济南企业建站哪家做的好,科技公司名称大全简单大气,深圳网a深圳网站建设,wordpress简单文章目录 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 … 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} \\ ⎩ ⎨ ⎧a11x1a12x2⋯a1nxnb1a21x1a22x2⋯a2nxnb2⋯⋯⋯⋯am1x1am2x2⋯amnxnbm 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} \\ ⎩ ⎨ ⎧a11x1a12x2⋯a1nxnb1a21x1a22x2⋯a2nxnb2⋯⋯⋯⋯am1x1am2x2⋯amnxnbm 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} \\ AB a11a21⋮am1a12a22⋮am2⋯⋯⋱⋯a1na2n⋮amn b11b21⋮bm1b12b22⋮bm2⋯⋯⋱⋯b1nb2n⋮bmn a11b11a21b21⋮am1bm1a12b12a22b22⋮am2bm2⋯⋯⋱⋯a1nb1na2nb2n⋮amnbmn 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}) ABBA(AB)CA(BC) 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} \\ cAc a11a21⋮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} cAAc(λμ)Aλ(μA)(λμ)AλAμAλ(AB)λAλB 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} CAB(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) \\ cijk1∑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)CA(BC)c(AB)(cA)BA(cB)A(BC)ABAC(BC)ABACA 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}) . ABBA (BE). 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)TA(AB)TATBT(λA)TλAT(AB)TBTAT 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} \\ ABBAE 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)−1A(λA)−1λ−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)−1B−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} \\ ⎩ ⎨ ⎧a11x1a12x2⋯b1a21x1a22x2⋯b2⋯⋯⋯⋯an1x1an2x2⋯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,i1⋮an,i1⋯⋯a1n⋮ann
