 0 \\end{array} \\right)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":78,"id":"c1573c51-725a-419b-bfc6-c89dd4d44735","page_size":[595,841],"block_position":"78-18"},{"bbox":[291,802,303,811],"type":"page_number","angle":0,"index":1000017,"text":"74 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":78,"is_discarded":true,"id":"8c502509-3be1-4081-b196-4cb157b013fc","page_size":[595,841],"block_position":"78-19"}],[{"bbox":[489,47,539,58],"type":"header","angle":0,"index":-99998,"text":"7.2 二次型 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":79,"is_discarded":true,"id":"dafe6dbe-a917-4664-a784-d81c71fa7670","page_size":[595,841],"block_position":"79-0"},{"bbox":[73,70,320,84],"type":"text","angle":0,"index":2,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"a051efd9-0527-493d-8286-caf734bd342c","page_size":[595,841],"block_position":"79-1"},{"bbox":[273,88,319,119],"type":"interline_equation","angle":0,"index":3,"text":"$$\n\\left( \\begin{array}{c c} 2 & 2 \\\\ 2 & 1 \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":79,"id":"0d17783f-cccf-4473-87a5-aacd46713aa0","page_size":[595,841],"block_position":"79-2"},{"bbox":[73,122,164,136],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"(3) 令 $A = (a_{ij})$ ，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"41431920-d76d-4fa0-8e70-5b3ac71aec95","page_size":[595,841],"block_position":"79-3"},{"bbox":[117,139,414,192],"type":"interline_equation","angle":0,"index":5,"text":"$$\nf = \\sum_ {i = 1} ^ {m} (a _ {i 1} x _ {1} + \\dots + a _ {i n} x _ {n}) ^ {2} = \\sum_ {i = 1} ^ {m} \\left[ \\left( \\begin{array}{c c c} x _ {1} & \\dots & x _ {n} \\end{array} \\right) \\left( \\begin{array}{c} a _ {i 1} \\\\ \\vdots \\\\ a _ {i n} \\end{array} \\right) \\right] ^ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":79,"id":"b9f09c14-7a51-4891-af16-b3dcf55d0249","page_size":[595,841],"block_position":"79-4"},{"bbox":[126,196,474,248],"type":"interline_equation","angle":0,"index":6,"text":"$$\n= \\sum_ {i = 1} ^ {m} \\left[ \\left( \\begin{array}{c c c} x _ {1} & \\dots & x _ {n} \\end{array} \\right) \\left( \\begin{array}{c} a _ {i 1} \\\\ \\vdots \\\\ a _ {i n} \\end{array} \\right) \\left( \\begin{array}{c c c} a _ {i 1} & \\dots & a _ {i n} \\end{array} \\right) \\left( \\begin{array}{c} x _ {1} \\\\ \\vdots \\\\ x _ {n} \\end{array} \\right) \\right] = X ^ {T} (A ^ {T} A) X\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":79,"id":"3c5b7c95-9743-49f4-bb59-89689ae8a282","page_size":[595,841],"block_position":"79-5"},{"bbox":[55,251,173,264],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"其中计算出 $A^T A$ 的过程为 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"44baa759-d9d0-418c-99b6-befe3ad3f6b7","page_size":[595,841],"block_position":"79-6"},{"bbox":[141,265,451,334],"type":"interline_equation","angle":0,"index":8,"text":"$$\nA = \\left[ \\begin{array}{c} A _ {1} \\\\ A _ {2} \\\\ \\vdots \\\\ A _ {m} \\end{array} \\right] \\Rightarrow A ^ {T} A = \\left[ \\begin{array}{c c c c} A _ {1} ^ {T} & A _ {2} ^ {T} & \\dots & A _ {m} ^ {T} \\end{array} \\right] \\left[ \\begin{array}{c} A _ {1} \\\\ A _ {2} \\\\ \\vdots \\\\ A _ {m} \\end{array} \\right] = \\sum_ {i = 1} ^ {m} A _ {i} ^ {T} A _ {i}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":79,"id":"f273de02-20fe-489f-b257-c064bc6f2d71","page_size":[595,841],"block_position":"79-7"},{"bbox":[73,348,170,360],"type":"title","angle":0,"index":9,"level":2,"text":"## 定义 7.7 (二次型等价)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"18b4f906-107b-4f6d-8a48-57e73d5527d6","page_size":[595,841],"block_position":"79-8"},{"bbox":[68,364,524,394],"type":"text","angle":0,"index":10,"merge_prev":false,"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） ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"95d90b64-eff6-41f0-b14a-c9404413313d","page_size":[595,841],"block_position":"79-9"},{"bbox":[74,415,170,428],"type":"title","angle":0,"index":11,"level":2,"text":"## 定理 7.4 (合同的性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"09ea3de0-30d2-4f05-9e04-a3675838164c","page_size":[595,841],"block_position":"79-10"},{"bbox":[69,432,155,444],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"若 $A = C^T BC$ ，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"45a649de-307d-4d1a-bf3a-c66254c135ee","page_size":[595,841],"block_position":"79-11"},{"bbox":[83,449,190,460],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"- 秩不变： $r(A)=r(B)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"4745f614-f9fe-4670-a1c1-6d9fceb64ad5","page_size":[595,841],"block_position":"79-12"},{"bbox":[84,464,158,475],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"- $C$ 不一定唯一 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"e1a3ae73-0960-4d15-bcf4-a2c08df9d56e","page_size":[595,841],"block_position":"79-13"},{"bbox":[83,478,446,492],"type":"text","angle":0,"index":15,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"20eb85a0-68cf-499e-ba84-e45e4ebeb52b","page_size":[595,841],"block_position":"79-14"},{"bbox":[54,508,358,521],"type":"text","angle":0,"index":16,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"ee7b8fa7-4331-4d34-a872-f6921730b108","page_size":[595,841],"block_position":"79-15"},{"bbox":[73,522,303,552],"type":"text","angle":0,"index":17,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"4535f22b-56e4-494f-8aa1-0ee2f25ec876","page_size":[595,841],"block_position":"79-16"},{"bbox":[108,555,485,589],"type":"interline_equation","angle":0,"index":18,"text":"$$\n\\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & - 1 \\end{array} \\right) = \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & i \\end{array} \\right) \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & 1 \\end{array} \\right) \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & i \\end{array} \\right) = \\left( \\begin{array}{c c} - 1 & 0 \\\\ 0 & i \\end{array} \\right) \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & 1 \\end{array} \\right) \\left( \\begin{array}{c c} - 1 & 0 \\\\ 0 & i \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":79,"id":"e527615b-feb4-4035-a302-be9eff20f2df","page_size":[595,841],"block_position":"79-17"},{"bbox":[36,600,399,629],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"c8832e78-14c4-4ee4-a16b-783bd82aace7","page_size":[595,841],"block_position":"79-18"},{"bbox":[56,647,246,661],"type":"title","angle":0,"index":20,"level":2,"text":"## 7.2.2 二次型的标准型存在与配方法","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"b43a99b0-3e14-4f7a-97ee-a881c49d914c","page_size":[595,841],"block_position":"79-19"},{"bbox":[73,674,220,687],"type":"title","angle":0,"index":21,"level":2,"text":"## 定理 7.5 (二次型标准型的存在性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"be4ec970-0d0b-44ef-83f3-9a64aafb38f4","page_size":[595,841],"block_position":"79-20"},{"bbox":[68,692,408,703],"type":"text","angle":0,"index":22,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"1519f51b-738d-4bdb-8266-a19aaeec6a86","page_size":[595,841],"block_position":"79-21"},{"bbox":[54,714,539,746],"type":"text","angle":0,"index":23,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":79,"id":"575834be-23f8-4177-b60e-354cee96ae9f","page_size":[595,841],"block_position":"79-22"},{"bbox":[291,802,302,812],"type":"page_number","angle":0,"index":999999,"text":"75 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":79,"is_discarded":true,"id":"f073031c-1d60-4c6b-a04b-d315865c17c7","page_size":[595,841],"block_position":"79-23"}],[{"bbox":[489,47,539,58],"type":"header","angle":0,"index":-99998,"text":"7.2 二次型 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":80,"is_discarded":true,"id":"74bc26d8-0246-4cff-ba75-3f582c935135","page_size":[595,841],"block_position":"80-0"},{"bbox":[35,349,47,365],"type":"image","angle":0,"index":0,"img_path":"/745302ee047e1008392d043c393193e07f26cf5103fa699b13fee41df18db0b2.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":80,"id":"5b49d943-f242-4e81-a106-bd4ba88d4eca","page_size":[595,841],"block_position":"80-1"},{"bbox":[55,69,310,107],"type":"text","angle":0,"index":2,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"18ec46ab-44c7-443e-a585-df0e6b5ba21f","page_size":[595,841],"block_position":"80-2"},{"bbox":[179,112,412,151],"type":"interline_equation","angle":0,"index":3,"text":"$$\nP ^ {T} A P = \\left( \\begin{array}{c c} a _ {1 1} & 0 \\\\ 0 & A _ {1} - \\frac {1}{a _ {1 1}} \\alpha \\alpha^ {T} \\end{array} \\right) := \\left( \\begin{array}{c c} a _ {1 1} & 0 \\\\ 0 & B _ {1} \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":80,"id":"dedb5ac9-2717-419e-baad-0d4862cec242","page_size":[595,841],"block_position":"80-3"},{"bbox":[54,154,394,185],"type":"text","angle":0,"index":4,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"b9649c46-03d0-4355-81d3-2098d04751da","page_size":[595,841],"block_position":"80-4"},{"bbox":[217,190,373,222],"type":"interline_equation","angle":0,"index":5,"text":"$$\nQ ^ {T} P ^ {T} A P Q = \\left( \\begin{array}{c c} a _ {1 1} & \\\\ & Q _ {1} ^ {T} B _ {1} Q _ {1} \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":80,"id":"259c648b-341a-4dab-98e3-84f1b0d0e7ce","page_size":[595,841],"block_position":"80-5"},{"bbox":[54,225,537,253],"type":"text","angle":0,"index":6,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"d0ec864f-2893-48ba-9e82-d6dc79f4a15a","page_size":[595,841],"block_position":"80-6"},{"bbox":[54,257,539,285],"type":"text","angle":0,"index":7,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"0d38f5d2-4f49-4110-9e74-9699c40890a1","page_size":[595,841],"block_position":"80-7"},{"bbox":[193,283,396,333],"type":"interline_equation","angle":0,"index":8,"text":"$$\nA = \\left( \\begin{array}{c c c} 0 & a _ {1 2} & * \\\\ a _ {1 2} & 0 & * \\\\ * & * & * \\end{array} \\right) \\to \\left( \\begin{array}{c c c} 2 a _ {1 2} & a _ {1 2} & * \\\\ a _ {1 2} & a _ {2 2} & * \\\\ * & * & * \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":80,"id":"4a5fe64b-f056-4bd3-a75a-01f0f4d9f111","page_size":[595,841],"block_position":"80-8"},{"bbox":[54,339,137,351],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"转换为第一种类型 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"6d071dfb-2baf-4900-96f7-7a63b6379a85","page_size":[595,841],"block_position":"80-9"},{"bbox":[52,354,364,367],"type":"text","angle":0,"index":10,"merge_prev":false,"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！ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"ae1efdea-64eb-4f62-923d-cbc613d5314f","page_size":[595,841],"block_position":"80-10"},{"bbox":[73,375,180,388],"type":"title","angle":0,"index":11,"level":2,"text":"## 定义 7.8 (二次型标准型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"7f9cc65b-ab9a-483e-a328-63b21b3191bc","page_size":[595,841],"block_position":"80-11"},{"bbox":[68,393,413,407],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"469144fd-a2b2-4850-93f4-c752fa59e81e","page_size":[595,841],"block_position":"80-12"},{"bbox":[73,427,151,439],"type":"title","angle":0,"index":13,"level":2,"text":"## 定理 7.6 (配方法)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"8a0481e8-24bd-4875-8ae6-2a3ff7618a32","page_size":[595,841],"block_position":"80-13"},{"bbox":[69,444,186,455],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"配方法按照以下步骤进行： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"bba4dc25-ec0e-4c73-9520-ee59f0e49d12","page_size":[595,841],"block_position":"80-14"},{"bbox":[80,460,429,472],"type":"text","angle":0,"index":15,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"3d2d836b-e7fa-4d67-93a3-fa6fb4cac0e2","page_size":[595,841],"block_position":"80-15"},{"bbox":[80,475,236,486],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"2. 把 $x_{1}$ 的项放在一起，全部配方。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"0087e214-dc13-4589-bbeb-7dd9c09ec6b2","page_size":[595,841],"block_position":"80-16"},{"bbox":[80,490,239,503],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"3. 把凑成的 $(x_{1}+\\cdots)^{2}$ 设为新变量 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"5e1c7dee-6c35-4485-9204-f0c43c0ba090","page_size":[595,841],"block_position":"80-17"},{"bbox":[80,507,137,518],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"4. 以此类推 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"ad3337d3-4a51-416d-b381-bc7b5389febe","page_size":[595,841],"block_position":"80-18"},{"bbox":[35,529,47,545],"type":"image","angle":0,"index":19,"img_path":"/0b92b81a7c670121596a9995aff39914c167ca625cc92dc802e14bf367d45534.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":80,"id":"5aba0291-a110-49cd-b0e4-a0caf4897f5b","page_size":[595,841],"block_position":"80-19"},{"bbox":[52,534,309,546],"type":"title","angle":0,"index":20,"level":2,"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","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"69cd4efc-466e-4c92-995e-5b66d42b07e4","page_size":[595,841],"block_position":"80-20"},{"bbox":[35,565,414,594],"type":"text","angle":0,"index":21,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"9fb653a7-4709-427a-988b-cfca925be24d","page_size":[595,841],"block_position":"80-21"},{"bbox":[215,600,377,614],"type":"interline_equation","angle":0,"index":22,"text":"$$\nf = 2 (y _ {1} - y _ {3}) ^ {2} - 2 y _ {2} ^ {2} - 2 y _ {3} ^ {2} + 8 y _ {2} y _ {3}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":80,"id":"cda82c46-e36a-4ed3-867d-8d453ce27d20","page_size":[595,841],"block_position":"80-22"},{"bbox":[54,619,539,647],"type":"text","angle":0,"index":23,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":80,"id":"6f15ed51-b99b-4284-82a0-df2260c95a37","page_size":[595,841],"block_position":"80-23"},{"bbox":[251,650,342,665],"type":"interline_equation","angle":0,"index":24,"text":"$$\nf = 2 y _ {1} ^ {2} - 2 y _ {2} ^ {2} + 6 y _ {3} ^ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":80,"id":"73d6aa3b-020a-4524-92b0-3399dd205a6f","page_size":[595,841],"block_position":"80-24"},{"bbox":[291,802,303,812],"type":"page_number","angle":0,"index":1000024,"text":"76 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":80,"is_discarded":true,"id":"02cfd522-1854-495a-a7c9-c5054e8b4b10","page_size":[595,841],"block_position":"80-25"}],[{"bbox":[489,47,539,58],"type":"header","angle":0,"index":-99998,"text":"7.2 二次型 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":81,"is_discarded":true,"id":"2ee0adbf-7ff9-4caf-a388-d7456007e50b","page_size":[595,841],"block_position":"81-0"},{"bbox":[35,207,47,223],"type":"image","angle":0,"index":0,"img_path":"/1fbb47a9c475507d8ef1e789ee4ed7772a02bb0019876a7edda07571148f915d.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":81,"id":"6ac42048-6ef1-4c07-b33f-33771d4427b7","page_size":[595,841],"block_position":"81-1"},{"bbox":[55,72,246,86],"type":"title","angle":0,"index":2,"level":2,"text":"## 7.2.3 二次型的标准型：初等变换法","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"0f4f6d90-ae25-47ed-a883-89e3e9198507","page_size":[595,841],"block_position":"81-2"},{"bbox":[73,100,241,113],"type":"title","angle":0,"index":3,"level":2,"text":"## 定理 7.7 (标准型的计算：初等变换法)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"74680e69-c96f-4eed-b750-9cfc0512c149","page_size":[595,841],"block_position":"81-3"},{"bbox":[68,116,398,129],"type":"text","angle":0,"index":4,"merge_prev":false,"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): ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"55981cba-05e1-44c2-94d8-938df55ed211","page_size":[595,841],"block_position":"81-4"},{"bbox":[147,132,447,164],"type":"interline_equation","angle":0,"index":5,"text":"$$\n\\binom{{{{A}}}}{{{{E}}}} \\rightarrow \\binom{{{{C ^ {T} A C}}}}{{{{C}}}} \\quad \\text {对} \\mathbf {A} \\text {行变换和对应列变换，对} \\mathbf {I} \\text {只列变换}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":81,"id":"81dc58af-4b4c-4cb5-8505-938ef3368b80","page_size":[595,841],"block_position":"81-5"},{"bbox":[67,167,525,196],"type":"text","angle":0,"index":6,"merge_prev":false,"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) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"5d8ec462-caec-4998-9a30-12dfb7e22aca","page_size":[595,841],"block_position":"81-6"},{"bbox":[52,211,539,240],"type":"text","angle":0,"index":7,"merge_prev":false,"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！ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"ae5ef347-8e30-4e54-becb-fc58395c9bf9","page_size":[595,841],"block_position":"81-7"},{"bbox":[35,258,478,272],"type":"text","angle":0,"index":8,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"1c46593a-f170-427c-883f-238577ab30d0","page_size":[595,841],"block_position":"81-8"},{"bbox":[72,275,491,287],"type":"text","angle":0,"index":9,"merge_prev":false,"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 $C$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"5dfb8d01-a17a-4435-99a0-15ccf8353526","page_size":[595,841],"block_position":"81-9"},{"bbox":[52,290,286,302],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"解(1)结果为 $f = y_{1}^{2} + y_{2}^{2}$ ，千万小心别行列变换错了 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"8f6cab76-813a-4913-bd82-b0f705192dc3","page_size":[595,841],"block_position":"81-10"},{"bbox":[54,305,539,362],"type":"text","angle":0,"index":11,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"cda9a7a8-b9a9-44c4-8c84-cc9cfcbbcc78","page_size":[595,841],"block_position":"81-11"},{"bbox":[146,365,448,459],"type":"interline_equation","angle":0,"index":12,"text":"$$\n\\left(\\begin{array}{c c c}0&1&1\\\\1&0&- 3\\\\1&- 3&0\\\\1&0&0\\\\0&1&0\\\\0&0&1\\end{array}\\right)\\rightarrow \\left(\\begin{array}{c c c}2&1&- 2\\\\1&0&- 3\\\\- 2&- 3&0\\\\1&0&0\\\\1&1&0\\\\0&0&1\\end{array}\\right)\\rightarrow \\left(\\begin{array}{c c c}2&1&0\\\\1&0&- 2\\\\0&- 2&- 2\\\\1&0&1\\\\1&1&1\\\\0&0&1\\end{array}\\right)\\rightarrow \\dots\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":81,"id":"fb02f163-6e7c-4243-8d43-3747e38f133b","page_size":[595,841],"block_position":"81-12"},{"bbox":[54,463,132,475],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"最终变为结果为: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"774c7098-d6e2-4b2e-b7c2-5079e495e627","page_size":[595,841],"block_position":"81-13"},{"bbox":[222,473,371,582],"type":"interline_equation","angle":0,"index":14,"text":"$$\n\\left( \\begin{array}{c c c} 2 & 0 & 0 \\\\ 0 & - \\frac {1}{2} & 0 \\\\ 0 & 0 & 6 \\\\ 1 & - \\frac {1}{2} & 3 \\\\ 1 & \\frac {1}{2} & - 1 \\\\ 0 & 0 & 1 \\end{array} \\right) = \\binom{C ^ {T} A C}{C}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":81,"id":"16976c45-1046-46da-8774-f3d36eb8d34d","page_size":[595,841],"block_position":"81-14"},{"bbox":[56,598,246,613],"type":"title","angle":0,"index":15,"level":2,"text":"## 7.2.4 二次型的标准型：正交变换法","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"292579bb-d016-4ca9-a3cf-41c28d5016df","page_size":[595,841],"block_position":"81-15"},{"bbox":[73,626,211,639],"type":"title","angle":0,"index":16,"level":2,"text":"## 定理 7.8 (正交对角化的可行性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"15d65c85-80ab-4f79-bf80-ac319c6c679d","page_size":[595,841],"block_position":"81-16"},{"bbox":[68,643,258,655],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"任何实对称矩阵 A 一定正交相似于对角阵。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"5aefa425-df03-4f13-ad42-4fce3d487f0d","page_size":[595,841],"block_position":"81-17"},{"bbox":[53,671,539,700],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"87ccc968-7091-4669-91c5-9e75f97d6072","page_size":[595,841],"block_position":"81-18"},{"bbox":[182,703,408,735],"type":"interline_equation","angle":0,"index":20,"text":"$$\nA T = T \\left( \\begin{array}{c c} \\lambda & \\alpha^ {T} \\\\ 0 & A _ {n - 1} \\end{array} \\right) \\Rightarrow T ^ {T} A T = \\left( \\begin{array}{c c} \\lambda & \\alpha^ {T} \\\\ 0 & A _ {n - 1} \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":81,"id":"483d3698-d6fb-42ee-a0eb-853dace7cfad","page_size":[595,841],"block_position":"81-19"},{"bbox":[54,738,539,751],"type":"text","angle":0,"index":21,"merge_prev":false,"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 $P =$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":81,"id":"9313f60b-af46-41e9-9e9d-caae402e9e33","page_size":[595,841],"block_position":"81-20"},{"bbox":[291,802,302,811],"type":"page_number","angle":0,"index":1000017,"text":"77 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":81,"is_discarded":true,"id":"6e7574dc-b501-4379-8024-cd2b3f643810","page_size":[595,841],"block_position":"81-21"}],[{"bbox":[489,47,539,58],"type":"header","angle":0,"index":-99998,"text":"7.2 二次型 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":82,"is_discarded":true,"id":"8daf7d3e-c9a3-463b-ad34-950efa883338","page_size":[595,841],"block_position":"82-0"},{"bbox":[54,72,134,84],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"$\\mathrm{diag}(1, P_1)$ ，此时 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"1762e064-d629-4986-bf04-6888f8dc66d2","page_size":[595,841],"block_position":"82-1"},{"bbox":[219,84,372,116],"type":"interline_equation","angle":0,"index":3,"text":"$$\nP ^ {T} T ^ {T} A T P = \\left( \\begin{array}{c c} \\lambda & 0 \\\\ 0 & P ^ {T} A _ {n - 1} P \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":82,"id":"8b0da3cd-1077-4824-b239-345c96491dfa","page_size":[595,841],"block_position":"82-2"},{"bbox":[55,123,204,136],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"显然 $TP = Q$ ，则 $Q$ 也是正交阵。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"51954ffb-b2d0-478d-8ab1-73354d0a2671","page_size":[595,841],"block_position":"82-3"},{"bbox":[72,144,221,157],"type":"title","angle":0,"index":5,"level":2,"text":"## 定理 7.9 (标准型计算：正交变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"c8e2188c-7d05-4adc-bfa7-b49e3cb3bb2c","page_size":[595,841],"block_position":"82-4"},{"bbox":[70,162,207,173],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"正交变换计算标准型过程如下： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"9899afb7-49ea-479c-9d27-bb37f9372e82","page_size":[595,841],"block_position":"82-5"},{"bbox":[83,177,217,190],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"- 计算 $A$ 的特征值 $\\lambda_1, \\cdots, \\lambda_n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"19304801-42ac-4756-b60e-d39b77380d45","page_size":[595,841],"block_position":"82-6"},{"bbox":[83,192,501,205],"type":"text","angle":0,"index":8,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"bd412fc5-94d8-4a9b-99ca-e230879d2a27","page_size":[595,841],"block_position":"82-7"},{"bbox":[83,208,281,221],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"- 将特征向量单位化，组成 $T = [\\mathbf{v}_1, \\dots, \\mathbf{v}_n]$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"75f7470b-bb98-4e00-a162-0cc70453f513","page_size":[595,841],"block_position":"82-8"},{"bbox":[35,253,539,281],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"6ffca587-bc11-4557-a8f7-7d3bc3b2518c","page_size":[595,841],"block_position":"82-9"},{"bbox":[243,280,348,328],"type":"interline_equation","angle":0,"index":11,"text":"$$\nA = \\left[ \\begin{array}{r r r} 2 & 2 & - 1 \\\\ 2 & 5 & - 4 \\\\ - 2 & - 4 & 5 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":82,"id":"f984abdf-7002-43ed-9760-c1a744057c92","page_size":[595,841],"block_position":"82-10"},{"bbox":[52,336,362,349],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"解 (1) $|\\lambda E - A| = (\\lambda - 1)^{2} (\\lambda - 10)$ ，特征值为 1, 10，正定要求 t > -1。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"1daf43b0-0c64-4a9d-8233-640e129c9adc","page_size":[595,841],"block_position":"82-11"},{"bbox":[73,352,198,364],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"先求 $(E - A)X = 0$ ，得到： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"c862e40a-1326-410c-951c-bc796a52ae58","page_size":[595,841],"block_position":"82-12"},{"bbox":[177,367,417,417],"type":"interline_equation","angle":0,"index":14,"text":"$$\n\\left[ \\begin{array}{c c c} 1 & 2 & - 2 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right] X = 0 \\Rightarrow X = k _ {1} \\left( \\begin{array}{c} - 2 \\\\ 1 \\\\ 0 \\end{array} \\right) + k _ {2} \\left( \\begin{array}{c} 2 \\\\ 0 \\\\ 1 \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":82,"id":"9da141e7-0cca-45c6-8137-7543f51aba9f","page_size":[595,841],"block_position":"82-13"},{"bbox":[54,419,229,444],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"单位化得到 $\\frac{1}{\\sqrt{5}} (-2,1,0)^T,\\frac{1}{\\sqrt{5}} (2,0,1)^T$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"2f18e5ce-20a1-4803-b421-aa5f07c50f24","page_size":[595,841],"block_position":"82-14"},{"bbox":[74,444,177,455],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"再求 $(10E - A)X = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"8a09c965-71f7-434f-9212-ffd09ebb9795","page_size":[595,841],"block_position":"82-15"},{"bbox":[208,454,385,505],"type":"interline_equation","angle":0,"index":17,"text":"$$\n\\left[ \\begin{array}{c c c} 2 & 4 & 5 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 0 \\end{array} \\right] X = 0 \\Rightarrow X = k _ {1} \\left( \\begin{array}{c} 1 \\\\ 2 \\\\ - 2 \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":82,"id":"4cfab1e7-a591-4742-911f-d12f8e9aed16","page_size":[595,841],"block_position":"82-16"},{"bbox":[55,511,160,533],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"单位化得到 $\\frac{1}{3} (1,2, - 2)$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"15d57b25-2db0-45a5-8830-d3bc26f7f61f","page_size":[595,841],"block_position":"82-17"},{"bbox":[74,533,132,544],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"最终结果为: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"51bf1e0f-f448-4009-a5e2-8d81f46d0d77","page_size":[595,841],"block_position":"82-18"},{"bbox":[177,541,414,612],"type":"interline_equation","angle":0,"index":20,"text":"$$\nT = \\left[ \\begin{array}{c c c} - \\frac {2}{\\sqrt {5}} & \\frac {2}{3 \\sqrt {5}} & \\frac {1}{3} \\\\ - \\frac {1}{\\sqrt {5}} & \\frac {4}{3 \\sqrt {5}} & \\frac {2}{3} \\\\ 0 & \\frac {5}{3 \\sqrt {5}} & - \\frac {2}{3} \\end{array} \\right], T ^ {T} A T = \\left[ \\begin{array}{c c c} 1 & & \\\\ & 1 & \\\\ & & 1 0 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":82,"id":"7d16532d-f561-4367-8694-1b3d408f9426","page_size":[595,841],"block_position":"82-19"},{"bbox":[73,629,196,642],"type":"title","angle":0,"index":21,"level":2,"text":"## 定理 7.10 (二次型与特征值)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"64c8817b-4a47-40da-8e06-b758b600ecac","page_size":[595,841],"block_position":"82-20"},{"bbox":[69,647,368,659],"type":"text","angle":0,"index":22,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"d73d8844-f44b-4999-9c08-a787d3492467","page_size":[595,841],"block_position":"82-21"},{"bbox":[210,661,383,687],"type":"interline_equation","angle":0,"index":23,"text":"$$\n\\max _ {X \\in \\mathbb {R} ^ {n}} \\frac {X ^ {T} A X}{X ^ {T} X} = \\lambda_ {1}, \\min _ {X \\in \\mathbb {R} ^ {n}} \\frac {X ^ {T} A X}{X ^ {T} X} = \\lambda_ {n}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":82,"id":"e4adeac7-2bb8-41d9-bbc0-8fcb9501a47f","page_size":[595,841],"block_position":"82-22"},{"bbox":[52,703,539,732],"type":"text","angle":0,"index":24,"merge_prev":false,"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 $X_0^T (\\lambda_1E - A)X_0 = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"8ef47c97-b12c-4ae5-ae42-26ed6966bcc6","page_size":[595,841],"block_position":"82-23"},{"bbox":[35,750,540,762],"type":"text","angle":0,"index":25,"merge_prev":false,"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, ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":82,"id":"2db2d960-5626-438a-b2f2-bf4855e6fa3f","page_size":[595,841],"block_position":"82-24"},{"bbox":[291,802,302,812],"type":"page_number","angle":0,"index":999999,"text":"78 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":82,"is_discarded":true,"id":"b1e001c2-7696-46ab-bbd1-fb8f48ad6c85","page_size":[595,841],"block_position":"82-25"}],[{"bbox":[420,47,539,58],"type":"header","angle":0,"index":-99998,"text":"7.3 (半) 正定二次型的研究 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":83,"is_discarded":true,"id":"16eb56ad-0097-49ad-8db2-a79eff86e841","page_size":[595,841],"block_position":"83-0"},{"bbox":[54,72,205,84],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"证明 $A + B$ 的特征值都大于 $a + b$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"4e7bd2f7-829a-4b4c-9be7-3958d0fb3796","page_size":[595,841],"block_position":"83-1"},{"bbox":[53,87,539,116],"type":"text","angle":0,"index":3,"merge_prev":false,"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 $\\sqrt{\\lambda_{n}} \\leq |\\mu| \\leq \\sqrt{\\lambda_{1}}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"b30c06b7-b062-4fa6-8506-818ad44bc65f","page_size":[595,841],"block_position":"83-2"},{"bbox":[53,117,539,146],"type":"text","angle":0,"index":4,"merge_prev":false,"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 $a + b$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"f041d0a3-b366-48f2-a21d-5fd3266fa7a6","page_size":[595,841],"block_position":"83-3"},{"bbox":[73,148,276,162],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"(2) 由于 $B\\alpha = \\mu \\alpha$ ，因此 $\\alpha^T B^T = \\mu \\alpha^T$ ，得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"77d18187-d347-4a66-a4f1-e645cea96976","page_size":[595,841],"block_position":"83-4"},{"bbox":[174,168,418,182],"type":"interline_equation","angle":0,"index":6,"text":"$$\n\\alpha^ {T} B ^ {T} B \\alpha = \\mu^ {2} \\alpha^ {T} \\alpha \\in [ \\lambda_ {n} \\alpha^ {T} \\alpha , \\lambda_ {1} \\alpha^ {T} \\alpha ] \\Rightarrow \\lambda_ {n} \\leq \\mu^ {2} \\leq \\lambda_ {1}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":83,"id":"10f0cbc3-71f9-4373-9db4-930711c38f6d","page_size":[595,841],"block_position":"83-5"},{"bbox":[56,204,176,217],"type":"title","angle":0,"index":7,"level":2,"text":"## 7.2.5 R, C 上的规范型","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"0d38f533-9df9-4a75-8fd0-6c23b0d14de1","page_size":[595,841],"block_position":"83-6"},{"bbox":[54,227,539,255],"type":"text","angle":0,"index":8,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"856367b4-0e4a-46eb-9e46-9a1d06dd16db","page_size":[595,841],"block_position":"83-7"},{"bbox":[73,263,186,275],"type":"title","angle":0,"index":9,"level":2,"text":"## 定理 7.11 (C 上的规范型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"0c6d4da6-06d5-429d-9d05-0a88602ebda9","page_size":[595,841],"block_position":"83-8"},{"bbox":[68,280,190,292],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"$\\mathbb{C}$ 上任一二次型 $f$ 都等价于 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"92f16479-e3d9-40cd-ba6f-89fefa9f23fc","page_size":[595,841],"block_position":"83-9"},{"bbox":[257,295,336,310],"type":"interline_equation","angle":0,"index":11,"text":"$$\ng = u _ {1} ^ {2} + \\dots + u _ {r} ^ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":83,"id":"cde48fe0-da28-436a-b068-ff2a353bec29","page_size":[595,841],"block_position":"83-10"},{"bbox":[72,330,201,343],"type":"title","angle":0,"index":12,"level":2,"text":"## 推论 7.1 (C 上合同充要条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"10042a17-b228-412c-85db-424aec7ea325","page_size":[595,841],"block_position":"83-11"},{"bbox":[68,347,295,359],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"C 上两个对称矩阵合同的充要条件为它们的秩相同。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"881b81b7-5d43-455f-98da-f16f1544bf6d","page_size":[595,841],"block_position":"83-12"},{"bbox":[73,380,186,391],"type":"title","angle":0,"index":14,"level":2,"text":"## 定理 7.12 (R 上的规范型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"83ff6098-4e12-42ef-a963-a94918facc14","page_size":[595,841],"block_position":"83-13"},{"bbox":[68,396,466,409],"type":"text","angle":0,"index":15,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"9f0408bb-9a73-4b6a-b8a4-a1e9f0f66566","page_size":[595,841],"block_position":"83-14"},{"bbox":[217,414,376,430],"type":"interline_equation","angle":0,"index":16,"text":"$$\ng = u _ {1} ^ {2} + \\dots + u _ {p} ^ {2} - u _ {p + 1} ^ {2} - \\dots - u _ {r} ^ {2}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":83,"id":"28917a4c-ac8c-4e67-86af-c863cef9ff5e","page_size":[595,841],"block_position":"83-15"},{"bbox":[73,451,201,464],"type":"title","angle":0,"index":17,"level":2,"text":"## 推论 7.2 ( $\\mathbb{R}$ 上合同充要条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"6f3cc530-30d6-489c-bff6-accd5b87b300","page_size":[595,841],"block_position":"83-16"},{"bbox":[68,469,325,480],"type":"text","angle":0,"index":18,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"cd9b5bc2-cb3e-494c-b273-dc5e6dafb4a8","page_size":[595,841],"block_position":"83-17"},{"bbox":[35,497,473,509],"type":"text","angle":0,"index":19,"merge_prev":false,"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 $X^T A X < 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"7f526f84-b389-4b93-abb9-7dba5c6dd0b3","page_size":[595,841],"block_position":"83-18"},{"bbox":[53,512,538,541],"type":"text","angle":0,"index":20,"merge_prev":false,"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 $X_0^TAX_0 = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"bf8add66-3311-4f44-8ba4-4e3c35d2a5be","page_size":[595,841],"block_position":"83-19"},{"bbox":[52,543,539,570],"type":"text","angle":0,"index":21,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"2a488648-a97b-4ff8-bc90-9bc18a45125c","page_size":[595,841],"block_position":"83-20"},{"bbox":[201,574,390,587],"type":"interline_equation","angle":0,"index":22,"text":"$$\nC ^ {T} A C = \\operatorname{diag} \\{1, 1, \\dots , 1, - 1, - 1, \\dots , - 1 \\}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":83,"id":"d903395f-4318-461f-b465-8fcb98340e42","page_size":[595,841],"block_position":"83-21"},{"bbox":[55,597,193,610],"type":"text","angle":0,"index":23,"merge_prev":false,"text":"取 $Y = e_{n}$ 时 $Y^{T}C^{T}ACY = -1$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"0371b648-2c56-4d54-bf99-0be48975d4e4","page_size":[595,841],"block_position":"83-22"},{"bbox":[54,613,539,641],"type":"text","angle":0,"index":24,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"1e61037c-3914-4cdf-b7a8-645bec68cedf","page_size":[595,841],"block_position":"83-23"},{"bbox":[56,664,228,681],"type":"title","angle":0,"index":25,"level":2,"text":"## 7.3 (半) 正定二次型的研究","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"04d29f58-4c0a-4150-bdb2-82d44a7228e9","page_size":[595,841],"block_position":"83-24"},{"bbox":[56,697,223,711],"type":"title","angle":0,"index":26,"level":2,"text":"## 7.3.1 正定二次型的概念与判别","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"a3a18930-2c68-4112-91b1-7fc564e329a0","page_size":[595,841],"block_position":"83-25"},{"bbox":[74,720,336,734],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"正定二次型是 $\\mathbb{R}$ 中二次型中的特殊等价类，其代表为 $y^{T}Ey$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":83,"id":"a1dc5c3c-a9d7-4c4b-b29c-c3a72ff6e0e7","page_size":[595,841],"block_position":"83-26"},{"bbox":[291,802,302,811],"type":"page_number","angle":0,"index":999999,"text":"79 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":83,"is_discarded":true,"id":"745c104c-f457-4958-b321-ce8c3dc92bae","page_size":[595,841],"block_position":"83-27"}],[{"bbox":[420,47,539,58],"type":"header","angle":0,"index":-99998,"text":"7.3 (半) 正定二次型的研究 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":84,"is_discarded":true,"id":"887dea7e-27fb-477a-b215-3c3a00d8f6eb","page_size":[595,841],"block_position":"84-0"},{"bbox":[73,73,171,86],"type":"title","angle":0,"index":2,"level":2,"text":"## 定义 7.9 (正定二次型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"71d04530-7078-4fbe-aba0-c2e72f05cf17","page_size":[595,841],"block_position":"84-1"},{"bbox":[67,89,526,121],"type":"text","angle":0,"index":3,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"cd24fb34-18a1-47db-b4d8-03bf904391ce","page_size":[595,841],"block_position":"84-2"},{"bbox":[72,138,216,151],"type":"title","angle":0,"index":4,"level":2,"text":"## 定理 7.13 (正定二次型的规范型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"b51ac718-bb56-423e-bf18-4b6cea77696a","page_size":[595,841],"block_position":"84-3"},{"bbox":[69,155,290,170],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"正定二次型的规范型是 $y^{T}Ey=y_{1}^{2}+y_{2}^{2}+\\cdots+y_{n}^{2}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"6ba70374-5c3a-4b50-a76e-b5f5cc077448","page_size":[595,841],"block_position":"84-4"},{"bbox":[199,189,393,289],"type":"image","angle":0,"index":6,"img_path":"/c263772a56ac3460e77a1671c982bbce6d1734c60908b4b744d1e5be1797c031.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.32605042016806723,"page_idx":84,"id":"8da0ba59-731d-4b11-82cd-cf41d14e6715","page_size":[595,841],"block_position":"84-5"},{"bbox":[238,293,355,306],"type":"image_caption","angle":0,"index":7,"text":"图 7.1: 二次型分类示意图","id":"37449d26-7e24-4c4c-8005-f1c22d7c68d5","color":{"line":"rgba(13, 83, 222 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"page_size":[595,841],"block_position":"84-6"},{"bbox":[72,327,186,339],"type":"title","angle":0,"index":8,"level":2,"text":"## 定理 7.14 (正定充要条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"cfff468b-5264-47b6-b59c-7fbce3939c28","page_size":[595,841],"block_position":"84-7"},{"bbox":[67,343,223,356],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"$f = x^{T}Ax$ 是正定二次型当且仅当： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"4e4e2149-32ef-49cb-b9b7-dda86f2e3e86","page_size":[595,841],"block_position":"84-8"},{"bbox":[83,359,169,371],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"- $A$ 合同于矩阵 $E$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"cbcde533-9dc7-4d43-825a-ca00bd920034","page_size":[595,841],"block_position":"84-9"},{"bbox":[82,371,459,402],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"- 存在可逆矩阵 $C$ ，使得 $A = C^T C$ ，也可以分解为非方阵 $A = \\left( \\begin{array}{cc} C^T & O^T \\end{array} \\right) \\left( \\begin{array}{c} C \\\\ O \\end{array} \\right)$ . ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"74fbe42a-3983-4086-af99-e2d60dfed627","page_size":[595,841],"block_position":"84-10"},{"bbox":[83,401,186,414],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"所有特征值都大于0 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"a22f5a49-e02c-4d1b-b9ea-aed147dd543f","page_size":[595,841],"block_position":"84-11"},{"bbox":[83,417,298,430],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"- $A$ 的各阶顺序主子式/所有主子式行列式大于0 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"903a462b-bc23-4e8b-9db8-e59dd483190c","page_size":[595,841],"block_position":"84-12"},{"bbox":[54,445,319,458],"type":"text","angle":0,"index":14,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"46a584fe-6f83-41f6-87b9-5c0d6e5bc6b0","page_size":[595,841],"block_position":"84-13"},{"bbox":[254,461,337,491],"type":"interline_equation","angle":0,"index":15,"text":"$$\nA = \\left[ \\begin{array}{c c} A _ {1} & \\alpha \\\\ \\alpha^ {T} & a _ {n n} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":84,"id":"10f5fe0a-7711-4577-8d25-2eb09a2df066","page_size":[595,841],"block_position":"84-14"},{"bbox":[54,496,539,540],"type":"text","angle":0,"index":16,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"65c3cc8b-5191-46ce-be28-c13cb0657321","page_size":[595,841],"block_position":"84-15"},{"bbox":[201,542,389,576],"type":"interline_equation","angle":0,"index":17,"text":"$$\nP _ {2} = \\left[ \\begin{array}{c c} P _ {1} & \\\\ & 1 \\end{array} \\right], P _ {3} = \\left[ \\begin{array}{c c} E _ {n - 1} & - P _ {1} ^ {T} \\alpha \\\\ & 1 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":84,"id":"cba1dc61-fa16-4068-9d96-9524c3aaa032","page_size":[595,841],"block_position":"84-16"},{"bbox":[54,577,82,590],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"得到： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"8a2ef7ca-e018-411b-8a83-5ac90348c726","page_size":[595,841],"block_position":"84-17"},{"bbox":[195,588,395,620],"type":"interline_equation","angle":0,"index":19,"text":"$$\nP _ {3} ^ {T} P _ {2} ^ {T} A P _ {2} P _ {3} = \\left[ \\begin{array}{c c} E _ {n - 1} & \\\\ & a _ {n n} - \\alpha^ {T} P _ {1} P _ {1} ^ {T} \\alpha \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":84,"id":"111d1d87-e0c6-49a4-a331-20e088185f28","page_size":[595,841],"block_position":"84-18"},{"bbox":[54,628,207,640],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"而根据乘积矩阵的行列式性质得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"86d387c3-904f-4e01-8c5a-663069f18ff9","page_size":[595,841],"block_position":"84-19"},{"bbox":[215,646,379,661],"type":"interline_equation","angle":0,"index":21,"text":"$$\na _ {n n} - \\alpha^ {T} P _ {1} P _ {1} ^ {T} \\alpha = | A | | P _ {2} | ^ {2} | P _ {3} | ^ {2} > 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":84,"id":"4d6638a6-1cc4-47ea-9578-5fe5f7bb0acf","page_size":[595,841],"block_position":"84-20"},{"bbox":[54,662,276,690],"type":"text","angle":0,"index":22,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"c7b25779-3862-4d2a-a258-3d1b0dc2a95f","page_size":[595,841],"block_position":"84-21"},{"bbox":[269,692,324,704],"type":"interline_equation","angle":0,"index":23,"text":"$$\nP ^ {T} A P = E\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":84,"id":"3783357a-6748-4f43-adae-32238783cb82","page_size":[595,841],"block_position":"84-22"},{"bbox":[54,712,98,724],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"因此正定 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"800f0a12-038a-4b6f-8ccb-3f9cee407e45","page_size":[595,841],"block_position":"84-23"},{"bbox":[35,743,490,756],"type":"text","angle":0,"index":25,"merge_prev":false,"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？ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"51626f61-2756-4b5a-a90f-cf1700e50e16","page_size":[595,841],"block_position":"84-24"},{"bbox":[72,759,334,772],"type":"text","angle":0,"index":26,"merge_prev":false,"text":"(2) $A$ 为实对称矩阵，则 $\\exists M, \\forall t > M$ 有 $tE + A$ 为正定矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":84,"id":"5eb62ffb-6a6f-415b-8f7d-8d1bdd09853e","page_size":[595,841],"block_position":"84-25"},{"bbox":[291,802,303,812],"type":"page_number","angle":0,"index":999999,"text":"80 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":84,"is_discarded":true,"id":"57cad0c3-18ef-47a9-860a-6255a345a4b5","page_size":[595,841],"block_position":"84-26"}],[{"bbox":[420,47,539,58],"type":"header","angle":0,"index":-99999,"text":"7.3 (半) 正定二次型的研究 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":85,"is_discarded":true,"id":"8c9cc0ae-68e4-47ff-ba1c-ea9271057b44","page_size":[595,841],"block_position":"85-0"},{"bbox":[53,70,387,92],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"解(1)算顺序主子式即可，得到 $1-t^{2}>0,-5t^{2}-4t>0$ ，因此 $-\\frac{4}{5}<t<0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"13e053f4-88f1-4395-a816-2af1498e8947","page_size":[595,841],"block_position":"85-1"},{"bbox":[73,90,152,102],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"(2) 看特征值即可 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"4ce548f8-1996-4fe9-bf0c-b0803b70a081","page_size":[595,841],"block_position":"85-2"},{"bbox":[35,121,513,133],"type":"text","angle":0,"index":3,"merge_prev":false,"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 $r(B^{T}AB) = r(B)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"0812686f-20f2-4701-93d7-38aa5960b3aa","page_size":[595,841],"block_position":"85-3"},{"bbox":[52,137,539,181],"type":"text","angle":0,"index":4,"merge_prev":false,"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 $r(B^T AB) = r(B)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"340538a5-26e9-459c-b3a6-39b0f7b10413","page_size":[595,841],"block_position":"85-4"},{"bbox":[54,183,539,211],"type":"text","angle":0,"index":5,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"93146b7b-2d88-4543-9bf4-c9ec66063ecd","page_size":[595,841],"block_position":"85-5"},{"bbox":[73,234,205,247],"type":"title","angle":0,"index":6,"level":2,"text":"## 定理 7.15 (正定矩阵对角性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"30182baa-b337-4e16-a0ab-e8c6cb974c41","page_size":[595,841],"block_position":"85-6"},{"bbox":[69,251,173,262],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"$A$ 为 $n$ 阶正定矩阵，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"e7704abe-2790-42d1-b9ee-ba2f6d9e0100","page_size":[595,841],"block_position":"85-7"},{"bbox":[81,267,198,279],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"1. $a_{ii} > 0, \\forall i = 1, 2, \\cdots, n$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"403ac44e-3c0f-4d4a-88af-bf718c10fbad","page_size":[595,841],"block_position":"85-8"},{"bbox":[80,282,198,295],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"2. $2|a_{ij}| < a_{ii} + a_{jj} (i \\neq j)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"e7fe4b84-296d-4141-8b00-7926fa25f8c1","page_size":[595,841],"block_position":"85-9"},{"bbox":[80,297,415,310],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"f578b01b-8142-4d5c-bb5f-41671e371939","page_size":[595,841],"block_position":"85-10"},{"bbox":[53,326,257,338],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"证明 (1) 不妨取 $X = e_{i}$ ，则 $X^{T}AX = a_{ii} > 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"147f1630-0266-435c-a685-5ceabb6f8aa3","page_size":[595,841],"block_position":"85-11"},{"bbox":[54,341,539,370],"type":"text","angle":0,"index":12,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"8a26a4a8-cdac-4db5-a858-5d05f291951c","page_size":[595,841],"block_position":"85-12"},{"bbox":[73,373,349,386],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"(3) 根据 (2) 可知 $\\forall i, j$ ，有 $|a_{ij}| < \\max\\{a_{ii}, a_{jj}\\}$ ，因此结论成立 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"433d1ca9-984f-409f-a407-486e4eb40940","page_size":[595,841],"block_position":"85-13"},{"bbox":[56,403,234,417],"type":"title","angle":0,"index":14,"level":2,"text":"## 7.3.2 半正定二次型的概念与判别","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"4994bde1-bfe5-4a42-8306-acbdf92cfdaa","page_size":[595,841],"block_position":"85-14"},{"bbox":[73,431,186,444],"type":"title","angle":0,"index":15,"level":2,"text":"## 定义 7.10 (半正定二次型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"8124cb62-f949-4cd6-8535-d6be5225e93b","page_size":[595,841],"block_position":"85-15"},{"bbox":[68,448,524,476],"type":"text","angle":0,"index":16,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"d31f0d43-2a4d-4d86-a250-533ad92c9163","page_size":[595,841],"block_position":"85-16"},{"bbox":[73,496,256,508],"type":"title","angle":0,"index":17,"level":2,"text":"## 定理 7.16 (半正定二次型的基础充要条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"99df7e12-79de-4032-bd03-9f00b95ef341","page_size":[595,841],"block_position":"85-17"},{"bbox":[69,513,292,526],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"$f(x_{1},\\cdots,x_{n})=X^{T}AX$ 为半正定二次型当且仅当： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"a7265ea8-1456-4566-b5c3-7cbd00491266","page_size":[595,841],"block_position":"85-18"},{"bbox":[82,528,338,555],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"4f10e8a6-a98b-4cf9-a4c8-df53dfd07e74","page_size":[595,841],"block_position":"85-19"},{"bbox":[83,558,205,569],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"- $A$ 的特征值都大于等于0 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"21da4716-761f-4026-b906-0b4e49622580","page_size":[595,841],"block_position":"85-20"},{"bbox":[83,572,315,585],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"- $A$ 的所有主子式 (不等价于顺序主子式) 大于等于 0 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"e2098da6-8cb6-4c53-853d-1caa34ae50bf","page_size":[595,841],"block_position":"85-21"},{"bbox":[35,596,504,629],"type":"text","angle":0,"index":23,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"3bfea576-e19e-4246-a5be-563d534901b2","page_size":[595,841],"block_position":"85-22"},{"bbox":[73,639,233,651],"type":"title","angle":0,"index":24,"level":2,"text":"## 定理 7.17 (半正定矩阵的 $C^{T}C$ 分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"6594faa2-10e8-4276-86ee-4475c8342a21","page_size":[595,841],"block_position":"85-23"},{"bbox":[68,656,337,669],"type":"text","angle":0,"index":25,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"572f7afc-95b2-4ce8-8dcc-eb7f5869dd25","page_size":[595,841],"block_position":"85-24"},{"bbox":[83,671,302,684],"type":"text","angle":0,"index":26,"merge_prev":false,"text":"- 存在不列满秩的 $n \\times n$ 矩阵 $C$ , 使得 $A = C^T C$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"80509813-2ab1-47ef-b832-965cf3df9917","page_size":[595,841],"block_position":"85-25"},{"bbox":[83,687,373,700],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"- 存在行满秩 (列不满秩) 的 $r \\times n (r < n)$ 矩阵 $C$ ，使得 $A = C^T C$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":85,"id":"187e0fb9-af74-4497-8a15-2b90e0172ef4","page_size":[595,841],"block_position":"85-26"},{"bbox":[291,803,302,811],"type":"page_number","angle":0,"index":1000019,"text":"81 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":85,"is_discarded":true,"id":"022c8928-eaee-41da-bd5c-33f1d1127e5c","page_size":[595,841],"block_position":"85-27"}],[{"bbox":[420,47,539,58],"type":"header","angle":0,"index":-99999,"text":"7.3 (半) 正定二次型的研究 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":86,"is_discarded":true,"id":"04d7db7b-fb26-4a05-8918-9cfd9c7c526b","page_size":[595,841],"block_position":"86-0"},{"bbox":[54,70,229,101],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"证明 根据 $A = C^{T}\\left( \\begin{array}{cc}E_{r} & O\\\\ O & O \\end{array} \\right)C$ 以及 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"0a0c9541-ef1e-4fa0-b5b7-3252dc21fea7","page_size":[595,841],"block_position":"86-1"},{"bbox":[153,105,440,138],"type":"interline_equation","angle":0,"index":2,"text":"$$\n\\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) = \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) = \\binom{E _ {r}}{O} \\left( \\begin{array}{c c} E _ {r} & O \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":86,"id":"5d7d0834-3bae-4b74-8b87-265af62a0606","page_size":[595,841],"block_position":"86-2"},{"bbox":[55,141,141,153],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"可以得到两种分解： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"f49af2e8-bec8-47a5-b7ed-55cd9968e5e9","page_size":[595,841],"block_position":"86-3"},{"bbox":[209,152,373,211],"type":"interline_equation","angle":0,"index":4,"text":"$$\n\\left\\{ \\begin{array}{l} C _ {1} = \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) C \\Rightarrow A = C _ {1} ^ {T} C _ {1} \\\\ C _ {2} = \\left( \\begin{array}{c c} E _ {r} & O \\end{array} \\right) C \\Rightarrow A = C _ {2} ^ {T} C _ {2} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":86,"id":"cd449566-c5d2-47d3-80c6-72a498a5f3d2","page_size":[595,841],"block_position":"86-4"},{"bbox":[54,218,172,230],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"无论哪种分解列均不满秩。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"ef0cdd5b-6c27-483f-9c60-2c50072b3dd4","page_size":[595,841],"block_position":"86-5"},{"bbox":[73,253,233,266],"type":"title","angle":0,"index":6,"level":2,"text":"## 定理 7.18 (半正定矩阵的 $CC^{T}$ 分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"99974de7-7539-409e-b762-417bb853ba57","page_size":[595,841],"block_position":"86-6"},{"bbox":[68,270,337,284],"type":"text","angle":0,"index":7,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"3c72897a-df78-4d87-9329-381c377aa361","page_size":[595,841],"block_position":"86-7"},{"bbox":[83,286,301,298],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"- 存在不行满秩的 $n \\times n$ 矩阵 $C$ ，使得 $A = CC^T$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"ee44ff13-0451-4651-aa63-0d7fec496420","page_size":[595,841],"block_position":"86-8"},{"bbox":[83,301,372,314],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"- 存在列满秩 (行不满秩) 的 $n \\times r (r < n)$ 矩阵 $C$ ，使得 $A = CC^T$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"de976f91-ccae-4b2f-964f-15727ad9ff42","page_size":[595,841],"block_position":"86-9"},{"bbox":[53,329,539,375],"type":"text","angle":0,"index":10,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"e4c70dd7-b830-49a9-b56d-5398d7aa2db3","page_size":[595,841],"block_position":"86-10"},{"bbox":[35,392,385,405],"type":"text","angle":0,"index":11,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"87fd5a91-7494-472f-b320-bccf22016bcb","page_size":[595,841],"block_position":"86-11"},{"bbox":[72,407,447,426],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"d37c6b89-43fd-4c5b-8c52-bf416bebb6ed","page_size":[595,841],"block_position":"86-12"},{"bbox":[54,428,421,443],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"证明 (1) 显然 $\\alpha \\alpha^T$ 是实对称矩阵，从而 $X^T \\alpha \\alpha^T X = (\\alpha^T X)^T (\\alpha^T X) = (\\alpha^T X)^2 \\geq 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"fa45a903-c277-44ef-86d3-7b93c8363fea","page_size":[595,841],"block_position":"86-13"},{"bbox":[73,445,337,457],"type":"text","angle":0,"index":14,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"62229f36-a6b6-4d5f-b263-b8690e6b2a10","page_size":[595,841],"block_position":"86-14"},{"bbox":[153,462,438,536],"type":"interline_equation","angle":0,"index":15,"text":"$$\nA = \\left[ \\begin{array}{c} 1 \\\\ 2 \\\\ \\vdots \\\\ n \\end{array} \\right] \\left[ \\begin{array}{l l l l} 1 & 2 & \\dots & n \\end{array} \\right], \\quad B = \\left[ \\begin{array}{c} 1 \\\\ \\frac {1}{2} \\\\ \\vdots \\\\ \\frac {1}{n} \\end{array} \\right] \\left[ \\begin{array}{l l l l} 1 & \\frac {1}{2} & \\dots & \\frac {1}{n} \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":86,"id":"0978d8d9-4eb5-41e1-a9c9-4249e1a36cef","page_size":[595,841],"block_position":"86-15"},{"bbox":[73,551,215,564],"type":"title","angle":0,"index":16,"level":2,"text":"## 定理 7.19 (半正定矩阵对角性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"1145d4d3-2316-4e4d-a0e9-40e94bf2c551","page_size":[595,841],"block_position":"86-16"},{"bbox":[70,568,188,580],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"半正定矩阵 $A$ 有以下性质： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"25102d1c-5b42-4fc1-ba7d-dd797bddade8","page_size":[595,841],"block_position":"86-17"},{"bbox":[81,584,155,595],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"1. 对角元素非负 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"7fa5c51f-a528-4077-9f25-4da896566be8","page_size":[595,841],"block_position":"86-18"},{"bbox":[81,600,171,613],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"2. $2|a_{ij}| \\leq a_{ik} + a_{ki}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"3e91ba87-cc0c-4f4d-a0d7-2c543fc44398","page_size":[595,841],"block_position":"86-19"},{"bbox":[80,615,235,627],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"3. 若 $a_{ii} = 0$ ，则 $\\forall k, a_{ik} = a_{ki} = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"b712dd53-894a-4ad7-bc4d-81316131993b","page_size":[595,841],"block_position":"86-20"},{"bbox":[54,644,242,656],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"证明 (1) 只需要取 $x = e_i$ ， $x^T Ax \\geq 0$ 即可 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"a5ad87d2-1830-4351-9450-98ef7edfe782","page_size":[595,841],"block_position":"86-21"},{"bbox":[73,660,538,673],"type":"text","angle":0,"index":22,"merge_prev":false,"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 $2|a_{ij}| \\leq a_{ik} + a_{ki}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"0abdc850-53f9-4669-bc8e-62e635f7fbec","page_size":[595,841],"block_position":"86-22"},{"bbox":[73,675,167,687],"type":"text","angle":0,"index":23,"merge_prev":false,"text":"(3) 根据 (2) 可以得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"7b4359ab-20bf-4cf8-8a04-801f5df29c60","page_size":[595,841],"block_position":"86-23"},{"bbox":[56,705,192,719],"type":"title","angle":0,"index":25,"level":2,"text":"## 7.3.3 $C^T C$ 的 (半) 正定性","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"adbf1e52-f2f1-49d2-9f34-5255bde92a19","page_size":[595,841],"block_position":"86-24"},{"bbox":[54,728,539,772],"type":"text","angle":0,"index":26,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":86,"id":"3e86766f-0376-44d8-acc2-57e7de4d72e3","page_size":[595,841],"block_position":"86-25"},{"bbox":[291,802,302,812],"type":"page_number","angle":0,"index":1000023,"text":"82 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":86,"is_discarded":true,"id":"7aa545c9-c8be-431c-aad3-23b62f995c67","page_size":[595,841],"block_position":"86-26"}],[{"bbox":[420,47,539,58],"type":"header","angle":0,"index":-99998,"text":"7.3 (半) 正定二次型的研究 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":87,"is_discarded":true,"id":"5e3ee9ef-5e83-4b5a-8c98-2290bd64a4ab","page_size":[595,841],"block_position":"87-0"},{"bbox":[74,74,190,87],"type":"title","angle":0,"index":2,"level":2,"text":"## 引理7.1 $(r(A^T A)$ 与 $r(A))$","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"c6b218f4-6d2b-4dcc-8639-b6a0dd7b611f","page_size":[595,841],"block_position":"87-1"},{"bbox":[68,91,223,104],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"$A$ 为实数域上的 $m \\times n$ 的矩阵，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"4f42f005-631a-4fca-8b74-5cb2a01cafba","page_size":[595,841],"block_position":"87-2"},{"bbox":[261,106,332,120],"type":"interline_equation","angle":0,"index":4,"text":"$$\nr (A ^ {T} A) = r (A)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":87,"id":"d0e0aa46-ae54-4735-aee4-044e7543222d","page_size":[595,841],"block_position":"87-3"},{"bbox":[54,136,241,149],"type":"title","angle":0,"index":5,"level":2,"text":"## 证明 在最小二乘法一节中已经给出了证明","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"177da5b0-8bad-45b5-aa44-4d5f9e98e3e4","page_size":[595,841],"block_position":"87-4"},{"bbox":[73,158,189,171],"type":"title","angle":0,"index":6,"level":2,"text":"## 定理 7.20 ( $A^{T}A$ 的正定性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"8ca3867e-32b0-40fa-972f-24433fa6fbd8","page_size":[595,841],"block_position":"87-5"},{"bbox":[69,175,246,187],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"考虑 $A$ 为实数域上的 $m \\times n$ 的矩阵，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"e8091248-54b2-4ec5-8ad5-bd3482f3f3ba","page_size":[595,841],"block_position":"87-6"},{"bbox":[81,190,207,202],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"1. $A^T A, AA^T$ 是半正定矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"3d80234f-b31c-4f59-b62f-5decd2f1bb08","page_size":[595,841],"block_position":"87-7"},{"bbox":[80,206,389,218],"type":"text","angle":0,"index":9,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"4a7c66a0-380b-4667-9afa-d0d7a016e42f","page_size":[595,841],"block_position":"87-8"},{"bbox":[68,219,361,249],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"且 $A^{T}A$ 对应的二次型为 $f(X)=\\sum_{i=1}^{n}(a_{i1}x_{1}+a_{i2}x_{2}+\\cdots+a_{in}x_{n})^{2}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"7b2f7e36-1966-4f82-846d-fb5fa9e680b7","page_size":[595,841],"block_position":"87-9"},{"bbox":[54,264,539,294],"type":"text","angle":0,"index":11,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"564f9e80-58b6-4798-a565-b21ebc72f305","page_size":[595,841],"block_position":"87-10"},{"bbox":[74,296,312,308],"type":"text","angle":0,"index":12,"merge_prev":false,"text":"二次型表达式：在二次型矩阵一节已经作为练习给出。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"afddde64-84f8-4bb3-86dd-a4ec297d28bb","page_size":[595,841],"block_position":"87-11"},{"bbox":[56,327,198,342],"type":"title","angle":0,"index":13,"level":2,"text":"## 7.3.4 正定矩阵同时对角化","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"9c5966a5-f624-4fa2-b625-a84ec2871509","page_size":[595,841],"block_position":"87-12"},{"bbox":[72,354,196,367],"type":"title","angle":0,"index":14,"level":2,"text":"## 定理 7.21 (同时合同对角化)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"6b236774-2be6-49bb-b78e-93f4a08951f2","page_size":[595,841],"block_position":"87-13"},{"bbox":[67,371,486,385],"type":"text","angle":0,"index":15,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"01a48e53-0f39-45b1-8680-0af176e095a9","page_size":[595,841],"block_position":"87-14"},{"bbox":[53,401,356,414],"type":"text","angle":0,"index":16,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"cf65d807-8c4c-417b-a589-2b2a98f1b69a","page_size":[595,841],"block_position":"87-15"},{"bbox":[180,417,413,449],"type":"interline_equation","angle":0,"index":17,"text":"$$\nM ^ {T} (A + B) M = \\left( \\begin{array}{c c} E _ {r} & O \\\\ O & O \\end{array} \\right) = M ^ {T} A M + M ^ {T} B M\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":87,"id":"f951dfa6-7289-4e03-80e5-cf524d83f85a","page_size":[595,841],"block_position":"87-16"},{"bbox":[54,453,539,481],"type":"text","angle":0,"index":18,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"aec7f8c5-a156-4a73-8136-591734ca0cd0","page_size":[595,841],"block_position":"87-17"},{"bbox":[174,484,417,515],"type":"interline_equation","angle":0,"index":19,"text":"$$\nM ^ {T} A M = \\left( \\begin{array}{c c} H _ {r} & O \\\\ O & O \\end{array} \\right), M ^ {T} B M = \\left( \\begin{array}{c c} E _ {r} - H _ {r} & O \\\\ O & O \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":87,"id":"b95190a2-e124-4029-8ca9-8341085a7aa4","page_size":[595,841],"block_position":"87-18"},{"bbox":[54,519,448,533],"type":"text","angle":0,"index":20,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"a6e75e1f-7a95-42b7-9fb9-7e9928c54d7d","page_size":[595,841],"block_position":"87-19"},{"bbox":[185,536,406,567],"type":"interline_equation","angle":0,"index":21,"text":"$$\nP ^ {T} A P = \\left( \\begin{array}{c c} \\Lambda & O \\\\ O & O \\end{array} \\right), P ^ {T} B P = \\left( \\begin{array}{c c} E _ {r} - \\Lambda & O \\\\ O & O \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":87,"id":"6f4c4103-455e-49b5-a7be-1d59f8ef9d92","page_size":[595,841],"block_position":"87-20"},{"bbox":[55,571,127,583],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"同时为对角矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"67015a72-c143-486c-91a5-4abbaa7b1f5b","page_size":[595,841],"block_position":"87-21"},{"bbox":[72,592,242,605],"type":"title","angle":0,"index":23,"level":2,"text":"## 定理 7.22 (AB = BA 同时相似对角化)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"c0ef7348-4fce-4b49-b092-16800eb61bbb","page_size":[595,841],"block_position":"87-22"},{"bbox":[68,608,354,621],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"若 $A, B$ 可相似对角化，且 $AB = BA$ ，则存在可逆矩阵 $P$ ，使得 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"fd53a0b9-4e01-40aa-b168-3db44fe48b49","page_size":[595,841],"block_position":"87-23"},{"bbox":[258,627,334,640],"type":"interline_equation","angle":0,"index":25,"text":"$$\nP ^ {- 1} A P, P ^ {- 1} B P\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":87,"id":"1e0e96f7-bde5-42bc-a105-db30d1be6ad8","page_size":[595,841],"block_position":"87-24"},{"bbox":[69,647,142,660],"type":"text","angle":0,"index":26,"merge_prev":false,"text":"同时为对角矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"e0e79a3c-e227-40b0-bcdb-39c0273e7828","page_size":[595,841],"block_position":"87-25"},{"bbox":[54,677,455,689],"type":"text","angle":0,"index":27,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"19846ee7-e39d-4069-9263-5770702dd9d6","page_size":[595,841],"block_position":"87-26"},{"bbox":[131,695,461,708],"type":"interline_equation","angle":0,"index":28,"text":"$$\nM ^ {- 1} A B M = (M ^ {- 1} A M) (M ^ {- 1} B M) = M ^ {- 1} B A M = (M ^ {- 1} B M) (M ^ {- 1} A M)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":87,"id":"46722f2d-1e43-4ffb-a0e4-d1c3306c8a78","page_size":[595,841],"block_position":"87-27"},{"bbox":[54,714,538,742],"type":"text","angle":0,"index":29,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":87,"id":"d09d95ad-9626-4904-827a-cd7b67900f1b","page_size":[595,841],"block_position":"87-28"},{"bbox":[240,746,352,760],"type":"interline_equation","angle":0,"index":30,"text":"$$\nT = M \\operatorname{diag} \\left\\{Q _ {1}, \\dots , Q _ {r} \\right\\}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":87,"id":"25fe09dd-92c2-4d2e-9b4d-df6800cb33cc","page_size":[595,841],"block_position":"87-29"},{"bbox":[291,802,302,812],"type":"page_number","angle":0,"index":999999,"text":"83 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":87,"is_discarded":true,"id":"ffc5870f-44c4-42cd-b7e3-08b49e6291ba","page_size":[595,841],"block_position":"87-30"}],[{"bbox":[420,47,539,58],"type":"header","angle":0,"index":-99998,"text":"7.3 (半) 正定二次型的研究 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":88,"is_discarded":true,"id":"00f00842-39cf-4b87-9608-fe83ef4c94f5","page_size":[595,841],"block_position":"88-0"},{"bbox":[35,511,47,527],"type":"image","angle":0,"index":0,"img_path":"/18f1321b9a28cd1e7a912d0030429bba3abaea7c68f82d6ab7aab02fb1299de2.jpg","color":{"line":"rgba(89, 92, 220, 1)","fill":"rgba(89, 92, 220, 1)"},"img_ratio":0.2,"page_idx":88,"id":"1b6f749f-f5fe-4d7b-b99f-480fd3c249a5","page_size":[595,841],"block_position":"88-1"},{"bbox":[54,72,98,84],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"可以得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"d79dbfca-c3d0-4bb3-943e-4a0f2803d9c0","page_size":[595,841],"block_position":"88-2"},{"bbox":[164,86,428,100],"type":"interline_equation","angle":0,"index":3,"text":"$$\nT ^ {- 1} A T = \\operatorname{diag} \\left\\{\\lambda_ {1} E, \\dots , \\lambda_ {r} E \\right\\}, T ^ {- 1} B T = \\operatorname{diag} \\left\\{\\Lambda_ {1}, \\dots , \\Lambda_ {r} \\right\\}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":88,"id":"bcd6f9fe-71d3-488d-9140-9a62d38a08d7","page_size":[595,841],"block_position":"88-3"},{"bbox":[54,111,132,123],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"同时为对角矩阵。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"8eba7dee-f052-4427-adb0-14585ffbdd36","page_size":[595,841],"block_position":"88-4"},{"bbox":[55,142,222,156],"type":"title","angle":0,"index":5,"level":2,"text":"## 7.3.5 正定矩阵的四则运算性质","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"1a2ebb99-a6a8-4be6-b58a-4e6a6ed29da4","page_size":[595,841],"block_position":"88-5"},{"bbox":[73,169,196,181],"type":"title","angle":0,"index":6,"level":2,"text":"## 定理 7.23 (正定的四则运算)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"daedefd0-3a31-4e88-8c0e-b4053d9abbd2","page_size":[595,841],"block_position":"88-6"},{"bbox":[68,185,525,215],"type":"text","angle":0,"index":7,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"00c0665d-e1e4-40d3-b5b6-d1b1dafbfca1","page_size":[595,841],"block_position":"88-7"},{"bbox":[54,229,539,267],"type":"text","angle":0,"index":8,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"0cc7c179-96a9-4369-92c3-c0eeb008c6e9","page_size":[595,841],"block_position":"88-8"},{"bbox":[54,270,539,298],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"cacf35f3-3d46-40e3-9ade-33a33f91aa20","page_size":[595,841],"block_position":"88-9"},{"bbox":[53,301,539,330],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"b0b35bf4-78a3-44fb-a56d-4d9c885193bb","page_size":[595,841],"block_position":"88-10"},{"bbox":[188,335,405,350],"type":"interline_equation","angle":0,"index":11,"text":"$$\nQ A B Q ^ {- 1} = Q P ^ {T} P Q ^ {T} (Q Q ^ {- 1}) = (P Q ^ {T}) ^ {T} (P Q ^ {T})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":88,"id":"1e8d3167-1e26-4ddb-9fa9-805eb3bafc11","page_size":[595,841],"block_position":"88-11"},{"bbox":[54,355,422,369],"type":"text","angle":0,"index":12,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"93a89efb-f263-4703-8ef1-eaa7f8a01692","page_size":[595,841],"block_position":"88-12"},{"bbox":[56,387,186,401],"type":"title","angle":0,"index":13,"level":2,"text":"## 7.3.6 正定矩阵分解进阶","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"3f885b83-e342-45e3-9132-101cb4408819","page_size":[595,841],"block_position":"88-13"},{"bbox":[72,414,205,427],"type":"title","angle":0,"index":14,"level":2,"text":"## 定理 7.24 (正定矩阵平方分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"1f987a20-2d92-4d85-8820-03ffa9a8226d","page_size":[595,841],"block_position":"88-14"},{"bbox":[68,431,322,444],"type":"text","angle":0,"index":15,"merge_prev":false,"text":"$A$ 是 $n$ 阶正定矩阵，则存在实对称矩阵 $C_1$ ，使得 $A = C_1^2$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"f2cdc476-d2dd-464d-a204-bc4ef324b395","page_size":[595,841],"block_position":"88-15"},{"bbox":[54,461,458,475],"type":"text","angle":0,"index":16,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"60b23022-c6fe-45f9-99f6-fc96370ebc78","page_size":[595,841],"block_position":"88-16"},{"bbox":[264,480,330,493],"type":"interline_equation","angle":0,"index":18,"text":"$$\nC _ {1} = P ^ {T} \\sqrt {\\Lambda} P\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":88,"id":"2635d824-7c7e-478b-991e-5e7308386937","page_size":[595,841],"block_position":"88-17"},{"bbox":[55,500,133,513],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"即可得到 $A = C_1^2$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"531e2e9c-c00d-4553-8de0-98a8038d3673","page_size":[595,841],"block_position":"88-18"},{"bbox":[53,516,282,528],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"笔记 上述 $A = C_{1}^{2}$ 也是正定矩阵分解！且是唯一的。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"106a50df-8862-4927-b6ab-26ec446eb260","page_size":[595,841],"block_position":"88-19"},{"bbox":[73,539,206,550],"type":"title","angle":0,"index":21,"level":2,"text":"## 定理 7.25 (正定矩阵三角分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"dee31476-2c9e-43e6-accf-25add6054a24","page_size":[595,841],"block_position":"88-20"},{"bbox":[68,555,337,568],"type":"text","angle":0,"index":22,"merge_prev":false,"text":"$A$ 是 $n$ 阶正定矩阵，则存在可逆上三角阵 $R$ ，使得 $A = R^{T}R$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"d4f93937-1d79-453e-a284-1880cd8497d9","page_size":[595,841],"block_position":"88-21"},{"bbox":[54,584,539,597],"type":"text","angle":0,"index":23,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"4ce8268e-7ed8-44cb-a646-d79364be3a5c","page_size":[595,841],"block_position":"88-22"},{"bbox":[73,619,176,632],"type":"title","angle":0,"index":24,"level":2,"text":"## 定理 7.26 (极分解定理)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"24b90903-7ffc-47cd-9430-85fd0af576b8","page_size":[595,841],"block_position":"88-23"},{"bbox":[68,637,393,649],"type":"text","angle":0,"index":25,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"658dc465-5b16-4d29-87c4-26e026123930","page_size":[595,841],"block_position":"88-24"},{"bbox":[260,656,334,669],"type":"interline_equation","angle":0,"index":26,"text":"$$\nA = T S _ {1} = S _ {2} T\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":88,"id":"7ca8d8c2-bf3f-4ffb-9cf0-f488ee3fc984","page_size":[595,841],"block_position":"88-25"},{"bbox":[69,676,167,687],"type":"text","angle":0,"index":27,"merge_prev":false,"text":"且这种分解是唯一的。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"9d30f91a-77d9-4063-9090-288a9bba6c1a","page_size":[595,841],"block_position":"88-26"},{"bbox":[53,703,538,731],"type":"text","angle":0,"index":28,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"c5cbd0fc-5804-40d9-9969-41bb82a3e237","page_size":[595,841],"block_position":"88-27"},{"bbox":[247,734,345,748],"type":"interline_equation","angle":0,"index":29,"text":"$$\nA = (A ^ {T}) ^ {- 1} S _ {1} ^ {2} = T S _ {1}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":88,"id":"c3aa89c2-9be3-4e2d-8fc9-efacbad2333d","page_size":[595,841],"block_position":"88-28"},{"bbox":[54,756,502,772],"type":"text","angle":0,"index":30,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":88,"id":"1e32b550-1179-4ecc-a254-6a2823e0c80b","page_size":[595,841],"block_position":"88-29"},{"bbox":[291,803,303,811],"type":"page_number","angle":0,"index":1000016,"text":"84 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":88,"is_discarded":true,"id":"2efe43c8-ed39-4021-b412-af23e7a477cd","page_size":[595,841],"block_position":"88-30"}],[{"bbox":[420,47,539,58],"type":"header","angle":0,"index":-99999,"text":"7.3 (半) 正定二次型的研究 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":89,"is_discarded":true,"id":"a9594802-4de7-4ea6-880a-dda9f9ba8cf1","page_size":[595,841],"block_position":"89-0"},{"bbox":[74,71,381,84],"type":"text","angle":0,"index":1,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":89,"id":"a1410009-fd4c-45bb-a581-3a4377cb8140","page_size":[595,841],"block_position":"89-1"},{"bbox":[74,87,233,99],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"(2) 唯一性：设 $A = TS_{1} = \\tilde{T}\\tilde{S}_{1}$ ，则 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":89,"id":"04d64ca0-41ab-4217-8f25-c1d14190defc","page_size":[595,841],"block_position":"89-2"},{"bbox":[263,99,320,137],"type":"interline_equation","angle":0,"index":3,"text":"$$\n\\left\\{ \\begin{array}{l} A ^ {T} A = S _ {1} ^ {2} \\\\ A ^ {T} A = \\tilde {S} _ {1} ^ {2} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":89,"id":"5394403b-4b9b-4b37-b9f0-16ad20daa346","page_size":[595,841],"block_position":"89-3"},{"bbox":[54,143,232,156],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"根据前面正定矩阵分解唯一性可知唯一。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":89,"id":"7da71dba-4e65-4cdf-9c44-3e1b94d57116","page_size":[595,841],"block_position":"89-4"},{"bbox":[57,190,151,205],"type":"title","angle":0,"index":5,"level":2,"text":"## 7.3.7 负定二次型","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":89,"id":"9fc91d70-1f75-42d2-845a-2313ea5b5df6","page_size":[595,841],"block_position":"89-5"},{"bbox":[73,217,176,230],"type":"title","angle":0,"index":6,"level":2,"text":"## 定义 7.11 (负定二次型)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":89,"id":"18d86cea-2454-4e5f-82e5-763a539e114f","page_size":[595,841],"block_position":"89-6"},{"bbox":[69,234,482,248],"type":"text","angle":0,"index":7,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":89,"id":"0c7e85f6-ed8c-4a97-a5a2-149a3eede11d","page_size":[595,841],"block_position":"89-7"},{"bbox":[73,269,226,281],"type":"title","angle":0,"index":8,"level":2,"text":"## 定理 7.27 (负定二次型的充要条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":89,"id":"7d170d38-d654-4b7a-8b62-b7fbda20b17c","page_size":[595,841],"block_position":"89-8"},{"bbox":[67,285,525,314],"type":"text","angle":0,"index":9,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":89,"id":"12c6794c-3741-4821-b931-98c6f97e9522","page_size":[595,841],"block_position":"89-9"},{"bbox":[54,329,167,341],"type":"text","angle":0,"index":10,"merge_prev":false,"text":"证明 用特征值判断即可。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":89,"id":"240001dd-a7b7-44b8-92ee-fb5c9787a6df","page_size":[595,841],"block_position":"89-10"},{"bbox":[291,803,302,811],"type":"page_number","angle":0,"index":1000010,"text":"85 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":89,"is_discarded":true,"id":"aaa56c68-8c92-41cf-9b9c-6ed5f4d07d3a","page_size":[595,841],"block_position":"89-11"}],[{"bbox":[236,74,368,94],"type":"title","angle":0,"index":0,"level":2,"text":"## 第8章 内积空间","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"28156d25-5c5b-495a-8e9f-83862669e634","page_size":[595,841],"block_position":"90-0"},{"bbox":[55,123,158,138],"type":"title","angle":0,"index":1,"level":2,"text":"## 8.1 Euclid 空间","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"63d58348-37b9-4523-86b4-2340f0989c76","page_size":[595,841],"block_position":"90-1"},{"bbox":[55,156,186,169],"type":"title","angle":0,"index":2,"level":2,"text":"## 8.1.1 Euclid 空间与正交","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"0db2fe63-fd33-41aa-8bf1-93c9b990e195","page_size":[595,841],"block_position":"90-2"},{"bbox":[73,183,161,195],"type":"title","angle":0,"index":3,"level":2,"text":"## 定义 8.1 (欧式空间)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"45cc52d1-ef66-4080-912a-0a3ccfc7b188","page_size":[595,841],"block_position":"90-3"},{"bbox":[68,200,258,211],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"$V$ 是 $\\mathbb{R}$ 上线性空间，定义内积 $(\\alpha, \\beta)$ 满足： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"ace31c33-117b-4ae5-b075-3a0d57dfdf09","page_size":[595,841],"block_position":"90-4"},{"bbox":[83,216,311,227],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"- 线性性： $(k_{1}\\alpha_{1} + k_{2}\\alpha_{2}, \\beta) = k_{1}(\\alpha_{1}, \\beta) + k_{2}(\\alpha_{2}, \\beta)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"a3b13937-d912-4b24-8043-b1e08536c677","page_size":[595,841],"block_position":"90-5"},{"bbox":[83,231,200,243],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"- 对称性： $(\\alpha ,\\beta) = (\\alpha ,\\beta)$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"d8f8da94-e26d-4e1d-b94e-e5a6ac208dad","page_size":[595,841],"block_position":"90-6"},{"bbox":[83,247,250,259],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"- 正定性： $(\\alpha, \\alpha) = 0$ 当且仅当 $\\alpha = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"ed6ead70-cd5a-4304-ab84-d044ba90ab82","page_size":[595,841],"block_position":"90-7"},{"bbox":[73,280,160,292],"type":"title","angle":0,"index":8,"level":2,"text":"## 定义 8.2 (度量矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"9dbf54f7-6d78-4960-931f-6991382249c6","page_size":[595,841],"block_position":"90-8"},{"bbox":[68,297,389,310],"type":"text","angle":0,"index":9,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"f01d955d-939d-470e-bd3e-2632944972fb","page_size":[595,841],"block_position":"90-9"},{"bbox":[206,312,385,381],"type":"interline_equation","angle":0,"index":10,"text":"$$\nA = \\left[ \\begin{array}{c c c c} (\\epsilon_ {1}, \\epsilon_ {1}) & (\\epsilon_ {1}, \\epsilon_ {2}) & \\dots & (\\epsilon_ {1}, \\epsilon_ {n}) \\\\ (\\epsilon_ {2}, \\epsilon_ {1}) & (\\epsilon_ {2}, \\epsilon_ {2}) & \\dots & (\\epsilon_ {2}, \\epsilon_ {n}) \\\\ \\vdots & \\vdots & & \\vdots \\\\ (\\epsilon_ {n}, \\epsilon_ {1}) & (\\epsilon_ {n}, \\epsilon_ {2}) & \\dots & (\\epsilon_ {n}, \\epsilon_ {n}) \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":90,"id":"13d8a5c0-0781-42bc-a7f3-988bfcdf5278","page_size":[595,841],"block_position":"90-10"},{"bbox":[73,401,211,414],"type":"title","angle":0,"index":11,"level":2,"text":"## 定理 8.1 (内积的度量矩阵表示)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"b7ab39ed-da90-410e-8cd8-08c46c681483","page_size":[595,841],"block_position":"90-11"},{"bbox":[69,418,524,432],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"1012e3c5-8660-4d07-95b8-1d54f7d63672","page_size":[595,841],"block_position":"90-12"},{"bbox":[261,437,332,451],"type":"interline_equation","angle":0,"index":13,"text":"$$\n(\\alpha , \\beta) = X ^ {T} A Y\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":90,"id":"5fc4a53c-14c4-45f4-9d80-908d06e5601d","page_size":[595,841],"block_position":"90-13"},{"bbox":[35,483,277,496],"type":"text","angle":0,"index":14,"merge_prev":false,"text":"练习 8.1 度量矩阵 已知 Euclid 空间 V 的一组基为: ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"55300b8e-8869-48a6-bb36-93ff08e8d6ec","page_size":[595,841],"block_position":"90-14"},{"bbox":[161,506,431,516],"type":"interline_equation","angle":0,"index":15,"text":"$$\n\\epsilon_ {1} = e _ {1} + e _ {2}, \\epsilon_ {2} = e _ {1} + e _ {3}, \\epsilon_ {3} = e _ {4} - e _ {1}, \\epsilon_ {4} = e _ {1} - e _ {2} - e _ {3} + e _ {4}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":90,"id":"b699c22c-3535-4aea-b091-71b8d5194238","page_size":[595,841],"block_position":"90-15"},{"bbox":[52,522,534,550],"type":"text","angle":0,"index":16,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"b068758d-2b08-4779-8155-bcc455767f0c","page_size":[595,841],"block_position":"90-16"},{"bbox":[247,551,336,624],"type":"interline_equation","angle":0,"index":17,"text":"$$\n\\left\\{ \\begin{array}{l} \\epsilon_ {1} = (1, 1, 0, 0) \\\\ \\epsilon_ {2} = (1, 0, 1, 0) \\\\ \\epsilon_ {3} = (- 1, 0, 0, 1) \\\\ \\epsilon_ {4} = (1, - 1, - 1, 1) \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":90,"id":"998520ae-63c5-4834-b421-9dbad492352c","page_size":[595,841],"block_position":"90-17"},{"bbox":[54,632,281,644],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"因此根据向量的坐标内积公式可以得到度量矩阵为： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"c420969c-e4ff-408c-a380-9dd9881aaa75","page_size":[595,841],"block_position":"90-18"},{"bbox":[211,647,382,709],"type":"interline_equation","angle":0,"index":19,"text":"$$\nA = \\left[ \\begin{array}{r r r r} 2 & 1 & - 1 & 0 \\\\ 1 & 2 & - 1 & 0 \\\\ - 1 & - 1 & 2 & 0 \\\\ 0 & 0 & 0 & 4 \\end{array} \\right], (\\alpha , \\beta) = 5\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":90,"id":"1e322d2a-367a-49c8-9d1f-26c4909d4aad","page_size":[595,841],"block_position":"90-19"},{"bbox":[73,724,200,737],"type":"title","angle":0,"index":20,"level":2,"text":"## 定理 8.2 (正交向量线性无关)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"ed91bf83-4e54-4365-be25-8e0f49d477f1","page_size":[595,841],"block_position":"90-20"},{"bbox":[69,741,356,754],"type":"text","angle":0,"index":21,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":90,"id":"8a5ffbdd-acdd-4330-9d3d-dc32ce4c3dc6","page_size":[595,841],"block_position":"90-21"}],[{"bbox":[471,47,539,58],"type":"header","angle":0,"index":-99998,"text":"8.1 Euclid 空间 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":91,"is_discarded":true,"id":"1477bbd2-a6db-4a59-924d-dab6b714731e","page_size":[595,841],"block_position":"91-0"},{"bbox":[52,72,327,84],"type":"text","angle":0,"index":2,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"8ff42e46-ebe1-4ed3-8202-522fb1caa7f0","page_size":[595,841],"block_position":"91-1"},{"bbox":[217,91,377,104],"type":"interline_equation","angle":0,"index":3,"text":"$$\n\\left(c _ {1} \\alpha_ {1} + \\dots + c _ {s} \\alpha_ {s}, \\alpha_ {i}\\right) = (0, \\alpha_ {i}) = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":91,"id":"8f56c804-184c-425b-9b7c-9f1125a99cf6","page_size":[595,841],"block_position":"91-2"},{"bbox":[54,111,492,123],"type":"text","angle":0,"index":4,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"a602d61f-8af6-4574-9da1-af0d6854d794","page_size":[595,841],"block_position":"91-3"},{"bbox":[55,142,151,156],"type":"title","angle":0,"index":5,"level":2,"text":"## 8.1.2 标准正交基","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"55d0c57d-5785-4f0e-a3aa-b4023e804856","page_size":[595,841],"block_position":"91-4"},{"bbox":[54,164,197,178],"type":"title","angle":0,"index":6,"level":2,"text":"## 一、标准正交基的概念与存在性","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"b03bdf76-96d5-4b5d-9e0f-97ce37291a85","page_size":[595,841],"block_position":"91-5"},{"bbox":[73,185,171,198],"type":"title","angle":0,"index":7,"level":2,"text":"## 定义 8.3 (标准正交基)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"9851e716-c7dd-4599-b352-ba4ce9ede6da","page_size":[595,841],"block_position":"91-6"},{"bbox":[67,202,339,216],"type":"text","angle":0,"index":8,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"a5b0593b-6534-478c-bb3c-3963329936ca","page_size":[595,841],"block_position":"91-7"},{"bbox":[54,231,197,244],"type":"title","angle":0,"index":9,"level":2,"text":"## 二、标准正交基之间的过渡矩阵","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"b07058a7-3bb3-4131-bd9a-5634cd4bf022","page_size":[595,841],"block_position":"91-8"},{"bbox":[73,253,221,265],"type":"title","angle":0,"index":10,"level":2,"text":"## 定理 8.3 (标准正交基的过渡矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"b885c707-c5c2-45a7-94b8-d83f73030928","page_size":[595,841],"block_position":"91-9"},{"bbox":[67,269,524,296],"type":"text","angle":0,"index":11,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"9fbd55bc-5355-4f99-a034-b8af252fe0b5","page_size":[595,841],"block_position":"91-10"},{"bbox":[234,301,360,314],"type":"interline_equation","angle":0,"index":12,"text":"$$\n(\\eta_ {1}, \\dots , \\eta_ {n}) = (\\epsilon_ {1}, \\dots , \\epsilon_ {n}) T\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":91,"id":"91ee5fe0-ecc1-4282-b965-517413113794","page_size":[595,841],"block_position":"91-11"},{"bbox":[54,329,233,343],"type":"title","angle":0,"index":13,"level":2,"text":"## 三、标准正交基的计算：Schmidt 正交化","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"8b2b7e5c-3a3c-48f5-b66d-a6e026128688","page_size":[595,841],"block_position":"91-12"},{"bbox":[72,351,189,364],"type":"title","angle":0,"index":14,"level":2,"text":"## 定理 8.4 (Schmidt 正交化)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"7884e039-71fb-48e9-8ab5-8cbcdf54ebc0","page_size":[595,841],"block_position":"91-13"},{"bbox":[68,368,525,396],"type":"text","angle":0,"index":15,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"16e93071-c1d2-400d-a0b9-c9cfc1a2b143","page_size":[595,841],"block_position":"91-14"},{"bbox":[244,399,280,409],"type":"interline_equation","angle":0,"index":16,"text":"$$\n\\epsilon_ {1} = \\alpha_ {1}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":91,"id":"6574bb8c-2a11-4bea-ba20-63e9ae8298c7","page_size":[595,841],"block_position":"91-15"},{"bbox":[243,407,335,436],"type":"interline_equation","angle":0,"index":17,"text":"$$\n\\epsilon_ {2} = \\alpha_ {2} - \\frac {\\left(\\alpha_ {2} , \\epsilon_ {1}\\right)}{\\left(\\epsilon_ {1} , \\epsilon_ {1}\\right)} \\epsilon_ {1}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":91,"id":"e473d733-7b59-44f6-8eab-429ce2bff31a","page_size":[595,841],"block_position":"91-16"},{"bbox":[243,453,349,486],"type":"interline_equation","angle":0,"index":18,"text":"$$\n\\epsilon_ {s} = \\alpha_ {s} - \\sum_ {i = 1} ^ {s - 1} \\frac {(\\alpha_ {s} , \\epsilon_ {i})}{(\\epsilon_ {i} , \\epsilon_ {i})} \\epsilon_ {i}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":91,"id":"ea9b0083-687a-41fc-a3ad-75f89629fa83","page_size":[595,841],"block_position":"91-17"},{"bbox":[35,517,539,544],"type":"text","angle":0,"index":19,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"4bb243f6-e095-4115-9fb0-6e78e7b04c6e","page_size":[595,841],"block_position":"91-18"},{"bbox":[52,543,376,570],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"解 最终结果为 $\\eta_{1} = \\frac{1}{\\sqrt{6}} (1,2, - 1)^{T},\\eta_{2} = \\frac{1}{\\sqrt{3}} (-1,1,1)^{T},\\eta_{3} = \\frac{1}{\\sqrt{2}} (1,0,1)^{T}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"4009b903-dc72-4381-9bbb-51dda17e2b92","page_size":[595,841],"block_position":"91-19"},{"bbox":[73,584,159,597],"type":"title","angle":0,"index":21,"level":2,"text":"## 定理 8.5 (QR 分解)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"1f228e80-55be-46f7-bdc6-e9120ab066c7","page_size":[595,841],"block_position":"91-20"},{"bbox":[53,629,483,643],"type":"text","angle":0,"index":22,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":91,"id":"8315b650-3070-4763-b576-ea98c41fb579","page_size":[595,841],"block_position":"91-21"},{"bbox":[219,648,364,722],"type":"interline_equation","angle":0,"index":23,"text":"$$\n\\left\\{ \\begin{array}{l} \\alpha_ {1} = \\beta_ {1} \\\\ \\alpha_ {2} = a _ {1 2} \\beta_ {1} + \\beta_ {2} \\\\ \\quad \\vdots \\\\ \\alpha_ {n} = a _ {1 n} \\beta_ {1} + a _ {2 n} \\beta_ {2} + \\dots + \\beta_ {n} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":91,"id":"554e1181-a051-47a0-9417-ab09d9127995","page_size":[595,841],"block_position":"91-22"},{"bbox":[291,802,302,811],"type":"page_number","angle":0,"index":999999,"text":"87 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":91,"is_discarded":true,"id":"f4373a51-8459-43b1-999b-bbca17710de0","page_size":[595,841],"block_position":"91-23"}],[{"bbox":[471,47,539,58],"type":"header","angle":0,"index":-99999,"text":"8.1 Euclid 空间 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":92,"is_discarded":true,"id":"09a1bce3-e90e-4ae9-80d8-9491571cb6a2","page_size":[595,841],"block_position":"92-0"},{"bbox":[54,68,248,95],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"再将 $\\beta$ 进行单位化得到 $\\eta_{i} = \\frac{\\beta_{i}}{|\\beta_{i}|}$ ，此时写为 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":92,"id":"719bea9b-6585-46a3-b4dd-cb0eacd3d92e","page_size":[595,841],"block_position":"92-1"},{"bbox":[180,98,411,165],"type":"interline_equation","angle":0,"index":2,"text":"$$\n(\\alpha_ {1}, \\dots , \\alpha_ {n}) = (\\eta_ {1}, \\dots , \\eta_ {n}) \\left( \\begin{array}{c c c c} b _ {1 1} & b _ {1 2} & \\dots & b _ {1 n} \\\\ & b _ {2 2} & \\dots & b _ {2 n} \\\\ & & \\ddots & \\vdots \\\\ & & & b _ {n n} \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":92,"id":"3afdfb2b-6165-4897-860e-04780ea005d1","page_size":[595,841],"block_position":"92-2"},{"bbox":[54,169,475,182],"type":"text","angle":0,"index":3,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":92,"id":"ffb54d1c-231b-4aff-840d-a65bdb79d212","page_size":[595,841],"block_position":"92-3"},{"bbox":[35,200,461,212],"type":"text","angle":0,"index":4,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":92,"id":"27a3e54e-fcb7-43df-a93a-5793d9076f70","page_size":[595,841],"block_position":"92-4"},{"bbox":[255,215,336,265],"type":"interline_equation","angle":0,"index":5,"text":"$$\nA = \\left[ \\begin{array}{l l l} 1 & 1 & 1 \\\\ 1 & 0 & 1 \\\\ 1 & 2 & 0 \\end{array} \\right]\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":92,"id":"0c6981cb-fa17-46c1-89c1-5c15835be305","page_size":[595,841],"block_position":"92-5"},{"bbox":[53,268,232,280],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"解 先进行 $\\alpha_{1}, \\alpha_{2}, \\alpha_{3}$ 正交化到 $\\beta_{1}, \\beta_{2}, \\beta_{3}$ : ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":92,"id":"f3c2b70f-35a8-4966-8c40-9c9fd8ee7d8e","page_size":[595,841],"block_position":"92-6"},{"bbox":[180,285,404,349],"type":"interline_equation","angle":0,"index":8,"text":"$$\n\\left\\{ \\begin{array}{l} \\beta_ {1} = \\alpha_ {1} = (1, 1, 1) ^ {T}, | \\beta_ {1} | = \\sqrt {3} \\\\ \\beta_ {2} = \\alpha_ {2} - \\frac {3}{3} (1, 1, 1) ^ {T} = (0, - 1, 1) ^ {T}, | \\beta_ {2} | = \\sqrt {2} \\\\ \\beta_ {3} = \\alpha_ {3} + \\frac {1}{2} \\beta_ {2} - \\frac {2}{3} \\beta_ {1} = (\\frac {1}{3}, - \\frac {1}{6}, - \\frac {1}{6}), | \\beta_ {3} | = \\sqrt {\\frac {1}{6}} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":92,"id":"d9038d52-d354-4ff5-88ab-8346ddb825ac","page_size":[595,841],"block_position":"92-7"},{"bbox":[54,350,127,362],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"进行单位化得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":92,"id":"3db0951c-d016-45df-b6e4-72b957119411","page_size":[595,841],"block_position":"92-8"},{"bbox":[236,359,348,432],"type":"interline_equation","angle":0,"index":10,"text":"$$\n\\left\\{ \\begin{array}{l} \\eta_ {1} = (\\frac {\\sqrt {3}}{3}, \\frac {\\sqrt {3}}{3}, \\frac {\\sqrt {3}}{3}) \\\\ \\eta_ {2} = (0, - \\frac {\\sqrt {2}}{2}, \\frac {\\sqrt {2}}{2}) \\\\ \\eta_ {3} = (\\frac {\\sqrt {6}}{3}, - \\frac {\\sqrt {6}}{6}, - \\frac {\\sqrt {6}}{6}) \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":92,"id":"9e51953b-f3c7-41d0-a226-a40392eb9d77","page_size":[595,841],"block_position":"92-9"},{"bbox":[54,437,97,449],"type":"text","angle":0,"index":11,"merge_prev":false,"text":"反向得到 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":92,"id":"ee7abde1-c2b2-489f-945a-ff06427380f3","page_size":[595,841],"block_position":"92-10"},{"bbox":[239,449,344,508],"type":"interline_equation","angle":0,"index":12,"text":"$$\n\\left\\{ \\begin{array}{l} \\alpha_ {1} = \\beta_ {1} \\\\ \\alpha_ {2} = \\beta_ {1} + \\beta_ {2} \\\\ \\alpha_ {3} = \\frac {2}{3} \\beta_ {1} - \\frac {1}{2} \\beta_ {2} + \\beta_ {3} \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":92,"id":"99e9e8ca-9d83-4eeb-9eef-5cf12f763260","page_size":[595,841],"block_position":"92-11"},{"bbox":[55,513,78,524],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"因此 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":92,"id":"23180002-3406-4f63-9444-fc68fb8c6c14","page_size":[595,841],"block_position":"92-12"},{"bbox":[89,525,502,583],"type":"interline_equation","angle":0,"index":14,"text":"$$\n(\\alpha_ {1}, \\alpha_ {2}, \\alpha_ {3}) = (\\beta_ {1}, \\beta_ {2}, \\beta_ {3}) \\left( \\begin{array}{c c c} 1 & 1 & \\frac {2}{3} \\\\ 0 & 1 & - \\frac {1}{2} \\\\ 0 & 0 & 1 \\end{array} \\right) = (\\eta_ {1}, \\eta_ {2}, \\eta_ {3}) \\left( \\begin{array}{c c c} | \\beta_ {1} | & & \\\\ & | \\beta_ {2} | & \\\\ & & | \\beta_ {3} | \\end{array} \\right) \\left( \\begin{array}{c c c} 1 & 1 & \\frac {2}{3} \\\\ 0 & 1 & - \\frac {1}{2} \\\\ 0 & 0 & 1 \\end{array} \\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":92,"id":"62306624-9c4e-4929-89db-cbc5a36e6cf4","page_size":[595,841],"block_position":"92-13"},{"bbox":[56,600,174,614],"type":"title","angle":0,"index":15,"level":2,"text":"## 8.1.3 正交和与正交补","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":92,"id":"1895753d-042a-4249-8977-96c6fab8cb13","page_size":[595,841],"block_position":"92-14"},{"bbox":[73,628,191,640],"type":"title","angle":0,"index":16,"level":2,"text":"## 定义 8.4 (正交和、正交补)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":92,"id":"9d8db829-fa8a-44b8-9d3b-565f7b01ce9f","page_size":[595,841],"block_position":"92-15"},{"bbox":[68,644,525,672],"type":"text","angle":0,"index":17,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":92,"id":"072c0543-c5e4-473d-8d55-3d7d205f87b2","page_size":[595,841],"block_position":"92-16"},{"bbox":[73,694,191,706],"type":"title","angle":0,"index":18,"level":2,"text":"## 定理 8.6 (正交和必为直和)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":92,"id":"1c4b3a6d-2233-4b5c-be42-76b82a2cb7b3","page_size":[595,841],"block_position":"92-17"},{"bbox":[69,710,257,723],"type":"text","angle":0,"index":19,"merge_prev":false,"text":"若 $V_{1} + \\cdots + V_{s}$ 为正交和，则其必为直和。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":92,"id":"b668cade-ae3d-4c70-849b-394f5a78b633","page_size":[595,841],"block_position":"92-18"},{"bbox":[54,740,275,751],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"证明 设 $0 = \\alpha_{1} + \\cdots + \\alpha_{s}$ ，两侧与 $\\alpha_{i}$ 做内积可知 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":92,"id":"178e699e-d875-44dd-be69-f66fdf6f288c","page_size":[595,841],"block_position":"92-19"},{"bbox":[269,759,323,772],"type":"interline_equation","angle":0,"index":21,"text":"$$\n0 = (\\alpha_ {i}, \\alpha_ {i})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":92,"id":"104de7af-38a5-4d80-9b13-2ddec9524af1","page_size":[595,841],"block_position":"92-20"},{"bbox":[291,803,302,812],"type":"page_number","angle":0,"index":1000006,"text":"88 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":92,"is_discarded":true,"id":"324aff33-4c6d-4548-870b-c17533b23018","page_size":[595,841],"block_position":"92-21"}],[{"bbox":[471,47,539,58],"type":"header","angle":0,"index":-99999,"text":"8.1 Euclid 空间 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":93,"is_discarded":true,"id":"3d3debae-ed7a-466e-9da1-fd51023fe97c","page_size":[595,841],"block_position":"93-0"},{"bbox":[54,72,242,84],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"因此 $\\alpha_{i}=0$ ，故零元素分解唯一，为直和。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"ad1de756-f56c-47e8-870d-a4dbfb960b20","page_size":[595,841],"block_position":"93-1"},{"bbox":[73,93,191,105],"type":"title","angle":0,"index":2,"level":2,"text":"## 定理 8.7 (正交补的唯一性)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"5ed234bb-38a6-48ff-99d7-c68ee1df0c54","page_size":[595,841],"block_position":"93-2"},{"bbox":[68,110,298,122],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"Euclid 空间 V 的任一子空间 $V_{1}$ 有唯一的正交补 $W^{\\perp}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"8a50f423-377d-48ec-aecc-fe899bfe5995","page_size":[595,841],"block_position":"93-3"},{"bbox":[54,138,512,152],"type":"text","angle":0,"index":4,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"1d8ea023-96d0-487a-94b2-eae9eb05479d","page_size":[595,841],"block_position":"93-4"},{"bbox":[255,158,337,171],"type":"interline_equation","angle":0,"index":5,"text":"$$\nV = L \\left(\\epsilon_ {1}, \\dots , \\epsilon_ {n}\\right)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":93,"id":"0db18113-8102-41f2-b68f-e30eeb341b64","page_size":[595,841],"block_position":"93-5"},{"bbox":[54,178,207,190],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"则 $V_{1}^{\\perp}=L(\\epsilon_{s+1},\\cdots,\\epsilon_{n})$ 为正交补 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"76712862-0cbf-4b09-8a80-61858697dfa0","page_size":[595,841],"block_position":"93-6"},{"bbox":[54,193,539,222],"type":"text","angle":0,"index":8,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"a97017fe-0a0f-4fc9-a52b-a49214178b3e","page_size":[595,841],"block_position":"93-7"},{"bbox":[200,225,392,238],"type":"interline_equation","angle":0,"index":9,"text":"$$\n0 = (\\alpha , \\alpha_ {1}) = (\\alpha_ {1}, \\alpha_ {1}) + (\\alpha_ {3}, \\alpha_ {1}) = (\\alpha_ {1}, \\alpha_ {1})\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":93,"id":"a8512bd0-fb06-4b03-9f68-a713df48510d","page_size":[595,841],"block_position":"93-8"},{"bbox":[54,248,327,260],"type":"text","angle":0,"index":10,"merge_prev":false,"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 $V_{2} = V_{3}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"e4f0748c-3487-4c89-b7c0-8563d100fd90","page_size":[595,841],"block_position":"93-9"},{"bbox":[73,269,211,282],"type":"title","angle":0,"index":11,"level":2,"text":"## 定理 8.8 (正交补的具体表达式)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"2020e24b-205e-4f55-92fc-1cad80e77b9d","page_size":[595,841],"block_position":"93-10"},{"bbox":[69,287,382,300],"type":"text","angle":0,"index":12,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"31ba5a2a-caff-4da4-a5c8-d0e48f33a68c","page_size":[595,841],"block_position":"93-11"},{"bbox":[246,306,346,320],"type":"interline_equation","angle":0,"index":13,"text":"$$\nV _ {1} ^ {\\perp} = \\{\\alpha \\in V, \\alpha \\perp V _ {1} \\}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":93,"id":"b5e19adb-3727-4990-bf91-c7b29a232981","page_size":[595,841],"block_position":"93-12"},{"bbox":[53,335,429,349],"type":"text","angle":0,"index":14,"merge_prev":false,"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 $V_1^\\perp \\subseteq W$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"74fff679-94a2-4476-a81f-466497b11909","page_size":[595,841],"block_position":"93-13"},{"bbox":[73,352,379,364],"type":"text","angle":0,"index":15,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"ad06436d-876a-4296-bad8-929142731a77","page_size":[595,841],"block_position":"93-14"},{"bbox":[191,371,402,385],"type":"interline_equation","angle":0,"index":16,"text":"$$\n\\left(\\alpha_ {1}, \\alpha_ {1}\\right) = \\left(\\alpha - \\alpha_ {2}, \\alpha_ {1}\\right) = \\left(\\alpha , \\alpha_ {1}\\right) - \\left(\\alpha_ {2}, \\alpha_ {1}\\right) = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":93,"id":"de4d0f2a-4adf-4eb9-b618-bc2bfcc8cc93","page_size":[595,841],"block_position":"93-15"},{"bbox":[55,391,164,403],"type":"text","angle":0,"index":17,"merge_prev":false,"text":"因此 $\\alpha_{1} = 0$ ，即 $\\alpha \\in V_1^\\perp$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"9950bf01-25f9-40de-b2b8-bd5ec3aa5f22","page_size":[595,841],"block_position":"93-16"},{"bbox":[73,413,180,425],"type":"title","angle":0,"index":18,"level":2,"text":"## 定理 8.9 (正交补的性质)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"0777b597-357b-4340-bc46-4f8e617984fc","page_size":[595,841],"block_position":"93-17"},{"bbox":[68,429,367,444],"type":"interline_equation","angle":0,"index":19,"text":"$$\n(1) (U ^ {\\perp}) ^ {\\perp} = U (2) (U _ {1} + U _ {2}) ^ {\\perp} = U _ {1} ^ {\\perp} \\cap U _ {2} ^ {\\perp} (3) (U _ {1} \\cap U _ {2}) ^ {\\perp} = U _ {1} ^ {\\perp} + U _ {2} ^ {\\perp}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":93,"id":"7fddec3d-273d-4ed8-ae5d-7317af794788","page_size":[595,841],"block_position":"93-18"},{"bbox":[54,460,246,473],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"证明 (1) 根据正交补的具体表达式一下得出 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"d82e319e-0ec4-4962-9583-159899583f33","page_size":[595,841],"block_position":"93-19"},{"bbox":[73,475,415,489],"type":"text","angle":0,"index":21,"merge_prev":false,"text":"(2) 显然 $(U_{1} + U_{2})^{\\perp}\\subseteq U_{1}^{\\perp}$ ， $(U_{1} + U_{2})^{\\perp}\\subseteq U_{2}^{\\perp}$ ，因此 $(U_{1} + U_{2})^{\\perp}\\subseteq U_{1}^{\\perp}\\cap U_{2}^{\\perp}$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"934fe7cf-a7e9-4df5-96a6-882241ff6717","page_size":[595,841],"block_position":"93-20"},{"bbox":[74,491,354,504],"type":"text","angle":0,"index":22,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"101a2186-2b77-4e1d-a51a-6d97bc4e0255","page_size":[595,841],"block_position":"93-21"},{"bbox":[199,511,394,524],"type":"interline_equation","angle":0,"index":23,"text":"$$\n(\\alpha , \\beta) = (\\alpha , \\beta_ {1} + \\beta_ {2}) = (\\alpha , \\beta_ {1}) + (\\alpha , \\beta_ {2}) = 0\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":93,"id":"5a68eafd-10fc-4600-bbb6-09b603690b98","page_size":[595,841],"block_position":"93-22"},{"bbox":[55,529,182,543],"type":"text","angle":0,"index":24,"merge_prev":false,"text":"因此 $U_{1}^{\\perp}\\cap U_{2}^{\\perp}\\subseteq (U_{1} + U_{2})^{\\perp}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"5651d5b7-d1c9-49cb-a842-f3154e311143","page_size":[595,841],"block_position":"93-23"},{"bbox":[74,546,145,558],"type":"text","angle":0,"index":25,"merge_prev":false,"text":"(3) 可转换为 (2) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"855428fe-b294-4ca4-8f13-61b372f11b1f","page_size":[595,841],"block_position":"93-24"},{"bbox":[56,576,138,590],"type":"title","angle":0,"index":26,"level":2,"text":"## 8.1.4 正交变换","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"af8231e2-329a-4a5c-82a4-bfe59a13df2b","page_size":[595,841],"block_position":"93-25"},{"bbox":[73,604,160,617],"type":"title","angle":0,"index":27,"level":2,"text":"## 定义 8.5 (正交变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"cb64101a-1e78-424e-80ce-d79995805945","page_size":[595,841],"block_position":"93-26"},{"bbox":[68,621,504,634],"type":"text","angle":0,"index":28,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"517e3500-4755-4e1a-9129-7e1d9bfe9476","page_size":[595,841],"block_position":"93-27"},{"bbox":[73,655,205,667],"type":"title","angle":0,"index":29,"level":2,"text":"## 定理 8.10 (正交变换充要条件)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"e1566e40-f642-4136-9113-012c7c0de809","page_size":[595,841],"block_position":"93-28"},{"bbox":[68,671,226,684],"type":"text","angle":0,"index":30,"merge_prev":false,"text":"$\\mathcal{A}$ 是正交变换与下面几个命题等价： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"74964515-2fa5-431d-87f7-af708dd92657","page_size":[595,841],"block_position":"93-29"},{"bbox":[83,687,247,699],"type":"text","angle":0,"index":31,"merge_prev":false,"text":"- $\\mathcal{A}$ 在标准正交基下矩阵为正交矩阵 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"f14726ca-4571-4780-9800-c746ccf322c0","page_size":[595,841],"block_position":"93-30"},{"bbox":[83,703,279,715],"type":"text","angle":0,"index":32,"merge_prev":false,"text":"- 标准正交基在 $\\mathcal{A}$ 的作用下仍为标准正交基 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"92497bf7-d226-424a-b0c4-a7f668cf86b4","page_size":[595,841],"block_position":"93-31"},{"bbox":[83,719,186,730],"type":"text","angle":0,"index":33,"merge_prev":false,"text":"- $\\mathcal{A}$ 保持向量长度不变 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"bd182434-8070-42e8-9ae9-7a136661fd1d","page_size":[595,841],"block_position":"93-32"},{"bbox":[83,734,427,747],"type":"text","angle":0,"index":34,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":93,"id":"64671bda-f306-490d-8ec7-102296f07f2c","page_size":[595,841],"block_position":"93-33"},{"bbox":[291,802,302,811],"type":"page_number","angle":0,"index":1000006,"text":"89 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":93,"is_discarded":true,"id":"3e71a0f2-3d31-4c90-b132-036d7bf95d15","page_size":[595,841],"block_position":"93-34"}],[{"bbox":[467,47,539,58],"type":"header","angle":0,"index":-99999,"text":"8.2 Unitary 空间 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":94,"is_discarded":true,"id":"ba6fb0da-85aa-40b7-b940-e51e3beaa411","page_size":[595,841],"block_position":"94-0"},{"bbox":[56,72,139,86],"type":"title","angle":0,"index":1,"level":2,"text":"## 8.1.5 镜面反射","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"b73c142c-2dff-4fb0-9e68-adca25e35261","page_size":[595,841],"block_position":"94-1"},{"bbox":[73,100,161,112],"type":"title","angle":0,"index":2,"level":2,"text":"## 定义 8.6 (镜面反射)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"0e38f82a-c41c-4541-bf6e-95eb0e4dcfc5","page_size":[595,841],"block_position":"94-2"},{"bbox":[70,116,206,129],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"若对 $\\forall \\alpha \\in V$ ， $\\mathcal{A}:V\\to V$ 满足 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"c73df68b-e497-46f5-a404-a3e7b2f1a971","page_size":[595,841],"block_position":"94-3"},{"bbox":[255,133,339,146],"type":"interline_equation","angle":0,"index":4,"text":"$$\n\\mathcal {A} \\alpha = \\alpha - 2 (\\eta , \\alpha) \\eta\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":94,"id":"558a2c75-753c-465d-ba73-de42fb2b4e8d","page_size":[595,841],"block_position":"94-4"},{"bbox":[69,156,152,169],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"其中 $\\eta$ 是单位向量 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"9d5f8472-f9f9-472d-ae26-383fabf37116","page_size":[595,841],"block_position":"94-5"},{"bbox":[73,189,210,201],"type":"title","angle":0,"index":6,"level":2,"text":"## 引理 8.1 (镜面反射是正交变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"f7511623-7224-497c-b81b-003a66a96d8f","page_size":[595,841],"block_position":"94-6"},{"bbox":[69,206,162,218],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"镜面反射是正交变换 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"6cd86bb7-1ebe-45da-9e4f-64304e8fde72","page_size":[595,841],"block_position":"94-7"},{"bbox":[55,234,77,246],"type":"text","angle":0,"index":8,"merge_prev":false,"text":"证明 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"47f45a29-e369-4b83-8d13-7e0fa0432d3a","page_size":[595,841],"block_position":"94-8"},{"bbox":[74,255,196,268],"type":"text","angle":0,"index":9,"merge_prev":false,"text":"定理 8.11 (镜面反射的特征) ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"a31d7f78-1b3e-47a5-a41f-dc59af62fa1f","page_size":[595,841],"block_position":"94-9"},{"bbox":[56,312,166,327],"type":"title","angle":0,"index":10,"level":2,"text":"## 8.2 Unitary 空间","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"aa1373c6-65cd-437f-a223-c9d72995ba28","page_size":[595,841],"block_position":"94-10"},{"bbox":[56,343,181,359],"type":"title","angle":0,"index":11,"level":2,"text":"## 8.2.1 Unitary 空间概念","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"1d6fe890-8941-40a0-9f42-ea5224ec3e2d","page_size":[595,841],"block_position":"94-11"},{"bbox":[73,371,176,384],"type":"title","angle":0,"index":12,"level":2,"text":"## 定义 8.7 (Unitary 空间)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"74ffe790-ca9f-4501-983a-f6ea81f76bcd","page_size":[595,841],"block_position":"94-12"},{"bbox":[69,389,326,401],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"$V$ 是 $\\mathbb{C}$ 上内积空间，其上定义内积 $(\\alpha, \\beta)$ 满足 $\\forall k_1, k_2 \\in \\mathbb{C}$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"5a396797-8e41-4c54-9da3-690ddd66fc48","page_size":[595,841],"block_position":"94-13"},{"bbox":[83,404,271,417],"type":"interline_equation","angle":0,"index":14,"text":"$$\n\\bullet \\left(k _ {1} \\alpha_ {1} + k _ {2} \\alpha_ {2}, \\beta\\right) = k _ {1} (\\alpha_ {1}, \\beta) + k _ {2} (\\alpha_ {2}, \\beta)\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":94,"id":"aa1ffa13-eb4d-4c96-b94f-8e87466a8b58","page_size":[595,841],"block_position":"94-14"},{"bbox":[85,420,160,433],"type":"interline_equation","angle":0,"index":15,"text":"$$\n\\bullet (\\alpha , \\beta) = \\overline {{(\\beta , \\alpha)}}\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":94,"id":"706ab80a-c8ee-4f80-8f49-a684c99b735e","page_size":[595,841],"block_position":"94-15"},{"bbox":[83,435,268,448],"type":"text","angle":0,"index":16,"merge_prev":false,"text":"$(\\alpha, \\alpha) \\geq 0$ 且 $(\\alpha, \\alpha) = 0$ 当且仅当 $\\alpha = 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"835194dc-949d-448e-a0e2-9f1aeee58d3a","page_size":[595,841],"block_position":"94-16"},{"bbox":[73,469,151,481],"type":"title","angle":0,"index":17,"level":2,"text":"## 定义 8.8 (酉矩阵)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"c5df83f3-7a81-4666-b5f7-0ee0515f63d8","page_size":[595,841],"block_position":"94-17"},{"bbox":[68,487,246,501],"type":"text","angle":0,"index":18,"merge_prev":false,"text":"$\\mathbb{C}$ 上矩阵 $U^{-1} = \\overline{U}^T$ ，则称 $U$ 为酉矩阵。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"4b72cd4d-09fb-41ee-a3aa-085ffde53196","page_size":[595,841],"block_position":"94-18"},{"bbox":[74,522,150,534],"type":"title","angle":0,"index":19,"level":2,"text":"## 定义 8.9 (酉变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"1e5e9c28-e6a7-4d14-a42a-4491ee8b01fe","page_size":[595,841],"block_position":"94-19"},{"bbox":[67,539,264,551],"type":"text","angle":0,"index":20,"merge_prev":false,"text":"$\\mathcal{U}$ 满足 $(\\mathcal{U}\\alpha, \\mathcal{U}\\beta) = (\\alpha, \\beta)$ ，则称 $\\mathcal{U}$ 为酉变换 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"e1c00d9a-04a2-474d-962a-d4a064c3facc","page_size":[595,841],"block_position":"94-20"},{"bbox":[74,572,166,585],"type":"title","angle":0,"index":21,"level":2,"text":"## 定义 8.10 (共轭变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"f89ba037-aa70-4032-bae5-251f6751d3f0","page_size":[595,841],"block_position":"94-21"},{"bbox":[68,589,449,602],"type":"text","angle":0,"index":22,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"bb1a48c0-0a14-4da0-8ffa-a6e0ebf154ce","page_size":[595,841],"block_position":"94-22"},{"bbox":[73,623,166,635],"type":"title","angle":0,"index":23,"level":2,"text":"## 定义 8.11 (正规变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"fe38c800-02ae-4375-beca-ba6835ed58ad","page_size":[595,841],"block_position":"94-23"},{"bbox":[69,640,392,652],"type":"text","angle":0,"index":24,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"a7420a32-069c-4e1c-9c15-f2842efc9a74","page_size":[595,841],"block_position":"94-24"},{"bbox":[73,673,184,686],"type":"title","angle":0,"index":25,"level":2,"text":"## 定义 8.12 (Hermite 变换)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"d2f17ff8-33ea-48d5-b5b2-0e45d2844fda","page_size":[595,841],"block_position":"94-25"},{"bbox":[69,689,355,703],"type":"text","angle":0,"index":26,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":94,"id":"1ef27344-4de0-482d-986c-651387de3416","page_size":[595,841],"block_position":"94-26"},{"bbox":[291,802,303,812],"type":"page_number","angle":0,"index":1000026,"text":"90 ","color":{"line":"rgba(164, 164, 164, 1)","fill":"rgba(164, 164, 164, 1)"},"page_idx":94,"is_discarded":true,"id":"caaa9f6d-55e9-48af-9333-6ce62b63fe21","page_size":[595,841],"block_position":"94-27"}],[{"bbox":[236,74,368,94],"type":"title","angle":0,"index":0,"level":2,"text":"## 第9章 对偶空间","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":95,"id":"e1dd55bc-4364-45b0-9d9b-1a09dfed07f9","page_size":[595,841],"block_position":"95-0"},{"bbox":[55,122,201,139],"type":"title","angle":0,"index":1,"level":2,"text":"## 9.1 对偶空间与对偶基","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":95,"id":"2b842730-84f0-4b37-b205-9fb9b5ecb9f8","page_size":[595,841],"block_position":"95-1"},{"bbox":[72,157,161,169],"type":"title","angle":0,"index":2,"level":2,"text":"## 定义 9.1 (对偶空间)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":95,"id":"da5ece83-bb24-4b35-bb79-745d15d81170","page_size":[595,841],"block_position":"95-2"},{"bbox":[68,173,458,186],"type":"text","angle":0,"index":3,"merge_prev":false,"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 $V^{*}$ 。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":95,"id":"ae830f14-ae16-416c-88e9-6e12f18cab85","page_size":[595,841],"block_position":"95-3"},{"bbox":[73,206,151,220],"type":"title","angle":0,"index":4,"level":2,"text":"## 定义 9.2 (对偶基)","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":95,"id":"160eaf63-d81e-4bbc-b41f-27a2698cecf0","page_size":[595,841],"block_position":"95-4"},{"bbox":[68,224,297,236],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"给定 V 的一组基 $\\epsilon_{1},\\cdots,\\epsilon_{n}$ ，定义线性泛函 $f_{i}$ 满足： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":95,"id":"8dc4dfba-9119-4299-9312-5acc293df33f","page_size":[595,841],"block_position":"95-5"},{"bbox":[251,240,341,277],"type":"interline_equation","angle":0,"index":6,"text":"$$\nf _ {i} (\\epsilon_ {j}) = \\left\\{ \\begin{array}{l l} 1, & i = j \\\\ 0, & i \\neq j \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":95,"id":"1fd5e0e9-f106-4121-9dad-b588d4a4c8dc","page_size":[595,841],"block_position":"95-6"},{"bbox":[69,280,339,294],"type":"text","angle":0,"index":7,"merge_prev":false,"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。 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":95,"id":"75b08c8c-1eda-4581-924c-a2a9cbbd8be5","page_size":[595,841],"block_position":"95-7"},{"bbox":[52,310,539,338],"type":"text","angle":0,"index":8,"merge_prev":false,"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： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":95,"id":"4c9c0a98-daf3-4e49-baa3-1726c1158993","page_size":[595,841],"block_position":"95-8"},{"bbox":[224,336,372,405],"type":"interline_equation","angle":0,"index":9,"text":"$$\n\\left[ \\begin{array}{c} f _ {1} ^ {T} \\\\ f _ {2} ^ {T} \\\\ \\vdots \\\\ f _ {n} ^ {T} \\end{array} \\right] \\left[ \\begin{array}{c c c c} \\epsilon_ {1} & \\epsilon_ {2} & \\dots & \\epsilon_ {n} \\end{array} \\right] = E\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":95,"id":"cc9a0169-002e-4c72-a67c-3e28b0e6a37e","page_size":[595,841],"block_position":"95-9"},{"bbox":[36,418,365,433],"type":"text","angle":0,"index":10,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":95,"id":"efdb4d54-28a0-4362-b202-37f560faf7aa","page_size":[595,841],"block_position":"95-10"},{"bbox":[52,434,458,448],"type":"text","angle":0,"index":11,"merge_prev":false,"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":95,"id":"53627fec-86a6-4214-bd9e-786bbb65d0d9","page_size":[595,841],"block_position":"95-11"},{"bbox":[231,449,363,501],"type":"interline_equation","angle":0,"index":12,"text":"$$\nA P = \\left[ \\begin{array}{l} f _ {1} ^ {T} \\\\ f _ {2} ^ {T} \\\\ f _ {3} ^ {T} \\end{array} \\right] [ \\epsilon_ {1}, \\epsilon_ {2}, \\epsilon_ {3} ] = E\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":95,"id":"12cff6e5-9a74-487b-bc54-94847c8b1009","page_size":[595,841],"block_position":"95-12"},{"bbox":[54,502,267,515],"type":"text","angle":0,"index":13,"merge_prev":false,"text":"因此 $A = P^{-1}$ ，只需求出 $P^{-1}$ 即可。最终求出： ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":95,"id":"cc18abaf-2ff6-48b0-8746-f088f665c041","page_size":[595,841],"block_position":"95-13"},{"bbox":[180,520,401,580],"type":"interline_equation","angle":0,"index":14,"text":"$$\nP ^ {- 1} = \\left[ \\begin{array}{c c c} 1 & 0 & 0 \\\\ - \\frac {1}{2} & \\frac {1}{2} & \\frac {1}{2} \\\\ - \\frac {1}{2} & \\frac {1}{2} & - \\frac {1}{2} \\end{array} \\right] \\Rightarrow \\left\\{ \\begin{array}{l} f _ {1} ^ {T} = (1, 0, 0) \\\\ f _ {2} ^ {T} = (- \\frac {1}{2}, \\frac {1}{2}, \\frac {1}{2}) \\\\ f _ {3} ^ {T} = (- \\frac {1}{2}, \\frac {1}{2}, - \\frac {1}{2}) \\end{array} \\right.\n$$","color":{"line":"rgba(230, 122, 171, 1)","fill":"rgba(230, 122, 171, 1)"},"page_idx":95,"id":"445fa465-dc58-49f4-8fd7-96e3da4e9d51","page_size":[595,841],"block_position":"95-14"}]],"mergeConnections":[{"id":"88385798-7043-478f-8871-fe60f41d2cda","blocks":["10-19","10-20"],"type":"merge"},{"id":"06aa906b-ad1e-4175-ad32-a7deb8996066","blocks":["65-24","66-1"],"type":"merge"}]}