按行分块和按列分块
前置知识：
- 【定义】分块矩阵
- 分块矩阵的运算规则
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m×n 矩阵
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\boldsymbol{A}
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A 的
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\boldsymbol{a}_j = \begin{pmatrix} a_{1j} \\ a_{2j} \\ \vdots \\ a_{mj} \end{pmatrix}
aj=⎝
⎛a1ja2j⋮amj⎠
⎞
则
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\boldsymbol{A}
A 可按列分块为
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\boldsymbol{A} = (\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n)
A=(a1,a2,⋯,an)
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m \times n
m×n 矩阵
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\boldsymbol{A}
A 有
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m 行，称为矩阵
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\boldsymbol{A}
A 的
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\boldsymbol{\alpha}_i^T = (a_{i1},a_{i2},\cdots,a_{in})
αiT=(ai1,ai2,⋯,ain)
则
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\boldsymbol{A}
A 可按行分块为
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\boldsymbol{A} = \begin{pmatrix} \boldsymbol{\alpha}_1^T \\ \boldsymbol{\alpha}_2^T \\ \vdots \\ \boldsymbol{\alpha}_m^T \\ \end{pmatrix}
A=⎝
⎛α1Tα2T⋮αmT⎠
⎞
（1）通过矩阵的按行分块和按列分块，可以直观地表示，在矩阵乘法 A B = C = ( c i j ) m × n \boldsymbol{A} \boldsymbol{B} = \boldsymbol{C} = (c_{ij})_{m \times n} AB=C=(cij)m×n 中， c i j c_{ij} cij 为 A \boldsymbol{A} A 中的一个行向量与 B \boldsymbol{B} B 中的一个列向量的乘积。
对于矩阵
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\boldsymbol{A} = (a_{ij})_{m \times s}
A=(aij)m×s 与矩阵
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\boldsymbol{B} = (b_{ij})_{s \times n}
B=(bij)s×n 的乘积矩阵
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\boldsymbol{A} \boldsymbol{B} = \boldsymbol{C} = (c_{ij})_{m \times n}
AB=C=(cij)m×n，若把
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\boldsymbol{A}
A 按行分成
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m 块，将
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\boldsymbol{B}
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\boldsymbol{A} \boldsymbol{B} = \begin{pmatrix} \boldsymbol{\alpha}_1^T \\ \boldsymbol{\alpha}_2^T \\ \vdots \\ \boldsymbol{\alpha}_m^T \end{pmatrix} (\boldsymbol{b}_1, \boldsymbol{b}_2, \cdots, \boldsymbol{b}_n) = \begin{pmatrix} \boldsymbol{\alpha}_1^T \boldsymbol{b}_1 & \boldsymbol{\alpha}_1^T \boldsymbol{b}_2 & \cdots & \boldsymbol{\alpha}_1^T \boldsymbol{b}_n \\ \boldsymbol{\alpha}_2^T \boldsymbol{b}_1 & \boldsymbol{\alpha}_2^T \boldsymbol{b}_2 & \cdots & \boldsymbol{\alpha}_2^T \boldsymbol{b}_n \\ \vdots & \vdots & & \vdots \\ \boldsymbol{\alpha}_m^T \boldsymbol{b}_1 & \boldsymbol{\alpha}_m^T \boldsymbol{b}_2 & \cdots & \boldsymbol{\alpha}_m^T \boldsymbol{b}_n \\ \end{pmatrix} = (c_{ij})_{m \times n}
AB=⎝
⎛α1Tα2T⋮αmT⎠
⎞(b1,b2,⋯,bn)=⎝
⎛α1Tb1α2Tb1⋮αmTb1α1Tb2α2Tb2⋮αmTb2⋯⋯⋯α1Tbnα2Tbn⋮αmTbn⎠
⎞=(cij)m×n
其中
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c_{ij} = \boldsymbol{\alpha}_i^T \boldsymbol{b}_j = (a_{i1},a_{i2},\cdots,a_{is}) \begin{pmatrix} b_{1j} \\ b_{2j} \\ \vdots \\ b_{sj} \end{pmatrix} = \sum_{k=1}^s a_{ik} b_{kj}
cij=αiTbj=(ai1,ai2,⋯,ais)⎝
⎛b1jb2j⋮bsj⎠
⎞=k=1∑saikbkj
（2）通过将矩阵乘法中的矩阵按行分块和按列分块，可以得到线性方程组的另一个表达式。
已知线性方程组
{
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(1)
\begin{cases} a_{11} x_1 + a_{12} x_2 + \cdots a_{1n} x_n = b_1 \\ a_{21} x_1 + a_{22} x_2 + \cdots a_{2n} x_n = b_2 \\ \cdots \\ a_{m1} x_1 + a_{m2} x_2 + \cdots a_{mn} x_n = b_m \end{cases} \tag{1}
⎩
⎨
⎧a11x1+a12x2+⋯a1nxn=b1a21x1+a22x2+⋯a2nxn=b2⋯am1x1+am2x2+⋯amnxn=bm(1)
它的矩阵乘积形式为
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\boldsymbol{A}_{m \times n} \boldsymbol{x}_{n \times 1} = \boldsymbol{b}_{m \times 1}
Am×nxn×1=bm×1
将上式中
A
\boldsymbol{A}
A 按列分块，
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\boldsymbol{x}
x 按行分块，由分块矩阵的乘法有
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(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n) \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} = \boldsymbol{b}
(a1,a2,⋯,an)⎝
⎛x1x2⋮xn⎠
⎞=b
即
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x_1 \boldsymbol{a}_1 + x_2 \boldsymbol{a}_2 + \cdots x_n \boldsymbol{a}_n = \boldsymbol{b}
x1a1+x2a2+⋯xnan=b
这等价于将方程组表示为
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\begin{pmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{pmatrix} x_1 + \begin{pmatrix} a_{12} \\ a_{22} \\ \vdots \\ a_{m2} \end{pmatrix} x_2 + \cdots + \begin{pmatrix} a_{1n} \\ a_{2n} \\ \vdots \\ a_{mn} \end{pmatrix} x_n = \begin{pmatrix} b_{1} \\ b_{2} \\ \vdots \\ b_{m} \end{pmatrix}
⎝
⎛a11a21⋮am1⎠
⎞x1+⎝
⎛a12a22⋮am2⎠
⎞x2+⋯+⎝
⎛a1na2n⋮amn⎠
⎞xn=⎝
⎛b1b2⋮bm⎠
⎞