Image Processing Reference
In-Depth Information
3.9.5 M
ATRIX
D
IAGONALIZATION
Let A be an n
n matrix with n distinct eigenvalues
l
1
,
l
2
,
...
,
l
n
. Assume that the
corresponding independent eigenvectors are x
1
, x
2
,
...
, x
n
. Therefore, we have
Ax
1
¼ l
1
x
1
(
3
:
125
)
Ax
2
¼ l
2
x
2
(
3
:
126
)
.
Ax
n
¼ l
n
x
n
(
3
:
127
)
These equations can be put together in matrix form to obtain
½
¼ l
1
x
1
½
l
2
x
2
l
n
x
n
(
:
)
Ax
1
Ax
2
Ax
n
3
128
Equation 3.128 can be written as
2
3
l
1
0
0
4
5
0
l
2
0
Ax
1
½
x
2
x
n
¼
x
1
½
x
2
x
n
.
.
.
.
(
3
:
129
)
.
.
l
n
00
De
to be the diagonal matrix of eigenvalues, then the above matrix equation
can be written in terms of
ne
L
L
and modal matrix M as
AM
¼
M
L
(
3
:
130
)
This equation is valid regardless of whether the eigenvectors are linearly independent
or not. However, if the eigenvectors are linearly independent, then M is full rank
and has an inverse and in this case, we can post-multiply the above equation by M
1
to obtain
A
¼
M
L
M
1
(
3
:
131
)
or
L ¼
M
1
AM
(
3
:
132
)
If matrix A has repeated eigenvalues, it is diagonalizable if and only if the eigen-
vectors are linearly independent.
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