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where Q
(l)
and R
(l)
are the quotient and remainder polynomial after dividing f
(l)
by P
(l)
, respectively. The degree of the remainder polynomial is at most n
1 since
the degree of characteristic polynomial is n. Since P
(
A
) ¼
0 by the Cayley
Hamilton
-
theorem, then we have
f
(
A
) ¼
Q
(
A
)
P
(
A
) þ
R
(
A
) ¼
R
(
A
)
(
3
:
179
)
Example 3.45
¼ A
8
8A
6
2A
5
2A
3
3A
2
Find the matrix polynomial
f (A)
þ
þ
þ A þ
7I
if
42
A ¼
70
S
OLUTION
The characteristic polynomial of A is
¼ l
¼jlI Aj¼
l
42
2
P(
l
)
4
l þ
6
3
l
Dividing f (
l
)byP(
l
), we have
8
6
5
3
2
l
8
l
þ
2
l
þ
2
l
3
l
þ l þ
7
6
5
4
3
2
¼ l
þ
4
l
þ
2
l
14
l
68
l
186
l
2
l
4
l þ
6
þ
239
l þ
2041
339
2
l
4
l þ
6
Therefore,
4
2
30
2041
10
01
f (A)
¼ R(A)
¼
239A þ
2041I ¼
239
þ
956 478
2041
0
1085
478
¼
þ
¼
717
0
0
2041
71
2041
3.11.4 F
UNCTION OF
M
ATRICES
There are several techniques to
find function of matrices. They include Cayley
-
Hamilton and matrix diagonalization techniques.
3.11.4.1 Cayley
-
Hamilton Technique
First we assume that the eigenvalues of matrix A are distinct. Using the results of the
Cayley
Hamilton theorem, we have
-
f
(
A
) ¼
R
(
A
) ¼
r
n
1
A
n
1
þ
r
n
2
A
n
2
þ
r
n
3
A
m
3
þþ
r
1
A
þ
r
0
I
(
3
:
180
)
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