Image Processing Reference
In-Depth Information
To
nd the p
¼
2 matrix norm, we must solve the following optimization problem:
k
Ax
k
2
¼
x
H
A
H
Ax
maximize
x
(
3
:
146
)
x
H
x
¼
subject to:
1
Using Lagrange multiplier technique, this is equivalent to the following optimization
problem:
J
¼
x
H
A
H
Ax
l(
x
H
x
maximize
x
1
)
(
3
:
147
)
Setting the gradient of J with respect to x equal to zero, we obtain the equation
q
J
q
x
¼
2A
H
Ax
!
A
H
Ax
¼ l
x
2
l
x
¼
0
(
3
:
148
)
Therefore, the solution for vector x must be an eigenvector of square matrix A
H
A
corresponding to eigenvalue
l
and the resulting norm is
k
Ax
k
2
¼
x
H
A
H
Ax
¼ l
x
H
x
¼ l
(
3
:
149
)
Since we are maximizing the norm,
l
must be chosen to be the maximum eigenvalue
nite matrix A
H
A, therefore,
of the positive de
k
A
k
2
¼ max l(
A
H
A
)
(
3
:
150
)
The p norm has the property that for any two matrices A and B, the following
inequality holds:
k
Ax
k
p
k
A
k
p
k
x
k
p
(
3
:
151
)
and
k
AB
k
p
k
A
k
p
k
B
k
p
(
3
:
152
)
Frobenius Norm
Frobenius norm is another matrix norm that is not a p norm. It is de
ned as
!
1
2
X
X
m
n
2
k
A
k
F
¼
a
ij
(
3
:
153
)
i
¼
1
j
¼
1
Frobenius norm is also called Euclidean norm. As a simple example, the Frobenius
norm of an n
n identity matrix is
p
. The Frobenius norm can also be
k
I
k
F
¼
expressed as
p
Trace(
A
H
A
)
k
A
k
F
¼
(
:
)
3
154
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