Digital Signal Processing Reference
In-Depth Information
The general matrix
A
is written as
Á
a
1n
a
11
a
12
Á
a
2n
a
21
a
22
A
=
D
T
= [a
ij
],
(G.5)
o
o
o
o
Á
a
m1
a
m2
a
mn
where is a convenient notation for the matrix
A
. This matrix has
m
rows and
n
columns and, thus, is an
[a
ij
]
m * n
matrix. The element
a
ij
is the element common to
the
i
th row and the
j
th column.
Some definitions will given next.
Identity Matrix
The identity matrix is an (square) matrix with all main diagonal elements
equal to 1 and all off-diagonal elements
n * n
a
ii
a
ij
equal to 0,
i Z j.
For example, the
3 * 3
identity matrix is
100
010
001
I
=
C
S
.
(G.6)
The MATLAB statement for generating the
3 * 3
identity matrix is
I3 = eye(3);
If the matrix
A
is also
n * n,
then
AI
=
IA
=
A
.
(G.7)
Diagonal Matrix
A diagonal matrix is an
n * n
matrix with all off-diagonal elements equal to zero:
d
11
00
0 d
22
0
00d
33
D
=
C
S
.
(G.8)
Symmetric Matrix
The square matrix
A
is symmetric if
a
ij
= a
ji
for all
i
and
j
.
Transpose of a Matrix
To take the transpose of a matrix, interchange the rows and the columns. For example,
a
11
a
12
a
13
a
11
a
21
a
31
and
A
T
=
A
=
C
a
21
a
22
a
23
S
C
a
12
a
22
a
32
S
,
(G.9)
a
31
a
32
a
33
a
13
a
23
a
33
A
T
where
denotes the transpose of
A
. A property of the transpose is
(
AB
)
T
=
B
T
A
T
.
(G.10)
