Definition of block/partitioned matrix

Definition. A block matrix (or partitioned matrix) is a matrix whose entries are themselves matrices, called blocks.

If we partition an $m \times n$ matrix $A$ by grouping rows and columns, we can write:
$$A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}$$
where each $A_{ij}$ is a submatrix.

Example. The $4 \times 4$ matrix can be partitioned into $2 \times 2$ blocks:
$$\begin{pmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{pmatrix} = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}$$
where $A_{11} = \begin{pmatrix} 1 & 2 \\ 5 & 6 \end{pmatrix}$, $A_{12} = \begin{pmatrix} 3 & 4 \\ 7 & 8 \end{pmatrix}$, etc.