Block matrices

Definition: A block matrix (or partitioned matrix) is a matrix whose entries are themselves matrices (called blocks or submatrices). It is written by partitioning the rows and columns of a larger matrix:

$$M = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$$

where $A, B, C, D$ are submatrices.

Example:
$$M = \begin{pmatrix} 1 & 2 & | & 3 & 4 \\ 5 & 6 & | & 7 & 8 \\ - & - & & - & - \\ 9 & 10 & | & 11 & 12 \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$$

Properties:

• Block addition: $\begin{pmatrix} A & B \\ C & D \end{pmatrix} + \begin{pmatrix} E & F \\ G & H \end{pmatrix} = \begin{pmatrix} A+E & B+F \\ C+G & D+H \end{pmatrix}$
• Block multiplication: follows standard matrix multiplication rules with blocks
• Block diagonal matrices: $\begin{pmatrix} A & 0 \\ 0 & D \end{pmatrix}$ have determinant $\det(A)\det(D)$

Block matrices are useful for organizing large systems and proving theoretical results.