线性代数：线性代数（Linear Algebra）综述

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。
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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。