conformal partitioning

Let $R$ be a ring. Let the matrices $A\in M_{m,n}(R)$ and $B\in M_{n,p}(R)$ be partitioned into submatrices $A^{i,j}$ and $B^{i,j}$ respectively as follows:

$A=\begin{matrix}\begin{matrix}n_{1}&n_{2}&\quad\cdots&n_{h}\end{matrix}&\begin% {matrix}\end{matrix}\\ \left[\begin{matrix}\overbrace{A^{1,1}}&\overbrace{A^{1,2}}&\cdots&\overbrace{% A^{1,h}}\\ A^{2,1}&A^{2,2}&\cdots&A^{2,h}\\ \vdots&\vdots&&\vdots\\ A^{g,1}&A^{g,2}&\cdots&A^{g,h}\end{matrix}\right]&\begin{matrix}\}m_{1}\\ \}m_{2}\\ \vdots\\ \}m_{g}\end{matrix}\end{matrix}$

where $A^{i,j}$ is $m_{i}\times n_{j},\sum_{i=1}^{g}m_{i}=m$ , $\sum_{j=1}^{h}n_{j}=n$ ;

$B=\begin{matrix}\begin{matrix}p_{1}&p_{2}&\quad\cdots&p_{k}\end{matrix}&\begin% {matrix}\end{matrix}\\ \left[\begin{matrix}\overbrace{B^{1,1}}&\overbrace{B^{1,2}}&\cdots&\overbrace{% B^{1,k}}\\ B^{2,1}&B^{2,2}&\cdots&B^{2,k}\\ \vdots&\vdots&&\vdots\\ B^{h,1}&B^{h,2}&\cdots&B^{h,k}\end{matrix}\right]&\begin{matrix}\}n_{1}\\ \}n_{2}\\ \vdots\\ \}n_{h}\end{matrix}\end{matrix}$

where $B^{i,j}$ is $n_{i}\times p_{j},$ $\sum_{j=1}^{k}p_{j}=p$ . Then $A$ and $B$ (in this ) are said to be conformally partitioned for multiplication.

Now suppose that $A$ and $B$ are conformally partitioned for multiplication. Let $C=AB$ be partitioned as follows:

$C=\begin{matrix}\begin{matrix}p_{1}&p_{2}&\quad\cdots&p_{k}\end{matrix}&\begin% {matrix}\end{matrix}\\ \left[\begin{matrix}\overbrace{C^{1,1}}&\overbrace{C^{1,2}}&\cdots&\overbrace{% C^{1,k}}\\ C^{2,1}&C^{2,2}&\cdots&C^{2,k}\\ \vdots&\vdots&&\vdots\\ C^{g,1}&C^{g,2}&\cdots&C^{g,k}\end{matrix}\right]&\begin{matrix}\}m_{1}\\ \}m_{2}\\ \vdots\\ \}m_{g}\end{matrix}\end{matrix}$

where $C^{i,j}$ is $m_{i}\times p_{j}$ , $i=1,\cdots,g$ , $j=1,\cdots,k$ . Then

$C^{i,j}=\sum_{t=1}^{k}A^{i,t}B^{t,j},\quad i=1,\cdots,g,\quad j=1,\cdots,k.$

This method of computing $A B$ is sometimes called block multiplication.

Title	conformal partitioning
Canonical name	ConformalPartitioning
Date of creation	2013-03-22 16:04:16
Last modified on	2013-03-22 16:04:16
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	9
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 15-00
Defines	block multiplication