Information Technology Reference
In-Depth Information
∈
R
The above three norms are members of the
L
p
norm family, i.e., for
p
and
p
≥
1,
p
1
p
n
x
p
=
1
|
x
i
|
.
i
=
One can also define norms for matrices. For example, the following norm is the
commonly-used Frobenius norm. For a matrix
A
m
×
n
,
∈
R
tr
A
T
A
.
m
n
A
F
=
1
|
A
ij
|=
i
=
j
=
1
21.2.2.8 Inverse
n
×
n
is called
invertible
or
nonsingular
if there exists a square
A square matrix
A
∈
R
n
×
n
matrix
B
∈
R
such that
AB
=
I
=
BA.
In this case,
B
is uniquely determined by
A
and is referred to as the
inverse
of
A
,
denoted by
A
−
1
. If such a kind of
B
(or
A
−
1
) does not exist, we call
A
a
non-
invertible
or
singular
matrix.
The properties of inverse are listed as below:
n
×
n
,wehave
(A
−
1
)
−
1
•
given
A
∈
R
=
A
n
×
n
,wehave
(A
−
1
)
T
=
(A
T
)
−
1
=
A
−
T
•
given
A
∈
R
n
×
n
,wehave
(AB)
−
1
B
−
1
A
−
1
•
given
A, B
∈
R
=
21.2.2.9 Orthogonal Matrix
n
,theyare
orthogonal
if
x
T
y
Given two vectors
x,y
∈
R
=
0.
n
×
n
is an
orthogonal matrix
if its columns are orthogonal
A square matrix
A
∈
R
unit vectors.
A
T
A
AA
T
.
=
I
=
The properties of orthogonal matrix are listed as below:
the inverse of an orthogonal matrix equals to its transpose, i.e.,
A
−
1
=
A
T
•
n
n
×
n
•
Ax
2
=
x
2
for any
x
∈
R
and orthogonal
A
∈
R