Multiplication of blockmatrices

Question

For my university studies I was given this statement to prove:

$\begin {pmatrix} A & B \\ C & D\end {pmatrix}\begin {pmatrix} W & X \\ Y & Z\end {pmatrix} = \begin {pmatrix} AW + BY & AX +BZ \\ CW + DY & CX + DZ\end {pmatrix}\ $

Where:

$A,W \in R^{n \times n}, \,\, D,Z \in R^{m \times m} \,\, B,X \in R^{n \times m} \,\, and \,\, C,Y \in R^{m \times n}$

My problem ist that I have no clue how to even start my proof. Since I'm very confused by that many elements from different matices inside these blockmatrices.

Answer 1

You can partition as follows: $$ \left[ \begin{array}{ccc|cc} a_{11}&\cdots&a_{1n}&b_{11}&\cdots&b_{1m}\\ \vdots&&\vdots&\vdots&&\vdots\\ a_{n1}&\cdots&a_{nn}&b_{n1}&\cdots&b_{nm}\\ c_{11}&\cdots&c_{1n}&d_{11}&\cdots&d_{1m}\\ \vdots&\cdots&\vdots&\vdots&&\vdots\\ c_{m1}&\cdots&c_{mn}&d_{m1}&\cdots&d_{mm}\\ \end{array} \right]\cdot \left[ \begin{array}{ccccc} w_{11}&\cdots&w_{1n}&x_{11}&\cdots&x_{1m}\\ \vdots&&\vdots&\vdots&&\vdots\\ w_{n1}&\cdots&w_{nn}&x_{n1}&\cdots&x_{nm}\\ \hline y_{11}&\cdots&y_{1n}&z_{11}&\cdots&z_{1m}\\ \vdots&\cdots&\vdots&\vdots&&\vdots\\ y_{m1}&\cdots&y_{mn}&z_{m1}&\cdots&z_{mm}\\ \end{array} \right]= \\ \left[ \begin{array}{ccc} a_{11}&\cdots&a_{1n}\\ \vdots&&\vdots\\ a_{n1}&\cdots&a_{nn}\\ \hline c_{11}&\cdots&c_{1n}\\ \vdots&\cdots&\vdots\\ c_{m1}&\cdots&c_{mn}\\ \end{array} \right]\cdot \left[ \begin{array}{ccc|cc} w_{11}&\cdots&w_{1n}&x_{11}&\cdots&x_{1m}\\ \vdots&&\vdots&\vdots&&\vdots\\ w_{n1}&\cdots&w_{nn}&x_{n1}&\cdots&x_{nm}\\ \end{array} \right]+\\ \left[ \begin{array}{ccc} b_{11}&\cdots&b_{1n}\\ \vdots&&\vdots\\ b_{n1}&\cdots&b_{nn}\\ \hline d_{11}&\cdots&d_{1n}\\ \vdots&\cdots&\vdots\\ d_{m1}&\cdots&d_{mn}\\ \end{array} \right]\cdot \left[ \begin{array}{ccc|cc} y_{11}&\cdots&y_{1n}&z_{11}&\cdots&z_{1m}\\ \vdots&&\vdots&\vdots&&\vdots\\ y_{n1}&\cdots&y_{nn}&z_{n1}&\cdots&z_{nm}\\ \end{array} \right]= \\ \left[ \begin{array}{c|c} AW&AX\\ \hline CW&CX\\ \end{array} \right]+ \left[ \begin{array}{c|c} BY&BZ\\ \hline DY&DZ\\ \end{array} \right]=\\ \left[ \begin{array}{c|c} AW+BY&AX+BZ\\ \hline CW+DY&CX+DZ\\ \end{array} \right]=\left[ \begin{array}{cc} AW+BY&AX+BZ\\ CW+DY&CX+DZ\\ \end{array} \right].$$

Refer to this source:

When two block matrices have the same shape and their diagonal blocks are square matrices, then they multiply similarly to matrix multiplication.

Answer 2

Just refer to the definition for matrix product, that is if $A$ is an $n\times m$ matrix and $B$ is an $m\times p$ matrix, then $C=A\cdot B$ is an $n\times p$ matrix with

$$c_{ij} = a_{i1}b_{1j} +\cdots + a_{im}b_{mj}= \sum_{k=1}^m a_{ik}b_{kj}$$

and by linearity

$$\begin {pmatrix} A & B \\ C & D\end {pmatrix}\begin {pmatrix} W & X \\ Y & Z\end {pmatrix}=\begin {pmatrix} A & B \\ C & D\end {pmatrix}\begin {pmatrix} W & 0 \\ 0 & 0\end {pmatrix} + \begin {pmatrix} A & B \\ C & D\end {pmatrix}\begin {pmatrix} 0 & X \\ 0 & 0\end {pmatrix}+$$$$+\begin {pmatrix} A & B \\ C & D\end {pmatrix}\begin {pmatrix} 0 & 0 \\ Y & 0\end {pmatrix}+\begin {pmatrix} A & B \\ C & D\end {pmatrix}\begin {pmatrix} 0 & 0 \\ 0 & Z\end {pmatrix} $$