分块矩阵的概念

日期: 2023-01-01 19:59 查看: 73 次

对于行数和列数较高的矩阵 $A$ ，运算时常用一些横线和坚线将矩阵 $A$ 分划成若干个小 矩阵，每一个小矩阵称为 $A$ 的子块，以子块为元素的形式上的矩阵称为分块矩阵.
一个矩阵的分块方式会有很多种， 例如，

$$
A=\left(\begin{array}{cc:ccc}
a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\
\hdashline a_{31} & a_{32} & \ldots \ldots & a_{33} & a_{35} \\
\hdashline a_{41} & a_{42} & a_{43} & a_{44} & a_{45}
\end{array}\right)
$$

记为
其中

$$
A_{11}=\left(\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right), \quad A_{12}=\left(\begin{array}{lll}
a_{13} & a_{14} & a_{15} \\
a_{23} & a_{24} & a_{25}
\end{array}\right), \quad A_{21}=\left(\begin{array}{ll}
a_{31} & a_{32} \\
a_{41} & a_{42}
\end{array}\right), \quad A_{22}=\left(\begin{array}{lll}
a_{33} & a_{34} & a_{35} \\
a_{43} & a_{44} & a_{45}
\end{array}\right) .
$$

$$
A=\left(\begin{array}{cc:cc:c}
a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
\hdashline a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\
a_{31} & a_{33} & a_{33} & a_{34} & a_{35} \\
\hdashline a_{41} & a_{42} & a_{43} & a_4 & a_{45} .
\end{array}\right)
$$

记为
$A=\left(\begin{array}{lll}A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33}\end{array}\right)$, $A=\left(A_{11}, A_{12}, A_{13}, A_{14}, A_{15}\right),$
其中

$$
\begin{aligned}
& \boldsymbol{A}_{11}=\left(a_{11}, a_{12}\right), \quad \boldsymbol{A}_{12}=\left(a_{13}, a_{14}\right), \quad \boldsymbol{A}_{13}=a_{15}, \\
& \boldsymbol{A}_{21}=\left(\begin{array}{ll}
a_{21} & a_{22} \\
a_{31} & a_{32}
\end{array}\right), \quad \boldsymbol{A}_{22}=\left(\begin{array}{ll}
a_{23} & a_{24} \\
a_{33} & a_{34}
\end{array}\right), \quad \boldsymbol{A}_{23}=\left(\begin{array}{l}
a_{25} \\
a_{35}
\end{array}\right) \\
& \boldsymbol{A}_{31}=\left(a_{41}, a_{42}\right), \quad \boldsymbol{A}_{32}=\left(a_{43}, a_{44}\right), \quad \boldsymbol{A}_{33}=a_{45} .
\end{aligned}
$$

其中

$$
\boldsymbol{A}_{11}=\left(\begin{array}{l}
a_{11} \\
a_{21} \\
a_{31} \\
a_{41}
\end{array}\right), \quad \boldsymbol{A}_{12}=\left(\begin{array}{l}
a_{12} \\
a_{22} \\
a_{32} \\
a_{42}
\end{array}\right), \quad \boldsymbol{A}_{13}=\left(\begin{array}{l}
a_{13} \\
a_{23} \\
a_{33} \\
a_{43}
\end{array}\right), \quad \boldsymbol{A}_{14}=\left(\begin{array}{l}
a_{14} \\
a_{24} \\
a_{34} \\
a_{44}
\end{array}\right), \quad \boldsymbol{A}_{15}=\left(\begin{array}{l}
a_{15} \\
a_{25} \\
a_{35} \\
a_{45}
\end{array}\right) .

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