Image Processing Reference
In-Depth Information
THEOREM 3.1
An n
n Hermitian matrix is positive de
nite if and only if all its eigenvalues are
positive.
Proof: First assume that A is Hermitian and all of its eigenvalues are positive, then
x
H
Ax
¼
x
H
M
L
M
H
x
(
:
)
3
133
Let y
¼
M
H
x, then Equation 3.133 can be written as
X
n
i
¼
1
l
i
j
y
i
j
2
x
H
Ax
¼
x
H
M
L
M
H
x
¼
y
H
L
y
¼
>
0
(
3
:
134
)
Next we need to show that if A is Hermitian and positive de
nite, then all of its
eigenvalues are positive.
Let x
i
be the eigenvector corresponding to the ith eigenvalue
l
i
of A, then
2
<
x
i
Ax
i
¼
x
i
l
i
x
i
¼ l
i
k
x
i
k
0
(
3
:
135
)
Therefore,
l
i
>
0.
Similar theorems can be stated for positive semi-de
nite, negative de
nite, and
negative semi-de
nite matrices. Here we state a summary of the results of these
theorems:
(a) Positive de
nite: All eigenvalues are positive.
(b) Negative de
nite: All eigenvalues are negative.
(c) Positive semi-de
nite: All eigenvalues are nonnegative (zero or positive).
(d) Negative semi-de
nite: All eigenvalues are nonpositive (zero or negative).
Example 3.37
Check the following symmetric matrices for their de
niteness:
5
2
(a) A ¼
25
21
1
(b) B ¼
2
6
4
(c) C ¼
4
6
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