Matrix Algebra - Cellular matrices
Let's consider some matrix A and we’ll split it into matrices of lower order:
which are called cells or blocks.
Here cells (blocks) are matrices:
Now the matrix A can be considered as cellular or block matrix:
which elements are cells (blocks).
Apparently, that splitting of any matrix into cells (blocks) is maybe executed by various ways. In that specific case the cellular matrix can be quasi-diagonal one:
where cells– square matrices (generally speaking, of different orders), and outside of cells zeros are. Note, that
Cellular matrices of the same dimensions and with identical splitting are called conform.
Operations with cellular matrices are carried out by the same rules, as with usual matrices.
1. Addition and subtraction of cellular matrices
Let there are two conform cellular matrices:
where p = r, q = s and cells of identical dimension. Then
Subtraction of cellular matrices is carried out similarly.
2. Multiplication of cellular matrices
Multiplication of a cellular matrix to a number (scalar)
Let A – a cellular matrix and h – a number, then we have:
Multiplication of cellular matrices
Let's consider two conform cellular matrices:
and q = r .
Let all cells such, that a number of columns of a cell is equal to a number of rows of a cell (For example, apparently, that it takes place in that specific case, when all cells – square matrices and have also the same order). Then it is easy to show, that a product of matrices A and B is too a cellular matrix:
where that is multiplication of cellular matrices is similar to multiplication of numerical matrices .
E x a m p l e . Multiply the cellular matrices
S o l u t i o n .