Digital Signal Processing Reference
In-Depth Information
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
4
3
2
1
4 8 12 16
369 2
246 8
123 4
1234
=
The computation is as follows:
[
4
;
3
;
2
;
1
]∗[
1
,
2
,
3
,
4
]=[
4
,
3
,
2
,
1
;
8
,
6
,
4
,
2
;
12
,
9
,
6
,
3
;
16
,
12
,
8
,
4
]
Note that each column in the output matrix is the column of the input column vector, scaled by a
column (which is a single value) in the row vector.
B.2.3 PRODUCT OF CORRESPONDING VALUES
Two vectors (or matrices) of exactly the same dimensions may be multiplied on a value-by-value basis by
using the notation "
.* "
(a period followed by an asterisk). Thus two row vectors or two column vectors
can be multiplied in this way, and result in a row vector or column vector having the same length as the
original two vectors. For example, for two column vectors, we get
[
1
;
2
;
3
]
.
∗[
4
;
5
;
6
]=[
4
;
10
;
18
]
and for row vectors, we get
[
1
,
2
,
3
]
.
∗[
4
,
5
,
6
]=[
4
,
10
,
18
]
B.3 MATRIX MULTIPLIED BY A VECTOR OR MATRIX
An
m
by
n
matrix, meaning a matrix having
m
rows and
n
columns, can be multiplied from the right by
an
n
by 1 column vector, which results in an
m
by 1 column vector. For example,
[
1
,
2
,
1
;
2
,
1
,
2
]∗[
4
;
5
;
6
]=[
20
;
25
]
Or, written in standard matrix form:
⎡
⎣
⎤
121
212
4
8
10
5
6
12
20
25
4
5
6
⎦
=
+
+
=
(B.1)
An
m
by
n
matrix can be multiplied from the right by an
n
by
p
matrix, resulting in an
m
by
p
matrix. Each column of the
n
by
p
matrix operates on the
m
by
n
matrix as shown in (B.1), and creates
another column in the
n
by
p
output matrix.
B.4 MATRIX INVERSE AND PSEUDO-INVERSE
Consider the matrix equation
14
3
a
b
−
2
3
=
(B.2)
−
2
