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Multiplication of block/partitioned matrices

Created over 8 years ago, updated 10 days ago

Theorem. Block matrices can be multiplied as if the blocks were scalar entries, provided the block dimensions are compatible.

If $A$ and $B$ are partitioned conformally:
$$A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}, \quad B = \begin{pmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{pmatrix}$$
then:
$$AB = \begin{pmatrix} A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\ A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22} \end{pmatrix}$$

The column partitioning of $A$ must match the row partitioning of $B$ for the block products to be defined.