# 线性代数笔记（4）——矩阵的运算

## 线性代数笔记（4）——矩阵的运算

Contents

### 高斯消元·续

#### 消去矩阵

$$E_{21}(-3) = \begin{pmatrix} 1 & 0 & 0 \\ -3 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} (a_{2,1} = -3)$$

#### 矩阵乘法

##### 定义

$$A_{m \times n} B_{n \times p} = A(\mathbf{b_1},\mathbf{b_2},\cdots,\mathbf{b_p}) := (A \mathbf{b_1},A \mathbf{b_2},\cdots,A \mathbf{b_p})$$

#### 置换阵

$$P_{12} = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$$

#### 第三种初等矩阵

$$D_2(c) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & c & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}(c \not = 0)$$

### 矩阵的运算

#### 矩阵的加法

$A + B := (a_{i,j} + B_{i,j})_{m \times n}$

#### 矩阵加法和数乘满足的运算法则

1. $A + B = B + A$
2. $(A + B) + C = A + (B + C)$
3. $\mathbf{0} + A = A$
4. 记 $-A = (-a_{i,j})_{m \times n},A + (-A) = \mathbf{0}$
5. $1 \cdot A = A$
6. $(kl)A = k(lA)$
7. $k(A + B) = kA + kB$
8. $(k + l)A = kA + lA$

#### 矩阵的减法

$A – B := A + (-B)$

#### 矩阵的乘法

$(AB)_{ij} = (A \mathbf{b}_j)_i = a_{i,1}b_{1,j} + a_{i,2}b_{2,j} + \cdots + a_{i,n}b_{n,j} = \sum\limits_{k = 1}^{n} a_{i,k}b_{k,j}$