Page 117 - Matrix Analysis & Applied Linear Algebra

3.6 Properties of Matrix Multiplication 111

Executing multiplication between two matrices by partitioning one or both
factors into submatrices—a matrix contained within another matrix—can be
a useful technique.

Block Matrix Multiplication
Suppose that A and B are partitioned into submatrices—often referred
to as blocks—as indicated below.

   
A 11 A 12 ··· A 1r B 11 B 12 ··· B 1t
 A 21 A 22 ··· A 2r   B 21 B 22 ··· B 2t 
A =  . . . .  , B =  . . . .  .
 . . . .   . . . . 
. . . . . . . .
A s1 A s2 ··· A sr B r1 B r2 ··· B rt
If the pairs (A ik , B kj ) are conformable, then A and B are said to
be conformably partitioned. For such matrices, the product AB is
formed by combining the blocks exactly the same way as the scalars are
combined in ordinary matrix multiplication. That is, the (i, j) -block in
AB is
A i1 B 1j + A i2 B 2j + ··· + A ir B rj .

Although a completely general proof is possible, looking at some examples
better serves the purpose of understanding this technique.
Example 3.6.6
Block multiplication is particularly useful when there are patterns in the matrices
to be multiplied. Consider the partitioned matrices
12 10 10 00
   
 34 01  C I  01 00  I 0
A =   = , B =   = ,
10 00 12 12
  I 0   CC
01 00 34 34
where

10 12
I = and C = .
01 34
Using block multiplication, the product AB is easily computed to be
24 12
 
 68
C I I 0 2CC 34 
AB = = =   .
I 0 CC I 0  10 00 
01 00