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分块矩阵的概念
日期:
2023-01-01 19:59
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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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