Image Processing Reference
In-Depth Information
Since P(l2)
(l
1
) ¼
P
(l
2
) ¼¼
P
(l
n
) ¼
0, then
P
(
A
) ¼
0
(
3
:
175
)
Example 3.44
Show that matrix A satis
es its characteristic polynomial:
2
7
51
A ¼
S
OLUTION
The characteristic polynomial of A is
¼ l
¼jlI Aj¼
l
27
2
P(
l
)
3
l þ
37
5
l
1
Then
¼ A
2
P(A)
3A þ
37I
2
2
7
51
7
51
2
7
51
37
10
00
¼
3
þ
31
21
6 1
37 0
0 7
00
00
¼
þ
þ
¼
¼
0
15
34
15
3
Hamilton theorem is used to reduce any matrix polynomial of n
n
matrix A to a polynomial of degree n
Cayley
-
1. This is proved in the following theorem.
THEOREM 3.4
Consider an n
n matrix A and a matrix polynomial in A of degree m
n. That is,
f
(
A
) ¼
a
m
A
m
þ
a
m
1
A
m
1
þ
a
m
2
A
m
2
þþ
a
1
A
þ
a
0
I
(
3
:
176
)
Then f
(
A
)
is reducible to a polynomial of degree n
1 given by
R
(
A
) ¼
r
n
1
A
n
1
þ
r
n
2
A
n
2
þ
r
n
3
A
n
3
þþ
r
1
A
þ
r
0
I
(
3
:
177
)
Proof: Consider the scalar polynomial f
(l)
. Divide this polynomial by P(l),
(l)
, the
characteristic polynomial of matrix A, then
f
(l) ¼
Q
(l)
P
(l) þ
R
(l)
(
3
:
178
)
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