矩阵分块法

对于行数和列数比较高的矩阵A，运算时通常采用分块法，使大矩阵的运算化成小矩阵的运算
我们将矩阵A用若干条纵线和横线分成许多个小子块，每个小字块被称为矩阵A的子块，以字块为元素的形式上的矩阵被称为分块矩阵

$$\left [\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14} \\a_{21} & a_{22} & a_{23} & a_{24} \\a_{31} & a_{32} & a_{33} & a_{34}\end{matrix}\right ]$$

i分块后的矩阵记

$$\left [\begin{matrix}A_{11} & A_{12} \\A_{21} & A_{22}\end{matrix}\right ]$$

• 设矩阵A与B的行数与列数相同。采用相同的分块法，分解后的子块为

$$\bold A =\left [\begin{matrix}A_{11} & A_{12} & \cdots & A_{1n} \\A_{21} & A_{22} & \cdots & A_{2n} \\\vdots & \vdots & \ddots & \vdots \\A_{m1} & A_{m2} & \cdots & A_{mn}\end{matrix}\right ]$$

$$\bold B =\left [\begin{matrix}B_{11} & B_{12} & \cdots & B_{1n} \\B_{21} & B_{22} & \cdots & B_{2n} \\\vdots & \vdots & \ddots & \vdots \\B_{m1} & B_{m2} & \cdots & B_{mn}\end{matrix}\right ]$$

$$\bold A + \bold B =\left [\begin{matrix}A_{11} + B_{11} & A_{12} + B_{12} & \cdots &A_{1n} + B_{1n} \\A_{21} + B_{21} & A_{22} + B_{22} & \cdots &A_{2n} + B_{2n} \\\vdots & \vdots & \ddots & \vdots \\A_{m1} + B_{m1} & A_{m2} + B_{m2} & \cdots &A_{mn} + B_{mn}\end{matrix}\right ]$$

• $\lambda$为实数

$$\lambda \bold A =\left [\begin{matrix}\lambda A_{11} & \lambda A_{12} & \cdots & \lambda A_{1n} \\\lambda A_{21} & \lambda A_{22} & \cdots & \lambda A_{2n} \\\vdots & \vdots & \ddots & \vdots \\\lambda A_{m1} & \lambda A_{m2} & \cdots & \lambda A_{mn}\end{matrix}\right ]$$

• 分块矩阵的转置

$$\bold A^T =\left [\begin{matrix}A_{11}^T & A_{12}^T & \cdots & A_{1m}^T \\A_{21}^T & A_{22}^T & \cdots & A_{2m}^T \\\vdots & \vdots & \ddots & \vdots \\A_{n1}^T & A_{n2}^T & \cdots & A_{nm}^T\end{matrix}\right ]$$

因计算机实现暂时不涉及分块矩阵，因此暂不对分块矩阵的类型进行详细介绍