高等数学：线性代数-第二章

柔情只为你懂 2023-10-15 09:08 148阅读 0赞

#### 文章目录 ####

*  第2章 矩阵及其运算
 *   *  2.1 线性方程组和矩阵
     *  2.2 矩阵的运算
     *  2.3 逆矩阵
     *  2.4 Cramer法则

## 第2章 矩阵及其运算 ##

### 2.1 线性方程组和矩阵 ###

n \\bm\{n\} n 元线性方程组 设有 n 个未知数 m 个方程的线性方程组  
 \{ a 11 x 1 + a 12 x 2 + ⋯ + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + ⋯ + a 2 n x n = b 2 ⋯ ⋯ ⋯ ⋯ a m 1 x 1 + a m 2 x 2 + ⋯ + a m n x n = b m \\begin\{cases\} a\_\{11\}x\_\{1\} + a\_\{12\}x\_\{2\} + \\cdots + a\_\{1n\}x\_\{n\} = b\_\{1\} \\\\ a\_\{21\}x\_\{1\} + a\_\{22\}x\_\{2\} + \\cdots + a\_\{2n\}x\_\{n\} = b\_\{2\} \\\\ \\cdots\\cdots\\cdots\\cdots \\\\ a\_\{m1\}x\_\{1\} + a\_\{m2\}x\_\{2\} + \\cdots + a\_\{mn\}x\_\{n\} = b\_\{m\} \\\\ \\end\{cases\} \\\\ ⎩⎨⎧a11x1\+a12x2\+⋯\+a1nxn=b1a21x1\+a22x2\+⋯\+a2nxn=b2⋯⋯⋯⋯am1x1\+am2x2\+⋯\+amnxn=bm  
当常数项  b i b\_\{i\} bi 不全为零时，称该方程组为n 元非齐次线性方程组，当  b i b\_\{i\} bi 全为零时，称该方程组为n 元齐次线性方程组。

矩阵 由  m × n m \\times n m×n 个数  a i j a\_\{ij\} aij 排成的 m 行 n 列的数表  
 a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n \\begin\{matrix\} a\_\{11\} & a\_\{12\} & \\cdots & a\_\{1n\} \\\\ a\_\{21\} & a\_\{22\} & \\cdots & a\_\{2n\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a\_\{m1\} & a\_\{m2\} & \\cdots & a\_\{mn\} \\\\ \\end\{matrix\} \\\\ a11a21⋮am1a12a22⋮am2⋯⋯⋱⋯a1na2n⋮amn  
称为  m × n m \\times n m×n矩阵，记作  
 A = ( a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ) \\bm\{A\} = \\begin\{pmatrix\} a\_\{11\} & a\_\{12\} & \\cdots & a\_\{1n\} \\\\ a\_\{21\} & a\_\{22\} & \\cdots & a\_\{2n\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a\_\{m1\} & a\_\{m2\} & \\cdots & a\_\{mn\} \\\\ \\end\{pmatrix\} \\\\ A=a11a21⋮am1a12a22⋮am2⋯⋯⋱⋯a1na2n⋮amn  
特别地，当 m = n 时，该矩阵叫做n 阶方阵。

增广矩阵 对于非齐次线性方程组  
 \{ a 11 x 1 + a 12 x 2 + ⋯ + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + ⋯ + a 2 n x n = b 2 ⋯ ⋯ ⋯ ⋯ a m 1 x 1 + a m 2 x 2 + ⋯ + a m n x n = b m \\begin\{cases\} a\_\{11\}x\_\{1\} + a\_\{12\}x\_\{2\} + \\cdots + a\_\{1n\}x\_\{n\} = b\_\{1\} \\\\ a\_\{21\}x\_\{1\} + a\_\{22\}x\_\{2\} + \\cdots + a\_\{2n\}x\_\{n\} = b\_\{2\} \\\\ \\cdots\\cdots\\cdots\\cdots \\\\ a\_\{m1\}x\_\{1\} + a\_\{m2\}x\_\{2\} + \\cdots + a\_\{mn\}x\_\{n\} = b\_\{m\} \\\\ \\end\{cases\} \\\\ ⎩⎨⎧a11x1\+a12x2\+⋯\+a1nxn=b1a21x1\+a22x2\+⋯\+a2nxn=b2⋯⋯⋯⋯am1x1\+am2x2\+⋯\+amnxn=bm  
它的系数矩阵、未知数矩阵和常数项矩阵分别如下：  
 A = ( a i j ) m × n x = ( x 1 x 2 ⋯ x n ) b = ( b 1 b 2 ⋯ b m ) \\begin\{align\} &\\bm\{A\} = (a\_\{ij\})\_\{m \\times n\} \\\\ &\\bm\{x\} = \\begin\{pmatrix\} x\_\{1\} & x\_\{2\} & \\cdots & x\_\{n\} \\\\ \\end\{pmatrix\} \\\\ &\\bm\{b\} = \\begin\{pmatrix\} b\_\{1\} & b\_\{2\} & \\cdots & b\_\{m\} \\\\ \\end\{pmatrix\} \\\\ \\end\{align\} \\\\ A=(aij)m×nx=(x1x2⋯xn)b=(b1b2⋯bm)  
它的增广矩阵定义为  
 B = ( A b ) = ( a 11 a 12 ⋯ a 1 n b 1 a 21 a 22 ⋯ a 2 n b 2 ⋮ ⋮ ⋱ ⋮ ⋮ a m 1 a m 2 ⋯ a m n b m ) \\bm\{B\} = ( \\begin\{array\}\{c|c\} \\bm\{A\} & \\bm\{b\} \\end\{array\} ) = \\begin\{pmatrix\} a\_\{11\} & a\_\{12\} & \\cdots & a\_\{1n\} & b\_\{1\} \\\\ a\_\{21\} & a\_\{22\} & \\cdots & a\_\{2n\} & b\_\{2\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ a\_\{m1\} & a\_\{m2\} & \\cdots & a\_\{mn\} & b\_\{m\} \\\\ \\end\{pmatrix\} \\\\ B=(Ab)=a11a21⋮am1a12a22⋮am2⋯⋯⋱⋯a1na2n⋮amnb1b2⋮bm  
对角矩阵 方阵

( λ 1 λ 2 ⋱ λ n ) \\begin\{pmatrix\} \\lambda\_\{1\} & & & \\\\ & \\lambda\_\{2\} & & \\\\ & & \\ddots & \\\\ & & & \\lambda\_\{n\} \\\\ \\end\{pmatrix\} \\\\ λ1λ2⋱λn  
叫做对角矩阵，简称对角阵，记作  d i a g ( λ 1 λ 2 ⋯ λ n ) \\mathrm\{diag\}(\\begin\{array\}\{ccc\} \\lambda\_\{1\} & \\lambda\_\{2\} & \\cdots & \\lambda\_\{n\} \\end\{array\}) diag(λ1λ2⋯λn) .

单位矩阵 对角矩阵  d i a g ( 1 1 ⋯ 1 ) \\mathrm\{diag\}(\\begin\{array\}\{ccc\} 1 & 1 & \\cdots & 1 \\end\{array\}) diag(11⋯1) 叫做 n 阶单位矩阵，简称单位阵，记作  E n \\bm\{E\}\_\{n\} En .

### 2.2 矩阵的运算 ###

矩阵加法  
 A + B = ( 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\{align\} \\bm\{A\} + \\bm\{B\} &= \\begin\{pmatrix\} a\_\{11\} & a\_\{12\} & \\cdots & a\_\{1n\} \\\\ a\_\{21\} & a\_\{22\} & \\cdots & a\_\{2n\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a\_\{m1\} & a\_\{m2\} & \\cdots & a\_\{mn\} \\\\ \\end\{pmatrix\} + \\begin\{pmatrix\} b\_\{11\} & b\_\{12\} & \\cdots & b\_\{1n\} \\\\ b\_\{21\} & b\_\{22\} & \\cdots & b\_\{2n\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ b\_\{m1\} & b\_\{m2\} & \\cdots & b\_\{mn\} \\\\ \\end\{pmatrix\} \\\\ &= \\begin\{pmatrix\} a\_\{11\} + b\_\{11\} & a\_\{12\} + b\_\{12\} & \\cdots & a\_\{1n\} + b\_\{1n\} \\\\ a\_\{21\} + b\_\{21\} & a\_\{22\} + b\_\{22\} & \\cdots & a\_\{2n\} + b\_\{2n\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a\_\{m1\} + b\_\{m1\} & a\_\{m2\} + b\_\{m2\} & \\cdots & a\_\{mn\} + b\_\{mn\} \\\\ \\end\{pmatrix\} \\\\ \\end\{align\} \\\\ A\+B=a11a21⋮am1a12a22⋮am2⋯⋯⋱⋯a1na2n⋮amn\+b11b21⋮bm1b12b22⋮bm2⋯⋯⋱⋯b1nb2n⋮bmn=a11\+b11a21\+b21⋮am1\+bm1a12\+b12a22\+b22⋮am2\+bm2⋯⋯⋱⋯a1n\+b1na2n\+b2n⋮amn\+bmn  
矩阵加法满足：  
 A + B = B + A ( A + B ) + C = A + ( B + C ) \\bm\{A\} + \\bm\{B\} = \\bm\{B\} + \\bm\{A\} (\\bm\{A\} + \\bm\{B\}) + \\bm\{C\} = \\bm\{A\} + (\\bm\{B\} + \\bm\{C\}) A\+B=B\+A(A\+B)\+C=A\+(B\+C)  
矩阵数乘  
 c A = c ( a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ) = ( c a 11 c a 12 ⋯ c a 1 n c a 21 c a 22 ⋯ c a 2 n ⋮ ⋮ ⋱ ⋮ c a m 1 c a m 2 ⋯ c a m n ) \\begin\{align\} c\\bm\{A\} &= c \\begin\{pmatrix\} a\_\{11\} & a\_\{12\} & \\cdots & a\_\{1n\} \\\\ a\_\{21\} & a\_\{22\} & \\cdots & a\_\{2n\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a\_\{m1\} & a\_\{m2\} & \\cdots & a\_\{mn\} \\\\ \\end\{pmatrix\} \\\\ &= \\begin\{pmatrix\} ca\_\{11\} & ca\_\{12\} & \\cdots & ca\_\{1n\} \\\\ ca\_\{21\} & ca\_\{22\} & \\cdots & ca\_\{2n\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ ca\_\{m1\} & ca\_\{m2\} & \\cdots & ca\_\{mn\} \\\\ \\end\{pmatrix\} \\\\ \\end\{align\} \\\\ cA=ca11a21⋮am1a12a22⋮am2⋯⋯⋱⋯a1na2n⋮amn=ca11ca21⋮cam1ca12ca22⋮cam2⋯⋯⋱⋯ca1nca2n⋮camn  
矩阵数乘满足：  
 c A = A c ( λ μ ) A = λ ( μ A ) ( λ + μ ) A = λ A + μ A λ ( A + B ) = λ A + λ B c\\bm\{A\} = \\bm\{A\}c (\\lambda\\mu)\\bm\{A\} = \\lambda(\\mu\\bm\{A\}) (\\lambda + \\mu)\\bm\{A\} = \\lambda\\bm\{A\} + \\mu\\bm\{A\} \\lambda(\\bm\{A\} + \\bm\{B\})=\\lambda\\bm\{A\} + \\lambda\\bm\{B\} cA=Ac(λμ)A=λ(μA)(λ\+μ)A=λA\+μAλ(A\+B)=λA\+λB  
矩阵乘法 对于  m × s m \\times s m×s矩阵  A \\bm\{A\} A 和  s × n s \\times n s×n矩阵  B \\bm\{B\} B ，它们的乘法定义为  C = A B = ( c i j ) m × n \\bm\{C\} = \\bm\{A\}\\bm\{B\} = (c\_\{ij\})\_\{m \\times n\} C=AB=(cij)m×n ，且满足  
 c i j = ∑ k = 1 s a i k b k j ( i ∈ Z ≤ m , j ∈ Z ≤ n ) c\_\{ij\} = \\sum\_\{k = 1\}^\{s\}a\_\{ik\}b\_\{kj\} ~~~~ (i \\in \\mathbb\{Z\} \\leq m, j \\in \\mathbb\{Z\} \\leq n) \\\\ cij=k=1∑saikbkj(i∈Z≤m,j∈Z≤n)  
矩阵乘法满足：  
 ( A B ) C = A ( B C ) c ( A B ) = ( c A ) B = A ( c B ) A ( B + C ) = A B + A C ( B + C ) A = B A + C A (\\bm\{A\}\\bm\{B\})\\bm\{C\} = \\bm\{A\}(\\bm\{B\}\\bm\{C\}) c(\\bm\{A\}\\bm\{B\}) = (c\\bm\{A\})\\bm\{B\} = \\bm\{A\}(c\\bm\{B\}) \\bm\{A\}(\\bm\{B\} + \\bm\{C\}) = \\bm\{A\}\\bm\{B\} + \\bm\{A\}\\bm\{C\} (\\bm\{B\} + \\bm\{C\})\\bm\{A\} = \\bm\{B\}\\bm\{A\} + \\bm\{C\}\\bm\{A\} (AB)C=A(BC)c(AB)=(cA)B=A(cB)A(B\+C)=AB\+AC(B\+C)A=BA\+CA  
需要注意的是，  
 A B ≠ B A ( B ≠ E ) . \\bm\{A\}\\bm\{B\} \\ne \\bm\{B\}\\bm\{A\} ~~~~ (\\bm\{B\} \\ne \\bm\{E\}) . AB=BA(B=E).  
矩阵转置 矩阵  A = ( a i j ) m × n \\bm\{A\} = (a\_\{ij\})\_\{m \\times n\} A=(aij)m×n的转置矩阵记作  A T \\bm\{A\}^\\mathrm\{T\} AT ，且满足  
 A T = ( a j i ) n × m \\bm\{A\}^\\mathrm\{T\} = (a\_\{ji\})\_\{n \\times m\} \\\\ AT=(aji)n×m  
矩阵转置满足：  
 ( A T ) T = A ( A + B ) T = A T + B T ( λ A ) T = λ A T ( A B ) T = B T A T (\\bm\{A\}^\{T\})^\{T\} = \\bm\{A\} (\\bm\{A\} + \\bm\{B\})^\\mathrm\{T\} = \\bm\{A\}^\\mathrm\{T\} + \\bm\{B\}^\\mathrm\{T\} (\\lambda \\bm\{A\})^\\mathrm\{T\} = \\lambda\\bm\{A\}^\\mathrm\{T\} (\\bm\{A\}\\bm\{B\})^\\mathrm\{T\} =\\bm\{B\}^\\mathrm\{T\}\\bm\{A\}^\\mathrm\{T\} (AT)T=A(A\+B)T=AT\+BT(λA)T=λAT(AB)T=BTAT  
方阵的行列式 由 n 阶方阵  A \\bm\{A\} A的元素所构成的行列式，称为方阵  A \\pmb\{A\} A 的行列式，记作  det ⁡ A \\det\\bm\{A\} detA或  ∣ A ∣ | \\bm\{A\} | ∣A∣

方阵的行列式满足：  
 ∣ A T ∣ = ∣ A ∣ ∣ λ A ∣ = λ n ∣ A ∣ | \\bm\{A\}^\\mathrm\{T\} | = | \\bm\{A\} | | \\lambda\\bm\{A\} | = \\lambda^\{n\} | \\bm\{A\} | ∣AT∣=∣A∣∣λA∣=λn∣A∣  
其中 n 为矩阵  A \\bm\{A\} A的阶数  
 ∣ A B ∣ = ∣ A ∣ ∣ B ∣ | \\pmb\{A\}\\bm\{B\} | = | \\pmb\{A\} || \\bm\{B\} | ∣AB∣=∣A∣∣B∣

### 2.3 逆矩阵 ###

伴随矩阵 行列式 | \\bm\{A\} | 的各个元素的代数余子式 A\_\{ij\} 所构成的如下的矩阵  
 A ∗ = ( A 11 A 21 ⋯ A n 1 A 12 A 22 ⋯ A n 2 ⋮ ⋮ ⋱ ⋮ A 1 n A 2 n ⋯ A n n ) \\bm\{A\}^\{\*\} = \\begin\{pmatrix\} A\_\{11\} & A\_\{21\} & \\cdots & A\_\{n1\} \\\\ A\_\{12\} & A\_\{22\} & \\cdots & A\_\{n2\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ A\_\{1n\} & A\_\{2n\} & \\cdots & A\_\{nn\} \\\\ \\end\{pmatrix\} \\\\ A∗=A11A12⋮A1nA21A22⋮A2n⋯⋯⋱⋯An1An2⋮Ann  
称为矩阵  A \\bm\{A\} A的伴随矩阵，简称伴随阵，记作  A ∗ \\bm\{A\}^\{\*\} A∗

矩阵  A \\bm\{A\} A和它的伴随矩阵  A ∗ \\bm\{A\}^\{\*\} A∗ 满足  
 A A ∗ = A ∗ A = ∣ A ∣ E \\bm\{A\}\\bm\{A\}^\{\*\}=\\bm\{A\}^\{\*\}\\bm\{A\}=|\\bm\{A\}|\\bm\{E\} \\\\ AA∗=A∗A=∣A∣E  
逆矩阵 对于 n 阶矩阵  A \\bm\{A\} A，如果有一个 n 阶矩阵  B \\bm\{B\} B ，使得  
 A B = B A = E \\bm\{A\}\\bm\{B\} = \\bm\{B\}\\bm\{A\} = \\bm\{E\} \\\\ AB=BA=E  
则说矩阵  A \\bm\{A\} A是可逆的，并把矩阵  B \\bm\{B\} B称为矩阵  A \\bm\{A\} A的逆矩阵，简称逆阵，记作  A − 1 \\bm\{A\}^\{-1\} A−1.

如果矩阵  A \\bm\{A\} A是可逆的，那么  A \\bm\{A\} A 的逆矩阵是惟一的。

矩阵  A \\bm\{A\} A 可逆的充分必要条件是  ∣ A ∣ ≠ 0 | \\bm\{A\} | \\ne 0 ∣A∣=0 。若 ∣ A ∣ ≠ 0 | \\bm\{A\} | \\ne 0 ∣A∣=0，则  
 A − 1 = 1 ∣ A ∣ A ∗ \\bm\{A\}^\{-1\} = \\frac\{1\}\{| \\bm\{A\} |\}\\bm\{A\}^\{\*\} \\\\ A−1=∣A∣1A∗  
逆矩阵满足：  
 ( A − 1 ) − 1 = A ( λ A ) − 1 = λ − 1 A − 1 (\\bm\{A\}^\{-1\})^\{-1\} = \\bm\{A\} (\\lambda \\bm\{A\})^\{-1\} = \\lambda^\{-1\}\\bm\{A\}^\{-1\} (A−1)−1=A(λA)−1=λ−1A−1  
若  A \\bm\{A\} A、  B \\bm\{B\} B 为同阶矩阵且均可逆，则  
 ( A B ) − 1 = B − 1 A − 1 (\\bm\{A\}\\bm\{B\})^\{-1\} = \\bm\{B\}^\{-1\}\\bm\{A\}^\{-1\} (AB)−1=B−1A−1  
奇异矩阵 不可逆矩阵叫做奇异矩阵。

非奇异矩阵 可逆矩阵叫做非奇异矩阵。

### 2.4 Cramer法则 ###

Cramer法则 如果线性方程组  
 \{ a 11 x 1 + a 12 x 2 + ⋯ = b 1 a 21 x 1 + a 22 x 2 + ⋯ = b 2 ⋯ ⋯ ⋯ ⋯ a n 1 x 1 + a n 2 x 2 + ⋯ = b n \\begin\{cases\} a\_\{11\}x\_\{1\} + a\_\{12\}x\_\{2\} + \\cdots = b\_\{1\} \\\\ a\_\{21\}x\_\{1\} + a\_\{22\}x\_\{2\} + \\cdots = b\_\{2\} \\\\ \\cdots\\cdots\\cdots\\cdots \\\\ a\_\{n1\}x\_\{1\} + a\_\{n2\}x\_\{2\} + \\cdots = b\_\{n\} \\\\ \\end\{cases\} \\\\ ⎩⎨⎧a11x1\+a12x2\+⋯=b1a21x1\+a22x2\+⋯=b2⋯⋯⋯⋯an1x1\+an2x2\+⋯=bn  
的系数矩阵 A 的行列式不等于零，即  
 ∣ A ∣ = ∣ a 11 ⋯ a 1 n ⋮ ⋮ a n 1 ⋯ a n n ∣ ≠ 0 \\left\\lvert A \\right\\rvert = \\begin\{vmatrix\} a\_\{11\} & \\cdots & a\_\{1n\} \\\\ \\vdots & & \\vdots \\\\ a\_\{n1\} & \\cdots & a\_\{nn\} \\\\ \\end\{vmatrix\} \\ne 0 \\\\ ∣A∣=a11⋮an1⋯⋯a1n⋮ann=0  
则该方程组有惟一解  
 x i = ∣ A i ∣ ∣ A ∣ x\_\{i\} = \\frac\{\\left\\lvert A\_\{i\} \\right\\rvert\}\{\\left\\lvert A \\right\\rvert\} \\\\ xi=∣A∣∣Ai∣  
其中  
 A i = ( a 11 ⋯ a 1 , i − 1 b 1 a 1 , i + 1 ⋯ a 1 n ⋮ ⋮ ⋮ ⋮ ⋮ a n 1 ⋯ a n , i − 1 b n a n , i + 1 ⋯ a n n ) A\_\{i\} = \\begin\{pmatrix\} a\_\{11\} & \\cdots & a\_\{1, i - 1\} & b\_\{1\} & a\_\{1, i + 1\} & \\cdots & a\_\{1n\} \\\\ \\vdots & & \\vdots & \\vdots & \\vdots & & \\vdots \\\\ a\_\{n1\} & \\cdots & a\_\{n, i - 1\} & b\_\{n\} & a\_\{n, i + 1\} & \\cdots & a\_\{nn\} \\\\ \\end\{pmatrix\} \\\\ Ai=a11⋮an1⋯⋯a1,i−1⋮an,i−1b1⋮bna1,i\+1⋮an,i\+1⋯⋯a1n⋮ann