# 线性代数/矩阵的运算与变换

## 矩陣的性質

${\displaystyle ((AB)C)_{ij}=\sum _{l=1}^{p}(AB)_{il}C_{lj}=\sum _{l=1}^{p}(\sum _{k=1}^{n}A_{ik}B_{kl})C_{lj}=\sum _{l=1}^{p}\sum _{k=1}^{n}(A_{ik}B_{kl}C_{lj})}$

## 特殊方陣

### 零方陣

${\displaystyle O_{n}=[0]_{n\times n}={\begin{bmatrix}0&0&\dots &0\\0&0&\dots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\dots &0\end{bmatrix}}}$

### 單位方陣

${\displaystyle I_{n}=[\delta _{ij}]_{n\times n}={\begin{bmatrix}1&0&\dots &0\\0&1&\dots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\dots &1\end{bmatrix}}}$

${\displaystyle \delta _{ij}={\begin{cases}1&{\text{ if }}i=j\\0&{\text{ if }}i\neq j\end{cases}}}$

## 向量

${\displaystyle x={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}}$

{\displaystyle \left\{{\begin{aligned}a_{11}x_{1}+a_{12}x_{2}+\dots +a_{1n}x_{n}&=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\dots +a_{2n}x_{n}&=b_{2}\\&\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\dots +a_{mn}x_{n}&=b_{m}\end{aligned}}\right.}

${\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&\dots &a_{1n}\\a_{21}&a_{22}&\dots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\dots &a_{mn}\end{bmatrix}}}$ ${\displaystyle x={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}}$ ${\displaystyle b={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}}$

${\displaystyle Ax=b}$

${\displaystyle A}$ ${\displaystyle b}$  是已知數，${\displaystyle x}$  是未知數。要特別注意的是此時的增廣矩陣是${\displaystyle {\begin{bmatrix}A&b\end{bmatrix}}}$ ，跟一次聯主方程式與增廣矩陣中的符號不要產生混淆了。