# 1.2. 矩阵

## 1.2. 矩阵

(1.13)

#### 1.2.2. 矩阵运算

1.加减法

(1.14)

1. 结合律：
2. 交换律：

\begin{eqnarray}
\left( {\begin{array}{*{20}{c}}{ – {a_{1,1}}}&{ – {a_{1,2}}}& \cdots &{ – {a_{1,n}}}\\
{ – {a_{2,1}}}&{ – {a_{2,2}}}& \ldots &{ – {a_{2,n}}}\\
\vdots & \vdots & \ddots & \vdots \\
{ – {a_{m,1}}}&{ – {a_{m,2}}}& \ldots &{ – {a_{m,n}}}\end{array}} \right)
\tag{1.15}
\end{eqnarray}

2. 乘法

\begin{eqnarray}
{c_{i,j}} = {a_{i,1}}{b_{1,j}} + {a_{i,2}}{b_{2,j}} + \ldots + {a_{i,n}}{b_{n,j}} = \sum\limits_{k = 1}^n {{a_{i,k}}{b_{k,j}}}
\tag{1.16}
\end{eqnarray}

3. 数量乘法

\begin{eqnarray}
\left( {\begin{array}{*{20}{c}}{k{a_{1,1}}}&{k{a_{1,2}}}& \cdots &{k{a_{1,n}}}\\
{k{a_{2,1}}}&{k{a_{2,2}}}& \ldots &{k{a_{2,n}}}\\
\vdots & \vdots & \ddots & \vdots \\
{k{a_{m,1}}}&{k{a_{m,2}}}& \ldots &{k{a_{m,n}}}\end{array}} \right)
\tag{1.17}
\end{eqnarray}

\begin{eqnarray}
\left( {\begin{array}{*{20}{c}}k&0& \cdots &0\\
0&k& \ldots &0\\
\vdots & \vdots & \ddots & \vdots \\
0&0& \ldots &k\end{array}} \right)
\tag{1.18}
\end{eqnarray}

4. 转置

\begin{eqnarray}
{A^T} = \left( {\begin{array}{*{20}{c}}{{a_{1,1}}}&{{a_{2,1}}}& \cdots &{{a_{m,1}}}\\
{{a_{1,2}}}&{{a_{2,2}}}& \ldots &{{a_{m,2}}}\\
\vdots & \vdots & \ddots & \vdots \\
{{a_{1,n}}}&{{a_{2,n}}}& \ldots &{{a_{m,n}}}\end{array}} \right)
\tag{1.19}
\end{eqnarray}