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 intosubmatrices $A^{i,j}$ 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 beconformally 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.