Summary Block multiplication of matrices

Block multiplication of matrices

In certain situations it is prudent to treat arrays as blocks of smaller arrays, called sub-
arrays, and then multiply block by block rather than component by component. Block
multiplication is very similar to normal matrix multiplication.

Example #1: Block multiplication

consider the product


( )( )


The reader should verify that this product is defined. Now partition these matrices using
dotted lines.


( )( ) ( )( )


There are other ways to form the partition. In this case

( ) ( )

and so on. Assuming that all matrix products and sums are defined, you can multiply
normally to get

( )( ) ( )


Now

( ) ( ) ( ) ( )
( )( ) ( )
( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )
( )( ) ( )
( ) ( ) ( ) ( )

( ) ( )

, And

( )( ) ( )


( )

Similarly

( ) ( )
( )( ) ( ) ( )
( ) ( )

( ) ( )
( )( ) ( ) ( )
( ) ( )

( ) ( ) ( )


( )

For CH+DK and EG+FJ we will do the same procedure as follows:

( ) ( ) ( ) ( )
( )( ) ( )
( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )
( )( ) ( )
( ) ( ) ( ) ( )

( ) ( )



( )( ) ( )


( )