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
Search WWH ::




Custom Search