线性代数：线性代数（Linear Algebra）综述

快来打我* 2023-10-03 16:08 56阅读 0赞

#### 文章目录 ####

*   *  矩阵 / Matrix
     *   *  元素
         *  运算
         *   *  加法  A + B A+B A\+B
             *  数量乘法  c A cA cA
             *  与向量之间的运算
             *   *  乘法  A b A\\mathbf\{b\} Ab
             *  与矩阵之间的运算
             *   *  矩阵乘法
             *  乘方
         *  性质
         *  方阵 / Square Matrix
         *  零矩阵
         *  对角矩阵 / Diagonal Matrix
         *  单位矩阵 / Identity Matrix
         *  转置 / Transpose
         *  逆矩阵 / Inverse Matrix
         *  三角矩阵
         *   *  上三角矩阵
             *  下三角矩阵
         *  正交矩阵 / Orthogonal Matrix
     *  置换矩阵 / Permutation Matrix
     *  向量
     *   *  零向量
         *  加法
         *  数量乘法
         *  性质
         *  基底
         *  维数
         *  内积 / Inner Product
         *  外积 / Outter Product
     *  范数 / norm
     *  分块矩阵
     *   *  定义
         *  加法
         *  数量乘法
         *  乘法
         *  分块矩阵与向量
         *   *  列向量
             *  行向量
             *  矩阵乘法
             *  分块对角矩阵
     *  行列式
     *  向量空间 / Vector Space
     *   *  零向量空间 / Null Space

### 矩阵 / Matrix ###

一个给定的矩阵  A A A，确定了从向量到另一个向量的映射。

对于行数与列数都相等的矩阵  A A A、 B B B，如果  A x = B x A\\boldsymbol\{x\} = B\\boldsymbol\{x\} Ax=Bx 对于任意的向量  x \\boldsymbol\{x\} x 都成立，则  A = B A = B A=B

#### 元素 ####

矩阵  A A A 中，第  i i i 行第  j j j 列的值，称为  A A A 的  **( i , j ) (i, j) (i,j) 元素**，或  **( i , j ) (i, j) (i,j) 元**。

一般用小写字母表示，如  a a a 或  a i , j a\_\{i, j\} ai,j

#### 运算 ####

##### 加法  A + B A+B A\+B #####

对应位置元素相加  
 \[ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋮ a m 1 a m 2 ⋯ a m n \] + \[ b 11 b 12 ⋯ b 1 n b 21 b 22 ⋯ b 2 n ⋮ ⋮ ⋮ b m 1 b m 2 ⋯ b m n \] = \[ a 11 + b 11 a 12 + b 12 ⋯ a 1 n + b 1 n a 21 + b 21 a 22 + b 22 ⋯ a 2 n + b 2 n ⋮ ⋮ ⋮ a m 1 + b m 1 a m 2 + b m 2 ⋯ a m n + b m n \] \\begin\{array\}\{l\} \\left\[\\begin\{array\}\{cccc\} a\_\{11\} & a\_\{12\} & \\cdots & a\_\{1 n\} \\\\ a\_\{21\} & a\_\{22\} & \\cdots & a\_\{2 n\} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a\_\{m 1\} & a\_\{m 2\} & \\cdots & a\_\{m n\} \\end\{array\}\\right\] + \\left\[\\begin\{array\}\{cccc\} b\_\{11\} & b\_\{12\} & \\cdots & b\_\{1 n\} \\\\ b\_\{21\} & b\_\{22\} & \\cdots & b\_\{2 n\} \\\\ \\vdots & \\vdots & & \\vdots \\\\ b\_\{m 1\} & b\_\{m 2\} & \\cdots & b\_\{m n\} \\end\{array\}\\right\] = \\left\[\\begin\{array\}\{cccc\} a\_\{11\} + b\_\{11\} & a\_\{12\} + b\_\{12\} & \\cdots & a\_\{1 n\} + b\_\{1 n\} \\\\ a\_\{21\} + b\_\{21\} & a\_\{22\} + b\_\{22\} & \\cdots & a\_\{2 n\} + b\_\{2 n\} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a\_\{m 1\} + b\_\{m 1\} & a\_\{m 2\} + b\_\{m 2\} & \\cdots & a\_\{m n\} + b\_\{m n\} \\end\{array\}\\right\] \\end\{array\} ⎣⎢⎢⎢⎡a11a21⋮am1a12a22⋮am2⋯⋯⋯a1na2n⋮amn⎦⎥⎥⎥⎤\+⎣⎢⎢⎢⎡b11b21⋮bm1b12b22⋮bm2⋯⋯⋯b1nb2n⋮bmn⎦⎥⎥⎥⎤=⎣⎢⎢⎢⎡a11\+b11a21\+b21⋮am1\+bm1a12\+b12a22\+b22⋮am2\+bm2⋯⋯⋯a1n\+b1na2n\+b2n⋮amn\+bmn⎦⎥⎥⎥⎤

##### 数量乘法  c A cA cA #####

λ \[ a 11 ⋯ a 1 n ⋮ ⋮ a m 1 ⋯ a m n \] = \[ λ a 11 ⋯ λ a 1 n ⋮ ⋮ λ a m 1 ⋯ λ a m n \] \\lambda\\begin\{array\}\{l\} \\left\[\\begin\{array\}\{cccc\} a\_\{11\} & \\cdots & a\_\{1 n\} \\\\ \\vdots & & \\vdots \\\\ a\_\{m 1\} & \\cdots & a\_\{m n\} \\end\{array\}\\right\] = \\left\[\\begin\{array\}\{cccc\} \\lambda a\_\{11\} & \\cdots & \\lambda a\_\{1 n\} \\\\ \\vdots & & \\vdots \\\\ \\lambda a\_\{m 1\} & \\cdots & \\lambda a\_\{m n\} \\end\{array\}\\right\] \\end\{array\} λ⎣⎢⎡a11⋮am1⋯⋯a1n⋮amn⎦⎥⎤=⎣⎢⎡λa11⋮λam1⋯⋯λa1n⋮λamn⎦⎥⎤

− A = ( − 1 ) A A − B = A + ( − B ) -A = (-1)A \\\\ A - B = A + (-B) −A=(−1)AA−B=A\+(−B)

##### 与向量之间的运算 #####

###### 乘法  A b A\\mathbf\{b\} Ab ######

\[ a 11 ⋯ a 1 n ⋮ ⋮ a m 1 ⋯ a m n \] \[ x 1 ⋮ x n \] = \[ a 11 x 1 + ⋯ + a 1 n x n ⋮ a m 1 x 1 + ⋯ + a m n x n \] \\begin\{array\}\{l\} \\left\[\\begin\{array\}\{ccc\} a\_\{11\} & \\cdots & a\_\{1 n\} \\\\ \\vdots & \\quad & \\vdots \\\\ a\_\{m 1\} & \\cdots & a\_\{m n\} \\end\{array\}\\right\] \\left\[\\begin\{array\}\{c\} x\_\{1\} \\\\ \\vdots \\\\ x\_\{n\} \\end\{array\}\\right\] = \\left\[\\begin\{array\}\{ccc\} a\_\{11\}x\_\{1\} + \\cdots + a\_\{1n\}x\_\{n\} \\\\ \\vdots \\\\ a\_\{m1\}x\_\{1\} + \\cdots + a\_\{mn\}x\_\{n\} \\end\{array\}\\right\] \\end\{array\} ⎣⎢⎡a11⋮am1⋯⋯a1n⋮amn⎦⎥⎤⎣⎢⎡x1⋮xn⎦⎥⎤=⎣⎢⎡a11x1\+⋯\+a1nxn⋮am1x1\+⋯\+amnxn⎦⎥⎤

矩阵与向量的乘积是向量

若有  x + y = z \\boldsymbol\{x\} + \\boldsymbol\{y\} = \\boldsymbol\{z\} x\+y=z，则有  
 A x + A y = A z A\\boldsymbol\{x\} + A\\boldsymbol\{y\} = A\\boldsymbol\{z\} Ax\+Ay=Az  
若有  c x = y c\\boldsymbol\{x\} = \\boldsymbol\{y\} cx=y，则有  
 c ( A x ) = A y c(A\\boldsymbol\{x\}) = A\\boldsymbol\{y\} c(Ax)=Ay

##### 与矩阵之间的运算 #####

###### 矩阵乘法 ######

\[ b 11 ⋯ b 1 m ⋮ ⋮ b k 1 ⋯ b k m \] k × m \[ a 11 ⋯ a 1 n ⋮ ⋮ a m 1 ⋯ a m n \] m × n = \[ ( b 11 a 11 + ⋯ + b 1 m a m 1 ) ⋯ ( b 11 a 11 + ⋯ + b 1 m a m n ) ⋮ ⋮ ( b k 1 a 11 + ⋯ + b k m a m 1 ) ⋯ ( b k 1 a 1 n + ⋯ + b k m a m n ) \] k × n \\left\[\\begin\{array\}\{ccc\} b\_\{11\} & \\cdots & b\_\{1 m\} \\\\ \\vdots & \\quad & \\vdots \\\\ b\_\{k 1\} & \\cdots & b\_\{k m\} \\end\{array\}\\right\]\_\{k \\times m\} \\left\[\\begin\{array\}\{ccc\} a\_\{11\} & \\cdots & a\_\{1 n\} \\\\ \\vdots & \\quad & \\vdots \\\\ a\_\{m 1\} & \\cdots & a\_\{m n\} \\end\{array\}\\right\]\_\{m \\times n\} = \\left\[\\begin\{array\}\{ccc\} \\left(b\_\{11\}a\_\{11\} + \\cdots + b\_\{1m\}a\_\{m1\}\\right) & \\cdots & \\left(b\_\{11\}a\_\{11\} + \\cdots + b\_\{1m\}a\_\{mn\}\\right) \\\\ \\vdots & \\quad & \\vdots \\\\ \\left(b\_\{k1\}a\_\{11\} + \\cdots + b\_\{km\}a\_\{m1\}\\right) & \\cdots & \\left(b\_\{k1\}a\_\{1n\} + \\cdots + b\_\{km\}a\_\{mn\}\\right) \\end\{array\}\\right\]\_\{k \\times n\} ⎣⎢⎡b11⋮bk1⋯⋯b1m⋮bkm⎦⎥⎤k×m⎣⎢⎡a11⋮am1⋯⋯a1n⋮amn⎦⎥⎤m×n=⎣⎢⎡(b11a11\+⋯\+b1mam1)⋮(bk1a11\+⋯\+bkmam1)⋯⋯(b11a11\+⋯\+b1mamn)⋮(bk1a1n\+⋯\+bkmamn)⎦⎥⎤k×n

将矩阵  A m × n A\_\{m \\times n\} Am×n 的各个元素记作 $ a\_\{ij\} $，矩阵  B n × p B\_\{n \\times p\} Bn×p 的各个元素记作  b i j b\_\{ij\} bij，则其二者相乘得到的矩阵记作  C m × p C\_\{m \\times p\} Cm×p，其各个元素记作  c i j c\_\{ij\} cij，则有  
 c i j = ∑ k = 1 n a i k b k j c\_\{i j\}=\\sum\_\{k=1\}^\{n\} a\_\{ik\} b\_\{k j\} cij=k=1∑naikbkj

\[ a 11 a 12 a 21 a 22 \] \[ b 11 b 12 b 21 b 22 \] = \[ a 11 b 11 + a 12 b 21 a 11 b 12 + a 12 b 22 a 21 b 11 + a 22 b 21 a 21 b 12 + a 22 b 22 \] \\left\[\\begin\{array\}\{ll\} a\_\{11\} & a\_\{12\} \\\\ a\_\{21\} & a\_\{22\} \\end\{array\}\\right\]\\left\[\\begin\{array\}\{ll\} b\_\{11\} & b\_\{12\} \\\\ b\_\{21\} & b\_\{22\} \\end\{array\}\\right\]=\\left\[\\begin\{array\}\{ll\} a\_\{11\} b\_\{11\}+a\_\{12\} b\_\{21\} & a\_\{11\} b\_\{12\}+a\_\{12\} b\_\{22\} \\\\ a\_\{21\} b\_\{11\}+a\_\{22\} b\_\{21\} & a\_\{21\} b\_\{12\}+a\_\{22\} b\_\{22\} \\end\{array\}\\right\] \[a11a21a12a22\]\[b11b21b12b22\]=\[a11b11\+a12b21a21b11\+a22b21a11b12\+a12b22a21b12\+a22b22\]

##### 乘方 #####

A A = A 2 , A A A = A 3 , ⋯ AA = A^2, \\quad AAA = A^3, \\quad \\cdots AA=A2,AAA=A3,⋯

乘方的优先级高于加、减和乘法运算  
 5 A 2 = 5 ( A 2 ) 5A^2 = 5(A^2) 5A2=5(A2)

A B 2 − C 3 = A ( B 2 ) − C 3 AB^2 - C^3 = A(B^2) - C^3 AB2−C3=A(B2)−C3

#### 性质 ####

A α A β = A α + β A^\{\\alpha\}A^\{\\beta\} = A^\{\\alpha + \\beta\} AαAβ=Aα\+β

( A α ) β = A α β (A^\{\\alpha\})^\{\\beta\} = A^\{ \{\\alpha\}\{\\beta\}\} (Aα)β=Aαβ

( c A ) x = c ( A x ) = A ( c x ) (cA)\\boldsymbol\{x\} = c(A\\boldsymbol\{x\}) = A(c\\boldsymbol\{x\}) (cA)x=c(Ax)=A(cx)

( A + B ) x = A x + B x (A + B)\\boldsymbol\{x\} = A\\boldsymbol\{x\} + B\\boldsymbol\{x\} (A\+B)x=Ax\+Bx

A + B = B + A A + B = B + A A\+B=B\+A

( A + B ) + C = A + ( B + C ) (A + B) + C = A + (B + C) (A\+B)\+C=A\+(B\+C)

( a + b ) A = a A + b A (a+b)A = aA + bA (a\+b)A=aA\+bA

a b A = a ( b A ) abA = a(bA) abA=a(bA)

A ( B + C ) = A B + A C A\\left(B + C\\right) = AB + AC A(B\+C)=AB\+AC

( A + B ) C = A C + B C (A+B)C = AC + BC (A\+B)C=AC\+BC

( c A ) B = c ( A B ) = A ( c B ) (cA)B = c(AB) = A(cB) (cA)B=c(AB)=A(cB)

( A + B ) 2 = A 2 + A B + B A + B 2 (A+B)^2 = A^2 + AB + BA + B^2 (A\+B)2=A2\+AB\+BA\+B2

( A + B ) ( A − B ) = A 2 − A B + B A − B 2 (A+B)(A-B) = A^2 - AB + BA - B^2 (A\+B)(A−B)=A2−AB\+BA−B2

( A B ) 2 = A B A B (AB)^2 = ABAB (AB)2=ABAB

A 0 = I A^0 = I A0=I

#### 方阵 / Square Matrix ####

行数与列数相同的矩阵，称为**方阵**。

n n n 行  n n n 列的方阵可称作  **n n n 阶方阵**。

#### 零矩阵 ####

所有元素都是  0 0 0 的矩阵，称为零矩阵，记作  O O O

m m m 行  n n n 列的零矩阵记作  O m , n O\_\{m, n\} Om,n

对于任意向量  x \\boldsymbol\{x\} x，都有  O x = o O\\boldsymbol\{x\} = \\boldsymbol\{o\} Ox=o

零矩阵表示的是将所有的点都映到零点的映射。  
KaTeX parse error: No such environment: align at position 8: \\begin\{̲a̲l̲i̲g̲n̲\}̲ A + O &= O + A…

#### 对角矩阵 / Diagonal Matrix ####

主对角线上元素称作**对角元素**。其余元素称作**非对角度元素**。

对角矩阵可省略非对角元素，并记作  d i a g ( a 1 , a 2 , a 3 , ⋯   ) diag(a1, a2, a3, \\cdots) diag(a1,a2,a3,⋯)，其中  d i a g diag diag 是  d i a g o n a l diagonal diagonal 的缩写。  
 ( a 1 a 2 ⋱ a n ) = diag ⁡ ( a 1 , a 2 , … , a n ) \\left(\\begin\{array\}\{cccc\} a\_\{1\} & & & \\\\ & a\_\{2\} & & \\\\ & & \\ddots & \\\\ & & & a\_\{n\} \\end\{array\}\\right) = \\operatorname\{diag\}(a\_1, a\_2, \\dots, a\_n) ⎝⎜⎜⎛a1a2⋱an⎠⎟⎟⎞=diag(a1,a2,…,an)  
对角矩阵乘法  
 ( a 1 a 2 ⋱ a n ) ( b 1 b 2 ⋱ b n ) = ( a 1 b 1 a 2 b 2 ⋱ a n b n ) \\left(\\begin\{array\}\{cccc\} a\_\{1\} & & & \\\\ & a\_\{2\} & & \\\\ & & \\ddots & \\\\ & & & a\_\{n\} \\end\{array\}\\right) \\left(\\begin\{array\}\{cccc\} b\_\{1\} & & & \\\\ & b\_\{2\} & & \\\\ & & \\ddots & \\\\ & & & b\_\{n\} \\end\{array\}\\right) = \\left(\\begin\{array\}\{cccc\} a\_\{1\}b\_\{1\} & & & \\\\ & a\_\{2\}b\_\{2\} & & \\\\ & & \\ddots & \\\\ & & & a\_\{n\}b\_\{n\} \\end\{array\}\\right) ⎝⎜⎜⎛a1a2⋱an⎠⎟⎟⎞⎝⎜⎜⎛b1b2⋱bn⎠⎟⎟⎞=⎝⎜⎜⎛a1b1a2b2⋱anbn⎠⎟⎟⎞

( a 1 a 2 ⋱ a n ) k = ( a 1 k a 2 k ⋱ a n k ) \\left(\\begin\{array\}\{cccc\} a\_\{1\} & & & \\\\ & a\_\{2\} & & \\\\ & & \\ddots & \\\\ & & & a\_\{n\} \\end\{array\}\\right)^k = \\left(\\begin\{array\}\{cccc\} a\_\{1\}^k & & & \\\\ & a\_\{2\}^k & & \\\\ & & \\ddots & \\\\ & & & a\_\{n\}^k \\end\{array\}\\right) ⎝⎜⎜⎛a1a2⋱an⎠⎟⎟⎞k=⎝⎜⎜⎛a1ka2k⋱ank⎠⎟⎟⎞

#### 单位矩阵 / Identity Matrix ####

方阵中，如果除了从左到右的主对角线方上的元素是  1 1 1 以外，其余元素都是  0 0 0，则该矩阵称作**单位矩阵**。  
 A I = I A = A AI = IA = A AI=IA=A  
单位矩阵是对角度矩阵的特殊形式，可记作： I = diag ⁡ ( 1 , ⋯   , 1 ) I = \\operatorname\{diag\}(1, \\cdots, 1) I=diag(1,⋯,1)

对角表示的映射是：沿着坐标轴伸缩。其中对角元素的值就是各轴伸缩的倍率。

#### 转置 / Transpose ####

对于矩阵  A A A ，其转置矩阵记作  A T A^T AT。其中，字母  T T T 是英文 $ \\text\{Transpose\} $ 的首字母。

记矩阵  A A A 为  
 A = \[ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋮ a m 1 a m 2 ⋯ a m n \] A=\\left\[\\begin\{array\}\{cccc\} a\_\{11\} & a\_\{12\} & \\cdots & a\_\{1 n\} \\\\ a\_\{21\} & a\_\{22\} & \\cdots & a\_\{2 n\} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a\_\{m 1\} & a\_\{m 2\} & \\cdots & a\_\{m n\} \\end\{array\}\\right\] A=⎣⎢⎢⎢⎡a11a21⋮am1a12a22⋮am2⋯⋯⋯a1na2n⋮amn⎦⎥⎥⎥⎤  
则其转置矩阵为  
 A T = \[ a 11 a 21 ⋯ a m 1 a 12 a 22 ⋯ a m 2 ⋮ ⋮ ⋮ a 1 n a 2 n ⋯ a m n \] A^\{T\} =\\left\[\\begin\{array\}\{cccc\} a\_\{11\} & a\_\{21\} & \\cdots & a\_\{m1\} \\\\ a\_\{12\} & a\_\{22\} & \\cdots & a\_\{m2\} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a\_\{1n\} & a\_\{2n\} & \\cdots & a\_\{m n\} \\end\{array\}\\right\] AT=⎣⎢⎢⎢⎡a11a12⋮a1na21a22⋮a2n⋯⋯⋯am1am2⋮amn⎦⎥⎥⎥⎤  
例  
 A = \[ − 5 0 − 5 0 − 1 − 1 3 6 9 \] , A T = \[ − 5 0 3 0 − 1 6 − 5 − 1 9 \] A = \\left\[\\begin\{array\}\{ccc\} -5 & 0 & -5 \\\\ 0 & -1 & -1 \\\\ 3 & 6 & 9 \\end\{array\}\\right\], \\quad A^T = \\left\[\\begin\{array\}\{ccc\} -5 & 0 & 3 \\\\ 0 & -1 & 6 \\\\ -5 & -1 & 9 \\end\{array\}\\right\] A=⎣⎡−5030−16−5−19⎦⎤,AT=⎣⎡−50−50−1−1369⎦⎤  
对于任一矩阵，恒有  
 ( A T ) T = A (A^T)^T = A (AT)T=A  
对于对角矩阵  D D D，恒有  
 D T = D D^T = D DT=D  
若  A B AB AB 有定义，则有  
 ( A B ) T = B T A T (AB)^T = B^T A^T (AB)T=BTAT  
同理，可推广至  
 ( A B C D ) T = D T C T B T A T (ABCD)^T = D^T C^T B^T A^T (ABCD)T=DTCTBTAT  
对于方阵  A A A，若其逆矩阵  A − 1 A^\{-1\} A−1 存在，则有  
 ( A − 1 ) T = ( A T ) − 1 (A^\{-1\})^\{T\} = (A^\{T\})^\{-1\} (A−1)T=(AT)−1

#### 逆矩阵 / Inverse Matrix ####

设  A A A 是一个  n n n 阶方阵，若存在另一个  n n n 阶方阵  B B B，使得  A B = B A = E AB=BA=E AB=BA=E ，则称方阵  A A A 可逆，并称方阵  B B B 是  A A A 的**逆矩阵**。

该逆矩阵可记作  A − 1 A^\{-1\} A−1，并且它是唯一的。

对于一个给定的矩阵，其逆矩阵并不一定总是存在。  
 ( A − 1 ) − 1 = A (A^\{-1\})^\{-1\} = A (A−1)−1=A

( A B ) − 1 = B − 1 A − 1 ( A B C D ) − 1 = D − 1 C − 1 B − 1 A − 1 (AB)^\{-1\} = B^\{-1\}A^\{-1\} \\\\ (ABCD)^\{-1\} = D^\{-1\}C^\{-1\}B^\{-1\}A^\{-1\} (AB)−1=B−1A−1(ABCD)−1=D−1C−1B−1A−1

( A T ) − 1 = ( A − 1 ) T \\left(A^T\\right)^\{-1\} = \\left(A^\{-1\}\\right)^T (AT)−1=(A−1)T

( A k ) − 1 = ( A − 1 ) k (A^\{k\})^\{-1\} = (A^\{-1\})^\{k\} (Ak)−1=(A−1)k

diag ⁡ ( a 1 , ⋯   , a n ) − 1 = diag ⁡ ( a 1 − 1 , ⋯   , a n − 1 ) \\operatorname\{diag\}\\left(a\_\{1\}, \\cdots, a\_\{n\}\\right)^\{-1\} = \\operatorname\{diag\}\\left(a\_\{1\}^\{-1\}, \\cdots, a\_\{n\}^\{-1\}\\right) diag(a1,⋯,an)−1=diag(a1−1,⋯,an−1)

若二阶方阵  A A A 可逆，则其逆矩阵为

A − 1 = 1 a d − b c \[ d − b − c a \] A^\{-1\} = \\frac\{1\}\{a d-b c\}\\left\[\\begin\{array\}\{cc\} d & -b \\\\ -c & a \\end\{array\}\\right\] A−1=ad−bc1\[d−c−ba\]

#### 三角矩阵 ####

##### 上三角矩阵 #####

A = \[ a 1 , 1 a 1 , 2 a 1 , 3 … a 1 , n a 2 , 2 a 2 , 3 … a 2 , n ⋱ ⋱ ⋮ ⋱ a n − 1 , n a n , n \] A = \\left\[\\begin\{array\}\{ccccc\} a\_\{1,1\} & a\_\{1,2\} & a\_\{1,3\} & \\ldots & a\_\{1, n\} \\\\ & a\_\{2,2\} & a\_\{2,3\} & \\ldots & a\_\{2, n\} \\\\ & & \\ddots & \\ddots & \\vdots \\\\ & & & \\ddots & a\_\{n-1, n\} \\\\ & & & & a\_\{n, n\} \\end\{array\}\\right\] A=⎣⎢⎢⎢⎢⎢⎡a1,1a1,2a2,2a1,3a2,3⋱……⋱⋱a1,na2,n⋮an−1,nan,n⎦⎥⎥⎥⎥⎥⎤

例  
 A = \[ 11 22 33 0 44 55 0 0 66 \] A = \\left\[\\begin\{array\}\{ccc\} 11 & 22 & 33 \\\\ 0 & 44 & 55 \\\\ 0 & 0 & 66 \\end\{array\}\\right\] A=⎣⎡110022440335566⎦⎤

##### 下三角矩阵 #####

A = \[ a 1 , 1 a 2 , 1 a 2 , 2 a 3 , 1 a 3 , 2 a 3 , 2 ⋮ ⋮ ⋱ ⋱ a n , 1 a n , 2 … a n , n − 1 a n , n \] A = \\left\[\\begin\{array\}\{ccccc\} a\_\{1,1\} \\\\ a\_\{2,1\} & a\_\{2,2\}\\\\ a\_\{3,1\} & a\_\{3,2\} & a\_\{3,2\}\\\\ \\vdots & \\vdots & \\ddots & \\ddots & \\\\ a\_\{n, 1\} & a\_\{n, 2\} & \\ldots & a\_\{n, n-1\} & a\_\{n, n\} \\end\{array\}\\right\] A=⎣⎢⎢⎢⎢⎢⎡a1,1a2,1a3,1⋮an,1a2,2a3,2⋮an,2a3,2⋱…⋱an,n−1an,n⎦⎥⎥⎥⎥⎥⎤

例  
 A = \[ 11 0 0 22 33 0 44 55 66 \] A = \\left\[\\begin\{array\}\{ccc\} 11 & 0 & 0 \\\\ 22 & 33 & 0 \\\\ 44 & 55 & 66 \\end\{array\}\\right\] A=⎣⎡112244033550066⎦⎤  
同类型的三角矩阵的相乘，得到的依然是该类型的三角矩阵。

#### 正交矩阵 / Orthogonal Matrix ####

若  A A T = E A A^T = E AAT=E 或 A T A = E A^T A = E ATA=E，则称矩阵  A A A 为正交矩阵，即  A − 1 = A T A^\{-1\} = A^T A−1=AT

### 置换矩阵 / Permutation Matrix ###

置换矩阵是一个方形二进制矩阵，它在每行和每列中只有一个  1 1 1，而在其他地方则为  0 0 0。

用一个置换矩阵**左乘**任意矩阵，则等价于对应的行交换。例如：  
 ( 0 1 1 0 ) ( a b c d ) = ( c d a b ) \\left(\\begin\{array\}\{ll\} 0 & 1 \\\\ 1 & 0 \\end\{array\}\\right)\\left(\\begin\{array\}\{ll\} a & b \\\\ c & d \\end\{array\}\\right)=\\left(\\begin\{array\}\{ll\} c & d \\\\ a & b \\end\{array\}\\right) (0110)(acbd)=(cadb)  
用一个置换矩阵**右乘**任意矩阵，则等价于对应的列交换。例如：  
 ( a b c d ) ( 0 1 1 0 ) = ( b a d c ) \\left(\\begin\{array\}\{ll\} a & b \\\\ c & d \\end\{array\}\\right)\\left(\\begin\{array\}\{ll\} 0 & 1 \\\\ 1 & 0 \\end\{array\}\\right)=\\left(\\begin\{array\}\{ll\} b & a \\\\ d & c \\end\{array\}\\right) (acbd)(0110)=(bdac)  
以上规则可记为**左行右列**。

### 向量 ###

n n n 维**列向量**可看作是  n × 1 n \\times 1 n×1 矩阵； n n n 维**行向量**可看作是  1 × n 1 \\times n 1×n 矩阵。

*  一列数
    
    此时可将向量写作： x \\boldsymbol\{x\} x、 v \\boldsymbol\{v\} v、 e \\boldsymbol\{e\} e
 *  位置向量
 *  有向线段
    
    此时可将向量写作： x ⃗ \\vec\{x\} x、 v ⃗ \\vec\{v\} v、 e ⃗ \\vec\{e\} e

#### 零向量 ####

o = \[ 0 ⋮ 0 \] \\boldsymbol\{o\} =\\begin\{array\}\{l\} \\left\[\\begin\{array\}\{c\} 0 \\\\ \\vdots \\\\ 0 \\end\{array\}\\right\] \\end\{array\} o=⎣⎢⎡0⋮0⎦⎥⎤

#### 加法 ####

如果将向量视为有向线段，则向量加法可理解为线段的连接。

\[ x 1 ⋮ x n \] + \[ y 1 ⋮ y n \] = \[ x 1 + y 1 ⋮ x n + y n \] \\begin\{array\}\{l\} \\left\[\\begin\{array\}\{c\} x\_\{1\} \\\\ \\vdots \\\\ x\_\{n\} \\end\{array\}\\right\] + \\left\[\\begin\{array\}\{c\} y\_\{1\} \\\\ \\vdots \\\\ y\_\{n\} \\end\{array\}\\right\] = \\left\[\\begin\{array\}\{cccc\} x\_\{1\} + y\_\{1\} \\\\ \\vdots \\\\ x\_\{n\} + y\_\{n\} \\end\{array\}\\right\] \\end\{array\} ⎣⎢⎡x1⋮xn⎦⎥⎤\+⎣⎢⎡y1⋮yn⎦⎥⎤=⎣⎢⎡x1\+y1⋮xn\+yn⎦⎥⎤

#### 数量乘法 ####

如果将向量视为有向线段，则向量加法可理解为线段的伸缩。  
 λ \[ x 1 ⋮ x n \] = \[ λ x 1 ⋮ λ x n \] \\lambda\\begin\{array\}\{l\} \\left\[\\begin\{array\}\{c\} x\_\{1\} \\\\ \\vdots \\\\ x\_\{n\} \\end\{array\}\\right\] =\\left\[\\begin\{array\}\{c\} \\lambda x\_\{1\} \\\\ \\vdots \\\\ \\lambda x\_\{n\} \\end\{array\}\\right\] \\end\{array\} λ⎣⎢⎡x1⋮xn⎦⎥⎤=⎣⎢⎡λx1⋮λxn⎦⎥⎤

#### 性质 ####

a b x = a ( b x ) ab\\boldsymbol\{x\} = a(b\\boldsymbol\{x\}) abx=a(bx)

1 x = x 1\\boldsymbol\{x\} = \\boldsymbol\{x\} 1x=x

x + y = x + y \\boldsymbol\{x\} + \\boldsymbol\{y\} =\\boldsymbol\{x\} + \\boldsymbol\{y\} x\+y=x\+y

( x + y ) + z = x + ( y + z ) \\left(\\boldsymbol\{x\} + \\boldsymbol\{y\}\\right) + \\boldsymbol\{z\} = \\boldsymbol\{x\} + \\left(\\boldsymbol\{y\} + \\boldsymbol\{z\}\\right) (x\+y)\+z=x\+(y\+z)

x + o = x \\boldsymbol\{x\} + \\boldsymbol\{o\} = \\boldsymbol\{x\} x\+o=x

x + ( − x ) = o \\boldsymbol\{x\} + \\left(\\boldsymbol\{-x\}\\right) = \\boldsymbol\{o\} x\+(−x)=o

x + o = x \\boldsymbol\{x\} + \\boldsymbol\{o\} = \\boldsymbol\{x\} x\+o=x

c ( x + y ) = c x + c y c\\left(\\boldsymbol\{x\} + \\boldsymbol\{y\}\\right) = c\\boldsymbol\{x\} + c\\boldsymbol\{y\} c(x\+y)=cx\+cy

( a + b ) x = a x + b x (a + b)\\boldsymbol\{x\} = a\\boldsymbol\{x\} + b\\boldsymbol\{x\} (a\+b)x=ax\+bx

#### 基底 ####

在线形空间中，选定的作为基准的一组向量，称作**基底**。基底中的向量叫作**基向量**。

只有当前空间中的任何向量  v ⃗ \\vec\{v\} v 都可以用向量组  ( e ⃗ 1 , ⋯   , e ⃗ 2 ) (\\vec\{e\}\_1, \\cdots, \\vec\{e\}\_2) (e1,⋯,e2) 唯一的表示时，该组向量才能被称为基底，此时向量可通过基底表示为：

v ⃗ = x 1 e ⃗ 1 + ⋯ + x n e ⃗ n \\vec\{v\} = x\_\{1\}\\vec\{e\}\_1 + \\cdots + x\_\{n\}\\vec\{e\}\_n v=x1e1\+⋯\+xnen  
或表述为：若任意向量  v ⃗ \\vec\{v\} v 都可以用  e ⃗ 1 , ⋯   , e ⃗ 2 \\vec\{e\}\_1, \\cdots, \\vec\{e\}\_2 e1,⋯,e2 的线形组合来表示，且表示方法唯一，则  ( e ⃗ 1 , ⋯   , e ⃗ 2 ) (\\vec\{e\}\_1, \\cdots, \\vec\{e\}\_2) (e1,⋯,e2) 可称为基底。

#### 维数 ####

*  基向量的个数
 *  坐标的分量数

#### 内积 / Inner Product ####

设有向量  A = \[ a 1 a 2 ⋮ a n \] A = \\left\[\\begin\{array\}\{c\} a\_1 \\\\ a\_2 \\\\ \\vdots \\\\ a\_n \\\\\\end\{array\}\\right\] A=⎣⎢⎢⎢⎡a1a2⋮an⎦⎥⎥⎥⎤ 和向量  B = \[ b 1 b 2 ⋮ b n \] B = \\left\[\\begin\{array\}\{c\} b\_1 \\\\ b\_2 \\\\ \\vdots \\\\ b\_n \\\\ \\end\{array\}\\right\] B=⎣⎢⎢⎢⎡b1b2⋮bn⎦⎥⎥⎥⎤，则其内积定义为  
 A T B = \[ a 1 a 2 ⋯ a n \] \[ b 1 b 2 ⋮ b n \] = a 1 b 1 + a 2 b 2 + ⋯ + a n b n A^\{T\} B = \\left\[\\begin\{array\}\{c\} a\_1 & a\_2 & \\cdots & a\_n \\end\{array\}\\right\] \\left\[\\begin\{array\}\{c\} b\_1 \\\\ b\_2 \\\\ \\vdots \\\\ b\_n \\\\ \\end\{array\}\\right\] = a\_1 b\_1 + a\_2 b\_2 + \\cdots + a\_n b\_n ATB=\[a1a2⋯an\]⎣⎢⎢⎢⎡b1b2⋮bn⎦⎥⎥⎥⎤=a1b1\+a2b2\+⋯\+anbn

#### 外积 / Outter Product ####

设有向量  A = \[ a 1 a 2 ⋮ a n \] A = \\left\[\\begin\{array\}\{c\} a\_1 \\\\ a\_2 \\\\ \\vdots \\\\ a\_n \\\\\\end\{array\}\\right\] A=⎣⎢⎢⎢⎡a1a2⋮an⎦⎥⎥⎥⎤ 和向量  B = \[ b 1 b 2 ⋮ b n \] B = \\left\[\\begin\{array\}\{c\} b\_1 \\\\ b\_2 \\\\ \\vdots \\\\ b\_n \\\\ \\end\{array\}\\right\] B=⎣⎢⎢⎢⎡b1b2⋮bn⎦⎥⎥⎥⎤，则其外积定义为  
 A B T = \[ a 1 a 2 ⋮ a n \] \[ b 1 b 2 ⋯ b n \] = \[ a 1 b 1 a 1 b 2 ⋯ a 1 b n a 2 b 1 a 2 b 2 ⋯ a 2 b n ⋮ ⋮ ⋮ a n b 1 a n b 2 ⋯ a n b n \] A B^T = \\left\[\\begin\{array\}\{c\} a\_1 \\\\ a\_2 \\\\ \\vdots \\\\ a\_n \\\\ \\end\{array\}\\right\] \\left\[\\begin\{array\}\{c\} b\_1 & b\_2 & \\cdots & b\_n \\\\ \\end\{array\}\\right\] = \\left\[\\begin\{array\}\{ccc\} a\_1 b\_1 & a\_1 b\_2 & \\cdots & a\_1 b\_n \\\\ a\_2 b\_1 & a\_2 b\_2 & \\cdots & a\_2 b\_n \\\\ \\vdots & \\vdots & & \\vdots \\\\ a\_n b\_1 & a\_n b\_2 & \\cdots & a\_n b\_n \\\\ \\end\{array\}\\right\] ABT=⎣⎢⎢⎢⎡a1a2⋮an⎦⎥⎥⎥⎤\[b1b2⋯bn\]=⎣⎢⎢⎢⎡a1b1a2b1⋮anb1a1b2a2b2⋮anb2⋯⋯⋯a1bna2bn⋮anbn⎦⎥⎥⎥⎤

### 范数 / norm ###

使用符号 $ L^p $ 表示，定义如下  
 ∣ ∣ x ∣ ∣ p = ( ∑ i ∣ x i ∣ p ) 1 p \{||x||\}\_p = \\left(\\sum\_i\{|x\_i|\}^p\\right)^\\frac\{1\}\{p\} ∣∣x∣∣p=(i∑∣xi∣p)p1  
特别的，当 $ p = 1 $ 时，即 $ L^1 $，有  
 ∣ ∣ x ∣ ∣ 1 = ∑ i ∣ x i ∣ = ∣ x 1 ∣ + ∣ x 2 ∣ + ⋯ + ∣ x n ∣ \{||x||\}\_1 = \\sum\_i\{|x\_i|\} = |x\_1| + |x\_2| + \\cdots + |x\_n| ∣∣x∣∣1=i∑∣xi∣=∣x1∣\+∣x2∣\+⋯\+∣xn∣  
特别的，当  p = 2 p = 2 p=2 时，即  L 2 L^2 L2，有  
 ∣ ∣ x ∣ ∣ 2 = ( ∑ i ∣ x i ∣ 2 ) 1 2 = x 1 2 + x 2 2 + ⋯ + x n 2 \{||x||\}\_2 = \\left(\\sum\_i\{|x\_i|\}^2\\right)^\\frac\{1\}\{2\} = \\sqrt\{x\_1^2 + x\_2^2 + \\cdots + x\_n^2\} ∣∣x∣∣2=(i∑∣xi∣2)21=x12\+x22\+⋯\+xn2  
该范数亦称**欧几里得范数**

### 分块矩阵 ###

#### 定义 ####

A = \[ 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 \] = \[ A 11 A 12 A 13 A 21 A 22 A 23 \] A=\\left\[\\begin\{array\}\{ccc|cc|cc\} 3 & 1 & 4 & 1 & 5 & 9 & 2 \\\\ 6 & 5 & 3 & 5 & 8 & 9 & 7 \\\\ \\hline 9 & 3 & 2 & 3 & 8 & 4 & 6 \\\\ 2 & 6 & 4 & 3 & 3 & 8 & 3 \\\\ 2 & 7 & 9 & 5 & 0 & 2 & 8 \\end\{array\}\\right\]=\\left\[\\begin\{array\}\{lll\} A\_\{11\} & A\_\{12\} & A\_\{13\} \\\\ A\_\{21\} & A\_\{22\} & A\_\{23\} \\end\{array\}\\right\] A=⎣⎢⎢⎢⎢⎡36922153674324915335588309948227638⎦⎥⎥⎥⎥⎤=\[A11A21A12A22A13A23\]

#### 加法 ####

\[ A 11 ⋯ A 1 n ⋮ ⋮ A m 1 ⋯ A m n \] + \[ B 11 ⋯ B 1 n ⋮ ⋮ B m 1 ⋯ B m n \] = \[ A 11 + B 11 ⋯ A 1 n + B 1 n ⋮ ⋮ A m 1 + B m 1 ⋯ A m n + B m n \] \\left\[\\begin\{array\}\{ccc\} A\_\{11\} & \\cdots & A\_\{1 n\} \\\\ \\vdots & & \\vdots \\\\ A\_\{m 1\} & \\cdots & A\_\{m n\} \\end\{array\}\\right\] + \\left\[\\begin\{array\}\{ccc\} B\_\{11\} & \\cdots & B\_\{1 n\} \\\\ \\vdots & & \\vdots \\\\ B\_\{m 1\} & \\cdots & B\_\{m n\} \\end\{array\}\\right\]=\\left\[\\begin\{array\}\{ccc\} A\_\{11\}+B\_\{11\} & \\cdots & A\_\{1 n\}+B\_\{1 n\} \\\\ \\vdots & & \\vdots \\\\ A\_\{m 1\}+B\_\{m 1\} & \\cdots & A\_\{m n\}+B\_\{m n\} \\end\{array\}\\right\] ⎣⎢⎡A11⋮Am1⋯⋯A1n⋮Amn⎦⎥⎤\+⎣⎢⎡B11⋮Bm1⋯⋯B1n⋮Bmn⎦⎥⎤=⎣⎢⎡A11\+B11⋮Am1\+Bm1⋯⋯A1n\+B1n⋮Amn\+Bmn⎦⎥⎤

#### 数量乘法 ####

c \[ A 11 ⋯ A 1 n ⋮ ⋮ A m 1 ⋯ A m n \] = \[ c A 11 ⋯ c A 1 n ⋮ ⋮ c A m 1 ⋯ c A m n \] c\\left\[\\begin\{array\}\{ccc\} A\_\{11\} & \\cdots & A\_\{1 n\} \\\\ \\vdots & & \\vdots \\\\ A\_\{m 1\} & \\cdots & A\_\{m n\} \\end\{array\}\\right\]=\\left\[\\begin\{array\}\{ccc\} c A\_\{11\} & \\cdots & c A\_\{1 n\} \\\\ \\vdots & & \\vdots \\\\ c A\_\{m 1\} & \\cdots & c A\_\{m n\} \\end\{array\}\\right\] c⎣⎢⎡A11⋮Am1⋯⋯A1n⋮Amn⎦⎥⎤=⎣⎢⎡cA11⋮cAm1⋯⋯cA1n⋮cAmn⎦⎥⎤

#### 乘法 ####

\[ B 11 ⋯ B 1 m ⋮ ⋮ B k 1 ⋯ B k m \] \[ A 11 ⋯ A 1 n ⋮ ⋮ A m 1 ⋯ A m n \] = \[ ( B 11 A 11 + ⋯ + B 1 m A m 1 ) ⋯ ( B 11 A 1 n + ⋯ + B 1 m A m n ) ⋮ ⋮ ( B k 1 A 11 + ⋯ + B k m A m 1 ) ⋯ ( B k 1 A 1 n + ⋯ + B k m A m n ) \] \\begin\{array\}\{l\} \\left\[\\begin\{array\}\{ccc\} B\_\{11\} & \\cdots & B\_\{1 m\} \\\\ \\vdots & & \\vdots \\\\ B\_\{k 1\} & \\cdots & B\_\{k m\} \\end\{array\}\\right\] \\left\[\\begin\{array\}\{ccc\} A\_\{11\} & \\cdots & A\_\{1 n\} \\\\ \\vdots & & \\vdots \\\\ A\_\{m 1\} & \\cdots & A\_\{m n\} \\end\{array\}\\right\] = \\left\[\\begin\{array\}\{ccc\} \\left(B\_\{11\} A\_\{11\}+\\cdots+B\_\{1 m\} A\_\{m 1\}\\right) & \\cdots & \\left(B\_\{11\} A\_\{1 n\}+\\cdots+B\_\{1 m\} A\_\{m n\}\\right) \\\\ \\vdots & \\vdots \\\\ \\left(B\_\{k 1\} A\_\{11\}+\\cdots+B\_\{k m\} A\_\{m 1\}\\right) & \\cdots & \\left(B\_\{k 1\} A\_\{1 n\}+\\cdots+B\_\{k m\} A\_\{m n\}\\right) \\end\{array\}\\right\] \\end\{array\} ⎣⎢⎡B11⋮Bk1⋯⋯B1m⋮Bkm⎦⎥⎤⎣⎢⎡A11⋮Am1⋯⋯A1n⋮Amn⎦⎥⎤=⎣⎢⎡(B11A11\+⋯\+B1mAm1)⋮(Bk1A11\+⋯\+BkmAm1)⋯⋮⋯(B11A1n\+⋯\+B1mAmn)(Bk1A1n\+⋯\+BkmAmn)⎦⎥⎤

#### 分块矩阵与向量 ####

##### 列向量 #####

A = \[ a 11 a 12 ⋯ a 1 m ⋮ ⋮ ⋮ a n 1 a n 2 ⋯ a n m \] = \[ a 1 , a 2 , ⋯   , a m \] A=\\left\[\\begin\{array\}\{c|c|c|c\} a\_\{11\} & a\_\{12\} & \\cdots & a\_\{1 m\} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a\_\{n 1\} & a\_\{n 2\} & \\cdots & a\_\{n m\} \\end\{array\}\\right\]=\\left\[a\_\{1\}, a\_\{2\}, \\cdots, a\_\{m\}\\right\] A=⎣⎢⎡a11⋮an1a12⋮an2⋯⋯a1m⋮anm⎦⎥⎤=\[a1,a2,⋯,am\]

##### 行向量 #####

B = \[ b 11 ⋯ b 1 n ′ b 21 ⋯ b 2 n ′ ⋮ ⋮ b m ′ 1 ⋯ b m ′ n ′ \] = \[ b 1 T b 2 T ⋮ b m ′ T \] B=\\left\[\\begin\{array\}\{ccc\} b\_\{11\} & \\cdots & b\_\{1 n^\{\\prime\}\} \\\\ \\hline b\_\{21\} & \\cdots & b\_\{2 n^\{\\prime\}\} \\\\ \\hline \\vdots & & \\vdots \\\\ \\hline b\_\{m^\{\\prime\} 1\} & \\cdots & b\_\{m^\{\\prime\} n^\{\\prime\}\} \\end\{array\}\\right\]=\\left\[\\begin\{array\}\{c\} b\_\{1\}^\{T\} \\\\ b\_\{2\}^\{T\} \\\\ \\vdots \\\\ b\_\{m^\{\\prime\}\}^\{T\} \\end\{array\}\\right\] B=⎣⎢⎢⎢⎡b11b21⋮bm′1⋯⋯⋯b1n′b2n′⋮bm′n′⎦⎥⎥⎥⎤=⎣⎢⎢⎢⎡b1Tb2T⋮bm′T⎦⎥⎥⎥⎤

A \[ c 1 c 2 ⋮ c m \] = ( a 1 , a 2 , ⋯   , a m ) \[ c 1 c 2 ⋮ c m \] = c 1 a 1 + c 2 a 2 + ⋯ + c m a m A\\left\[\\begin\{array\}\{c\} c\_\{1\} \\\\ c\_\{2\} \\\\ \\vdots \\\\ c\_\{m\} \\end\{array\}\\right\] = \\left(\\boldsymbol\{a\}\_\{1\}, \\boldsymbol\{a\}\_\{2\}, \\cdots, \\boldsymbol\{a\}\_\{m\}\\right)\\left\[\\begin\{array\}\{c\} c\_\{1\} \\\\ c\_\{2\} \\\\ \\vdots \\\\ c\_\{m\} \\end\{array\}\\right\]=c\_\{1\} \\boldsymbol\{a\}\_\{1\}+c\_\{2\} \\boldsymbol\{a\}\_\{2\}+\\cdots+c\_\{m\} \\boldsymbol\{a\}\_\{m\} A⎣⎢⎢⎢⎡c1c2⋮cm⎦⎥⎥⎥⎤=(a1,a2,⋯,am)⎣⎢⎢⎢⎡c1c2⋮cm⎦⎥⎥⎥⎤=c1a1\+c2a2\+⋯\+cmam

B d = \[ b 1 T b 2 T ⋮ b m ′ T \] d = \[ b 1 T d b 2 T d ⋮ b m ′ T d \] B \\boldsymbol\{d\}=\\left\[\\begin\{array\}\{c\} b\_\{1\}^\{T\} \\\\ b\_\{2\}^\{T\} \\\\ \\vdots \\\\ b\_\{m^\{\\prime\}\}^\{T\} \\end\{array\}\\right\]\\boldsymbol\{d\}=\\left\[\\begin\{array\}\{c\} b\_\{1\}^\{T\} \\boldsymbol\{d\} \\\\ b\_\{2\}^\{T\} \\boldsymbol\{d\} \\\\ \\vdots \\\\ b\_\{m^\{\\prime\}\}^\{T\} \\boldsymbol\{d\} \\end\{array\}\\right\] Bd=⎣⎢⎢⎢⎡b1Tb2T⋮bm′T⎦⎥⎥⎥⎤d=⎣⎢⎢⎢⎡b1Tdb2Td⋮bm′Td⎦⎥⎥⎥⎤

##### 矩阵乘法 #####

A B = ( a 1 , a 2 , ⋯   , a m ) ( b 1 T b 2 T ⋮ b n T ) = a 1 b 1 T + a 2 b 2 T + ⋯ + a m b n T ( m = n ) AB=\\left(\\boldsymbol\{a\}\_\{1\}, \\boldsymbol\{a\}\_\{2\}, \\cdots, \\boldsymbol\{a\}\_\{m\}\\right)\\left(\\begin\{array\}\{c\} \\boldsymbol\{b\}\_\{1\}^\{T\} \\\\ \\boldsymbol\{b\}\_\{2\}^\{T\} \\\\ \\vdots \\\\ \\boldsymbol\{b\}\_\{n\}^\{T\} \\end\{array\}\\right)=\\boldsymbol\{a\}\_\{1\} \\boldsymbol\{b\}\_\{1\}^\{T\}+\\boldsymbol\{a\}\_\{2\} \\boldsymbol\{b\}\_\{2\}^\{T\}+\\cdots+\\boldsymbol\{a\}\_\{m\} \\boldsymbol\{b\}\_\{n\}^\{T\} \\quad \\left(m=n\\right) AB=(a1,a2,⋯,am)⎝⎜⎜⎜⎛b1Tb2T⋮bnT⎠⎟⎟⎟⎞=a1b1T\+a2b2T\+⋯\+ambnT(m=n)

##### 分块对角矩阵 #####

\[ A 1 O O O O A 2 O O O O A 3 O O O O A 4 \] ≡ diag ⁡ ( A 1 , A 2 , A 3 , A 4 ) \\left\[\\begin\{array\}\{cccc\} A\_\{1\} & O & O & O \\\\ O & A\_\{2\} & O & O \\\\ O & O & A\_\{3\} & O \\\\ O & O & O & A\_\{4\} \\end\{array\}\\right\] \\equiv \\operatorname\{diag\}\\left(A\_\{1\}, A\_\{2\}, A\_\{3\}, A\_\{4\}\\right) ⎣⎢⎢⎡A1OOOOA2OOOOA3OOOOA4⎦⎥⎥⎤≡diag(A1,A2,A3,A4)

\[ A 1 0 0 O 0 A 2 0 O 0 0 A 3 0 0 0 0 A 4 \] k = \[ A 1 k O O O O A 2 k O O 0 O A 3 k O 0 O O A 4 k \] \\left\[\\begin\{array\}\{cccc\} A\_\{1\} & 0 & 0 & O \\\\ 0 & A\_\{2\} & 0 & O \\\\ 0 & 0 & A\_\{3\} & 0 \\\\ 0 & 0 & 0 & A\_\{4\} \\end\{array\}\\right\]^\{k\}=\\left\[\\begin\{array\}\{cccc\} A\_\{1\}^\{k\} & O & O & O \\\\ O & A\_\{2\}^\{k\} & O & O \\\\ 0 & O & A\_\{3\}^\{k\} & O \\\\ 0 & O & O & A\_\{4\}^\{k\} \\end\{array\}\\right\] ⎣⎢⎢⎡A10000A20000A30OO0A4⎦⎥⎥⎤k=⎣⎢⎢⎡A1kO00OA2kOOOOA3kOOOOA4k⎦⎥⎥⎤

\[ A 1 0 0 0 0 A 2 0 0 0 0 A 3 0 0 0 0 A 4 \] − 1 = \[ A 1 − 1 O O O 0 A 2 − 1 O O 0 O A 3 − 1 O 0 O O A 4 − 1 \] \\left\[\\begin\{array\}\{cccc\} A\_\{1\} & 0 & 0 & 0 \\\\ 0 & A\_\{2\} & 0 & 0 \\\\ 0 & 0 & A\_\{3\} & 0 \\\\ 0 & 0 & 0 & A\_\{4\} \\end\{array\}\\right\]^\{-1\}=\\left\[\\begin\{array\}\{cccc\} A\_\{1\}^\{-1\} & O & O & O \\\\ 0 & A\_\{2\}^\{-1\} & O & O \\\\ 0 & O & A\_\{3\}^\{-1\} & O \\\\ 0 & O & O & A\_\{4\}^\{-1\} \\end\{array\}\\right\] ⎣⎢⎢⎡A10000A20000A30000A4⎦⎥⎥⎤−1=⎣⎢⎢⎡A1−1000OA2−1OOOOA3−1OOOOA4−1⎦⎥⎥⎤

### 行列式 ###

行列式用  det ⁡ \\operatorname\{det\} det 表示，例  
 det ⁡ A = ∣ a 11 ⋯ a 1 n ⋮ ⋮ a m 1 ⋯ a m n ∣ \\operatorname\{det\} A =\\begin\{array\}\{l\} \\left|\\begin\{array\}\{cccc\} a\_\{11\} & \\cdots & a\_\{1 n\} \\\\ \\vdots & & \\vdots \\\\ a\_\{m 1\} & \\cdots & a\_\{m n\} \\end\{array\}\\right| \\end\{array\} detA=∣∣∣∣∣∣∣a11⋮am1⋯⋯a1n⋮amn∣∣∣∣∣∣∣

det ⁡ I = 1 \\operatorname\{det\} I = 1 detI=1

det ⁡ ( A B ) = ( det ⁡ A ) ( det ⁡ B ) \\operatorname\{det\}(A B) =(\\operatorname\{det\} A)(\\operatorname\{det\} B) det(AB)=(detA)(detB)

由  
 det ⁡ ( A A − 1 ) = ( det ⁡ A ) ( det ⁡ A − 1 ) = det ⁡ I = 1 \\operatorname\{det\}(A A^\{-1\}) = (\\operatorname\{det\} A)(\\operatorname\{det\} A^\{-1\}) = \\operatorname\{det\} I = 1 det(AA−1)=(detA)(detA−1)=detI=1  
可得  
 det ⁡ A − 1 = 1 det ⁡ A \\operatorname\{det\} A^\{-1\} = \\dfrac\{1\}\{\\operatorname\{det\} A\} detA−1=detA1  
当  det ⁡ A = 0 \\operatorname\{det\} A = 0 detA=0 时， A − 1 A^\{-1\} A−1 不存在。  
 det ⁡ ( diag ⁡ ( a 1 , ⋯   , a n ) ) = a 1 ⋯ a n \\operatorname\{det\} (\\operatorname\{diag\} \\left(a\_\{1\}, \\cdots, a\_\{n\}\\right)) = a\_\{1\} \\cdots a\_\{n\} det(diag(a1,⋯,an))=a1⋯an

det ⁡ ( A T ) = det ⁡ A \\operatorname\{det\} \\left(A^T\\right) = \\operatorname\{det\} A det(AT)=detA

det ⁡ ( c A ) = c n det ⁡ A \\operatorname\{det\} \\left(cA\\right) = c^\{n\}\\operatorname\{det\} A det(cA)=cndetA

对于上三角矩阵，其对应的行列式的值为主对角线元素之和  
 det ⁡ A = a 11 a 22 a 33 \\operatorname\{det\} A = a\_\{11\} a\_\{22\} a\_\{33\} detA=a11a22a33

### 向量空间 / Vector Space ###

#### 零向量空间 / Null Space ####

零向量空间是所有满足  A x = 0 Ax = 0 Ax=0 的向量  x x x 所构成的向量空间。