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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