矩阵论复习笔记-华中科技大学

亦凉 2022-12-25 10:59 633阅读 0赞

> 来华科之后又的第一门有挂科风险的考试  
> 也是复习时间时间最长的一门课  
> 昨天考完试 赶紧记下来  
> 再过一天可能就要忘记了  
> 回顾考试涉及到的  
> 填空题 矩阵范数 最小多项式 线性变换的特征值 K积  
> 计算题 奇异值分解 Jordan 向量化算子求解矩阵方程 加号逆  
> 证明题 考了奇异值相关的定义 加号逆的证明

### 目录 ###

*  1 线性空间与线性变换
 *   *  1.1 线性空间
     *  1.2 基与维数
     *  1.3 基变换矩阵
     *  1.4 子空间
     *  1.5 列空间和零空间
     *  1.6 和空间
     *  1.7 直和子空间
     *  1.8 线性变换以及在某一组基的矩阵
     *  1.9 不变子空间
     *  1.10 正交变换与酉变换
 *  2 Jordan标准形
 *   *  2.1 线性变换的特征值
     *  2.2 Jordan 标准形
     *  2.3 Jordan 标准形的求法
     *  2.4 化零多项式
     *  2.5 最小多项式
 *  3 矩阵的分解
 *   *  3.1 LU LDV分解
     *  3.2 LDV分解的充要条件
     *  3.3 满秩分解
     *  3.4 Hermite 标准形求满秩分解
     *  3.5 Schur分解
     *  3.6 正规矩阵
     *  3.7 奇异值
     *  3.8 奇异值分解
 *  4 矩阵的广义逆
 *   *  4.1 左逆与右逆
     *  4.2 MP广义逆
     *  4.2.1 MP广义逆的求法
     *  4.3 投影变换
     *  4.4 投影变换的矩阵是幂等矩阵
     *  4.5 正交投影变换与正交投影矩阵
     *  4.6 最佳最小二乘解
 *  5 矩阵分析
 *   *  5.1 向量范数
     *  5.2 有限维空间范数等价
     *  5.3 矩阵范数
     *  5.4 谱半径
     *  5.5 范数的下界
     *  5.6 矩阵函数
     *  5.7 矩阵函数的求法
     *  5.8 一阶常系数齐次微分方程组
 *  6 Kronecker 积与 Hadamard 积
 *   *  6.1 K积与H积分的定义
     *  6.2 K积的性质
     *  6.3 K积的特征值
     *  6.4 向量化算子

# 1 线性空间与线性变换 #

## 1.1 线性空间 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70]

## 1.2 基与维数 ##

![在这里插入图片描述][20201201172536239.png]

## 1.3 基变换矩阵 ##

![在这里插入图片描述][20201201172920730.png]

## 1.4 子空间 ##

![在这里插入图片描述][20201201172853701.png]

## 1.5 列空间和零空间 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 1]

## 1.6 和空间 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 2]

## 1.7 直和子空间 ##

![在这里插入图片描述][2020120117341223.png]

## 1.8 线性变换以及在某一组基的矩阵 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 3]

## 1.9 不变子空间 ##

![在这里插入图片描述][20201201173641255.png]  
**不变子空间求法 在某一组基下 线性变换的矩阵A是准对角形，或者求解A的Jordan标准形可以求得不变子空间**

## 1.10 正交变换与酉变换 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 4]

# 2 Jordan标准形 #

## 2.1 线性变换的特征值 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 5]  
**线性变换的特征值就是在某一组基下矩阵的特征值，但是特征向量不是矩阵的特征向量，需要与基做乘积进行还原**  
![在这里插入图片描述][20201201174332323.png]

## 2.2 Jordan 标准形 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 6]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 7]

## 2.3 Jordan 标准形的求法 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 8]

## 2.4 化零多项式 ##

![在这里插入图片描述][20201201174843731.png]  
矩阵的特征多项式就是一个化零多项式

## 2.5 最小多项式 ##

**特征多项式中特征值a对应的因子的次数为a对应的所有Jordan快的最高阶数 例如**  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 9]

# 3 矩阵的分解 #

## 3.1 LU LDV分解 ##

![在这里插入图片描述][20201201185803154.png]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 10]

## 3.2 LDV分解的充要条件 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 11]

## 3.3 满秩分解 ##

![在这里插入图片描述][20201201190008943.png]

## 3.4 Hermite 标准形求满秩分解 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 12]

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 13]

## 3.5 Schur分解 ##

![在这里插入图片描述][20201201190235473.png]

## 3.6 正规矩阵 ##

![在这里插入图片描述][2020120119033676.png]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 14]

## 3.7 奇异值 ##

![在这里插入图片描述][20201201190434503.png]

## 3.8 奇异值分解 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 15]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 16]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 17]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 18]

# 4 矩阵的广义逆 #

## 4.1 左逆与右逆 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 19]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 20]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 21]

## 4.2 MP广义逆 ##

**提出者Moore-Penrose：2020年诺贝尔物理学奖得主**  
**论文**![在这里插入图片描述][20201201191622151.png]  
[A generalized inverse for matrices][]

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 22]

## 4.2.1 MP广义逆的求法 ##

![在这里插入图片描述][20201201191012590.png]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 23]

## 4.3 投影变换 ##

![在这里插入图片描述][2020120119110346.png]

## 4.4 投影变换的矩阵是幂等矩阵 ##

![在这里插入图片描述][20201201191752883.png]

## 4.5 正交投影变换与正交投影矩阵 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 24]

## 4.6 最佳最小二乘解 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 25]

# 5 矩阵分析 #

## 5.1 向量范数 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 26]

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 27]

## 5.2 有限维空间范数等价 ##

![在这里插入图片描述][20201201192147659.png]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 28]

## 5.3 矩阵范数 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 29]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 30]

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 31]

## 5.4 谱半径 ##

![在这里插入图片描述][20201201192306775.png]

## 5.5 范数的下界 ##

![在这里插入图片描述][20201201192334805.png]

## 5.6 矩阵函数 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 32]

## 5.7 矩阵函数的求法 ##

**求法有两个 一个是Jordan标准形**  
**一个是最小多项式**  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 33]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 34]

## 5.8 一阶常系数齐次微分方程组 ##

![在这里插入图片描述][2020120119275441.png]

# 6 Kronecker 积与 Hadamard 积 #

## 6.1 K积与H积分的定义 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 35]  
![在这里插入图片描述][20201201193038362.png]

## 6.2 K积的性质 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 36]  
![在这里插入图片描述][20201201193107514.png]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 37]

## 6.3 K积的特征值 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 38]

## 6.4 向量化算子 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0wxNTU4MTk4NzI3_size_16_color_FFFFFF_t_70 39]  
![在这里插入图片描述][20201201193222216.png]

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[20201201172536239.png]: /images/20221120/cf78e991edf5472cae2595850dffa5a2.png
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[20201201174843731.png]: /images/20221120/6e6a31754e5e4ffabf00888e3fe5adba.png
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[A generalized inverse for matrices]: https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/generalized-inverse-for-matrices/5F4516D6B9989BB6563A4B267CC7D615?__cf_chl_jschl_tk__=23f9744feca8bb5669a43b5deac02293b93b7368-1606821314-0-AS-pvBAPaeB3AoI37zNtGbukTXPkhZxvzkAL2QtJSU0CnCjUeNOVUqq6UgNeP0UQOMcb9ukH-063FRDZ79jWMlQBxXUsLHKJeIX1Nnnwjek7zw7rRJMLfhI0XAhVxezpQ3wG5rvHHrzzRBwZpdFUteb1-P0WMOw7xauKJyOtEVpSmZN56SsV_ENEAfFzCaEZjoKc0F69y0AfJ56wRXMfH7cnBoPUxSKtY46furk6FW4EgWPAYKmQtULTUIVY_xOiHbwqgi91MBdyuFD1gauA2WYQF5rlloVMRzZ8YGBCdiBuF-bhaGClrpnqMkaP8GYq3kDwMuD_44jTq3ZdZFo42_A7lqTNQZ0Scgbkp7TkIUkntPY0efLOoZY5jttIN8dZBHsvEAXxfAiSpwyZXeYnBjUU0smvZ0-P_Pz3fDl_sLKWsvT7a8IFSQ-50P9ZzUlttgVsRty1CxrmQIrReQSvLCNUsJgdsQ2ql1jb_TzFrYg2ttETj8Ewp3ASRdjaOv8g2mC20ZxbPXyr870ozUpMRqh2Ly36v60DO768r17Pa0SCcQLZAOrQvIN2idtIN53NzA
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[20201201193222216.png]: https://img-blog.csdnimg.cn/20201201193222216.png