Digital Signal Processing Reference
In-Depth Information
G
REVIEW OF MATRICES
This appendix presents a brief review of definitions, properties, and the algebra of
matrices. It is assumed that the reader has a background in this area. Those readers
interested in more depth are referred to Refs. 1 through 3. MATLAB statements
are given for performing the mathematical operations, where appropriate.
The study of matrices originated in linear algebraic equations. As an example,
consider the equations
x
1
+ x
2
+ x
3
= 3;
x
1
+ x
2
- x
3
= 1;
(G.1)
2 x
1
+ x
2
+ 3x
3
= 6.
In a
vector-matrix
format, we write these equations as
11 1
11-1
21
x
1
x
2
x
3
3
1
6
C
SC
S
=
C
S
.
(G.2)
3
We define the following:
11 1
11-1
21
x
1
x
2
x
3
3
1
6
A
=
C
S
;
x
=
C
S
;
u
=
C
S
.
(G.3)
3
Then (G.2) can be expressed as
Ax
=
u
.
(G.4)
In this equation,
A
is a (3 rows, 3 columns)
matrix
,
x
is a
matrix
, and
u
is a
matrix
. Usually, matrices that contain only one row or only one column
are called
vectors
. A matrix of only one row and one column is a scalar. In (G.1), for
example, is a scalar. One statement for entering the matrix
A
into MATLAB is
|A = [1 1 1; 1 1
1; 2 1 3];
3 * 3
3 * 1
3 * 1
x
1
737
