Image Processing Reference
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converges for all eigenvalues of A. With the assumption that eigenvalues of A are
distinct, we must have
1
j
a
j
< li
i
<
1
j
a
j
j
a
l
i
j <
1
or
for
i
¼
1, 2,
...
, n
(
3
:
169
)
Another example is the exponential matrix polynomial de
ned by
1
A
2
2
A
3
3
A
k
k
!
e
A
¼
I
þ
A
þ
!
þ
!
þ¼
(
:
)
3
170
k
¼
0
3.11.3 C
AYLEY
-
H
AMILTON
T
HEOREM
THEOREM 3.3
Any n
n square matrix satis
es its own characteristic polynomial, that is,
P
(
A
) ¼
0
(
3
:
171
)
where P
(l) ¼jl
I
A
j
is the characteristic polynomial of matrix A.
Proof: We prove the theorem for a special case when matrix A has n distinct
eigenvalues. Let P(l)
(l)
be the characteristic polynomial of A. Then P(l)
(l)
is a polyno-
mial of degree n in
l
and is given by
n
n
1
n
2
P
(l) ¼ l
þ a
1
l
þ a
2
l
þþa
n
1
l þ a
n
(
3
:
172
)
Then
X
n
P
(
A
) ¼
A
n
þ a
1
A
n
1
þ a
2
A
n
2
0
a
i
A
n
i
þþa
n
1
A
þ a
n
I
¼
(
3
:
173
)
i
¼
k
M
1
, therefore,
Since A
k
¼
M
L
X
X
¼
M
X
n
i
¼
0
a
i
A
n
i
n
i
¼
0
a
i
M
L
n
i
¼
0
a
i
L
n
i
M
1
n
i
M
1
P
(
A
) ¼
¼
2
4
3
5
P
(l
1
)
0
0
0
P
(l
2
)
0
M
1
¼
M
(
:
)
3
174
.
.
.
.
.
.
0
0
P
(l
n
)
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