矩阵理论复习部分——线性代数(2)矩阵运算

时间:2022-10-03 08:52:14

一、矩阵类型

1、转置矩阵: A = ( 3 2 1 1 2 3 2 3 1 ) A = \begin{pmatrix} 3 & 2 & 1 \\ 1 & 2 & 3 \\ 2 & 3 & 1 \\ \end{pmatrix} A= 312223131 A T = ( 3 1 2 2 2 3 1 3 1 ) A^T = \begin{pmatrix} 3 & 1 & 2 \\ 2 & 2 & 3 \\ 1 & 3 & 1 \\ \end{pmatrix} AT= 321123231 T T T 表示矩阵的转置

2、对角矩阵: ( 2 0 0 0 3 0 0 0 1 ) \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} 200030001 对角矩阵(diagonal matrix)是一个主对角线之外的元素皆为0的矩阵,常写为 d i a g ( a 1 , a 2 , a 3 , . . . , a n ) \mathrm{diag}(a_1,a_2,a_3,...,a_n) diag(a1,a2,a3,...,an)

3、上下三角矩阵: ( 1 2 3 0 2 3 0 0 1 ) \begin{pmatrix} 1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 & 0 & 1 \\ \end{pmatrix} 100220331 ( 1 0 0 3 2 0 2 3 1 ) \begin{pmatrix} 1 & 0 & 0 \\ 3 & 2 & 0 \\ 2 & 3 & 1 \\ \end{pmatrix} 132023001

4、单位矩阵: ( 1 0 0 0 1 0 0 0 1 ) = E = I \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} = E = I 100010001 =E=I

5、正交矩阵:若 n n n 阶方阵 A A A,满足 A A T = E AA^T=E AAT=E,称 A A A 为正交矩阵

6、对称矩阵: A T = T A^T=T AT=T

二、矩阵的基本运算

1、加法、减法:矩阵对应元素位置直接进行加减法运算,矩阵形状不发生改变;
2、乘法:左行乘右列(矩阵能够进行乘法的前提是:矩阵的左行等于右列);

三、矩阵运算相关性质

矩阵运算不一定满足交换律: A B ≠ B A AB \not= BA AB=BA

数乘分配律:
( λ + μ ) A = λ A + μ A (\lambda + \mu)A = \lambda A + \mu A (λ+μ)A=λA+μA

λ ( A + B ) = λ A + λ B \lambda(A+ B) = \lambda A + \lambda B λ(A+B)=λA+λB

矩阵分配律:
( A B ) C = A ( B C ) (AB)C = A(BC) (AB)C=A(BC)

A ( B + C ) A B + A C A(B+C) AB + AC A(B+C)AB+AC

( B + C ) A = B A + C A (B + C)A = BA + CA (B+C)A=BA+CA

E A = A E = A EA =AE =A EA=AE=A

转置相关性质:
( A T ) T = A (A^T)^T = A (AT)T=A

( A + B ) T = A T + B T (A + B)^T = A^T + B^T (A+B)T=AT+BT

( A B ) T = B T A T (AB)^T = B^TA^T (AB)T=BTAT(有些特殊)

模的性质:
∣ A ⋅ B ∣ = ∣ A ∣ ⋅ ∣ B ∣ |A \cdot B| = |A| \cdot |B| AB=AB

∣ λ A ∣ = λ n ∣ A ∣ |\lambda A| = \lambda^n|A| λA=λnA n n n 为矩阵 A A A 的阶数)