Definition 2.2.1. A matrix which elements different from zero are only on the main and some adjacent diagonals is called a band matrix.
Definition 2.2.2. It is
said that the matrix 
is a band matrix with the lower bandwidth p if
and with the upper bandwidth q if
and with the bandwidth p+q+1.
Example 2.2.1. The matrix
is a band matrix because all the elements different
from zero are on the main and two lower and one upper diagonals. The lower
bandwidth of the matrix A is 2 because aik=0
as i>k+2, and the upper bandwidth
is 1 because aik=0 as k>i+1. The bandwidth
of the matrix is 2+1+1=4. The elements of the matrix that are necessarily
not zeros are denoted by crosses.
Some most important types of band
matrices are presented in table 2.2.1. If 
is a diagonal matrix, 
and 
then the notation 
will be used.
Table 2.2.1.
Problem 2.2.1.* Find the type, lower
bandwidth, upper bandwidth and bandwidth
of the matrix A if
Definition 2.2.3. A matrix
is called a 
block
matrix if
where 
and 
and 
is a 
matrix.
Example 2.2.2. The matrix
is a 
block
matrix, where 
and n2=2 and
Let
and C=A+B. Then
Proposition 2.2.1.
If 
and C=AB are block matrices:
where 
, then
Proof. Let
As 
is an element of the block 
of the matrix C standing in the i-th row and k-th
column of this block, and 
is an element of the block 
of the matrix A standing in the i-th row and j-th
column of this block, and 
is an element of the block 
of the matrix B standing in the j-th row and k-th
column, then
Therefore,
Therefore, all the corresponding elements of the matrices
and 
are equal, and our proposition holds. 
Corollary 2.2.1. If
and
and 
then
where 
=
1 : q
=
1 : r) .
Example 2.2.3. It holds
Example 2.2.4. It holds
where A=(a) is a 
matrix,
B=(b) is a 
matrix,
C=(c) is a 
matrix,
D=(d) is a 
matrix,
E=(e) is a matrix,
F=(f) is a 
matrix,
G=(g) is a 
matrix
and H=(h) is a 
matrix.
Example 2.2.5.* Let us find the product
AB of block matrices A and B, when A and B
are matrices
We denote
where
and
We note that the dimensions of the matrices are in accordance with the
conditions of multiplication of block matrices.
If we denote
then
and
Thus
Problem 2.2.2.* Find the product AB
of 
-matrix
A and 
matrix
B in block form, when
then
