Answer:
We present and practice block matrix multiplication.
MAT-0023: Block Matrix Multiplication
It is often useful to consider matrices whose entries are themselves matrices,
called blocks. A matrix viewed in this way is said to be partitioned into
blocks
For example, writing a matrix in the form
is such a block partition of . Here is another example.Consider the matrices
where the blocks have been labelled as indicated. This is a natural way to partition into blocks in view of the blocks and the two-by-three zero matrix, denoted by , that occur. This notation is particularly useful when we are multiplying the matrices and because the product can be computed in block form as follows: This is easily checked to be the product , computed in the conventional manner.In other words, we can compute the product by ordinary matrix multiplication, using blocks as entries. The only requirement is that the blocks be compatible. That is, the sizes of the blocks must be such that all matrix products of blocks that occur make sense. This means that the number of columns in each block of must equal the number of rows in the corresponding block of .
If matrices and are partitioned compatibly into blocks, the product can be
computed by matrix multiplication using blocks as entries.
We omit the proof.
Block multiplication has theoretical uses as we shall see. However, it is also useful in computing products of matrices in a computer with limited memory capacity. The matrices are partitioned into blocks in such a way that each product of blocks can be handled. Then the blocks are stored in auxiliary memory and their products are computed one by one.
Practice Problems
Text Source
The text in this module is an adaptation of Section 2.3 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)
W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, p 73-74.