Image Processing Reference
In-Depth Information
3.11.1 M
ATRIX
P
OLYNOMIAL
Consider the monic polynomial f
(
x
)
of degree m given by
f
(
x
) ¼
x
m
þ
a
1
x
m
1
þ
a
2
x
m
2
þþ
a
m
1
x
þ
a
m
(
3
:
161
)
If the scalar variable x is replaced by the n
n matrix A, then the corresponding
matrix polynomial f
(
A
)
is de
ned by
f
(
A
) ¼
A
m
þ
a
1
A
m
1
þ
a
2
A
m
2
þþ
a
m
1
A
þ
a
m
I
(
3
:
162
)
where
m
times
A
m
¼
A
A
A
I is an n
n identity matrix
The polynomial f
(
x
)
can be written in factor form
f
(
x
) ¼ (
x
a
1
)(
x
a
2
) (
x
a
n
)
(
3
:
163
)
where
a
1
,
a
2
,
...
,
a
n
are the roots of the polynomial f
(
x
)
. The matrix polynomial
f
(
A
)
can be factored as
f
(
A
) ¼ (
A
a
1
I
)(
A
a
2
I
) (
A
a
n
I
)
(
3
:
164
)
3.11.2 I
NFINITE
S
ERIES OF
M
ATRICES
An in
nite series of matrix A is de
ned as
1
S
(
A
) ¼
a
0
I
þ
a
1
A
þ
a
2
A
2
a
k
A
k
þ¼
(
3
:
165
)
k
¼
0
It can be shown that the matrix in
nite series S(li)
(
A
)
converges if and only if the scalar
in
nite series S(li)
(l
i
)
converges for all values of
i, where
l
i
is the i
th
eigenvalue of
matrix A. For example, the geometric matrix series
1
S
(
A
) ¼
I
þ
aA
þ
a
2
A
2
a
k
A
k
þ¼
(
3
:
166
)
k
¼
0
is a convergence series and converges to
S
(
A
) ¼ (
I
aA
)
1
(
3
:
167
)
If and only if the scalar geometric series
1
k
¼
0
(
a
l)
þ
a
l þ
a
2
2
k
S
(l) ¼
l
þ¼
(
:
)
1
3
168
Search WWH ::
Custom Search