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:

$$A=\begin{matrix} \begin{matrix} n_{1} & n_{2} & \quad \cdots & n_{h} \end{matrix} & \begin{matrix} \quad \end{matrix} \\ \left[ \begin{matrix} \overbrace{A^{1,1}} & \overbrace{A^{1,2}} & \cdots & \overbrace{A^{1,h}} \\ A^{2,1} & A^{2,2} & \cdots & A^{2,h} \\ \vdots & \vdots & \ \ & \vdots \\ A^{g,1} & A^{g,2} & \cdots & A^{g,h} \end{matrix} \right] & \begin{matrix} \}m_{1} \\ \}m_{2} \\ \vdots \\ \}m_{g} \end{matrix} \end{matrix} $$ where $A^{i,j}$ is $m_{i} \times n_{j}, \sum_{i=1}^{g} m_{i} = m$ $\sum_{j=1}^{h} n_{j} = n$ $$B=\begin{matrix} \begin{matrix} p_{1} & p_{2} & \quad \cdots & p_{k} \end{matrix} & \begin{matrix} \quad \end{matrix} \\ \left[ \begin{matrix} \overbrace{B^{1,1}} & \overbrace{B^{1,2}} & \cdots & \overbrace{B^{1,k}} \\ B^{2,1} & B^{2,2} & \cdots & B^{2,k} \\ \vdots & \vdots & \ \ & \vdots \\ B^{h,1} & B^{h,2} & \cdots & B^{h,k} \end{matrix} \right] & \begin{matrix} \}n_{1} \\ \}n_{2} \\ \vdots \\ \}n_{h} \end{matrix} \end{matrix} $$ where $B^{i,j}$ is $n_{i} \times p_{j}, $ $\sum_{j=1}^{k} p_{j} = p$ Then $A$ and $B$ (in this order) 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: $$C=\begin{matrix} \begin{matrix} p_{1} & p_{2} & \quad \cdots & p_{k} \end{matrix} & \begin{matrix} \quad \end{matrix} \\ \left[ \begin{matrix} \overbrace{C^{1,1}} & \overbrace{C^{1,2}} & \cdots & \overbrace{C^{1,k}} \\ C^{2,1} & C^{2,2} & \cdots & C^{2,k} \\ \vdots & \vdots & \ \ & \vdots \\ C^{g,1} & C^{g,2} & \cdots & C^{g,k} \end{matrix} \right] & \begin{matrix} \}m_{1} \\ \}m_{2} \\ \vdots \\ \}m_{g} \end{matrix} \end{matrix} $$ 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 $AB$ is sometimes called block multiplication.