Image Processing Reference
In-Depth Information
The repeated eigenvalues have independent eigenvectors, therefore, matrix A is
diagonalizable. Hence,
2
4
3
5
2
4
3
5
2
4
3
5
¼
2
4
3
5
1
10 3
510
240
10 3
300
060
006
M
1
AM ¼
2023
11 8
2023
11 8
116
Note that
if matrix A has repeated eigenvalues,
it
is not always possible to
diagonalize it as is shown in the following example.
Example 3.36
Diagonalize the 2
2 matrix A:
01
A ¼
4
4
S
OLUTION
The eigenvalues of A are
l
1
¼ l
2
¼
2. The eigenvectors of A are computed as
(
a
b
¼
!
b ¼
2a
01
a
b
Ax ¼ lx !
2
!
b ¼
2a
4
4
4a
4b ¼
2b
and thematrixA is not diagonalizable.
1
Therefore, there is oneeigenvector x
1
¼
2
3.9.6 D
EFINITE
M
ATRICES
Positive and negative de
nite) matrices are an important class of
matrices with applications in signal processing, image processing, and control
systems. They are particularly useful matrices in optimization problems. We now
de
nite (semi-de
ne four types of de
nite matrices:
(a)
Positive De
nite Matrices
: The n
n Hermitian matrix A is said to be
positive de
nite if for any nonzero vector x
2
R
n
, the quantity x
H
Ax
>
0.
Here
H
stands for conjugate transpose. Matrix A is said to be Hermitian if
A
¼
A
H
.
(b)
Positive Semi-De
nite Matrices
: The n
n Hermitian matrix A is said to
be positive semi-de
nite if for any nonzero vector x
2
R
n
, the quantity
x
H
Ax
0.
(c)
Negative De
nite Matrices
: The n
n Hermitian matrix A is said to be
negative de
nite if for any nonzero vector x
2
R
n
, the quantity x
H
Ax
<
0.
(d)
Negative Semi-De
nite Matrices
: The n
n Hermitian matrix A is said to
be negative semi-de
nite if for any nonzero vector x
2
R
n
, the quantity
x
H
Ax
0.
The following theorem states the condition for positiveness of a Hermitian matrix.
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