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Lecture 32
Andrei Antonenko
April 30, 2003
1 Powers of diagonalizable matrices
In this section we will give 2 algorithms of computing the
m
th power of a matrix.
1.1 Method 1
First method is based on diagonalization. Suppose
A
is a given matrix, and we want to ﬁnd
its
m
th power, i.e. we want to get a formula for
A
m
. We will suppose that the matrix
A
is diagonalizable. Let
λ
1
,λ
2
,...,λ
n
be the eigenvalues of
A
, and
e
1
,e
2
,...,e
n
be its linearly
independent eigenvectors. Then we know, that there exists a matrix
C
, whose columns are
eigenvectors, and a diagonal matrix
D
=
λ
1
0
...
0
0
λ
2
...
0
. . . . . . . . . . . . . . .
0
0
... λ
n
such that
D
=
C

1
AC,
or
A
=
CDC

1
.
Now we can see that
A
m
= (
CDC

1
)
m
= (
CDC

1
)(
CDC

1
)
···
(
CDC

1
)

{z
}
m
=
CD
(
C

1
C
)
D
(
C

1
...C
)
DC

1
=
CDID .
..IDC

1
=
CD
m
C

1
.
But
D
m
=
λ
1
0
...
0
0
λ
2
...
0
. . . . . . . . . . . . . . .
0
0
... λ
n
m
=
λ
m
1
0
...
0
0
λ
m
2
...
0
. . . . . . . . . . . . . . . . .
0
0
... λ
m
n
1
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View Full Document So,
A
m
=
C
λ
m
1
0
...
0
0
λ
m
2
...
0
. . . . . . . . . . . . . . . . .
0
0
... λ
m
n
C

1
.
Example 1.1.
Let’s ﬁnd the formula for
ˆ
3

2
1
0
!
m
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This note was uploaded on 05/28/2011 for the course AMS 2010 taught by Professor Andant during the Spring '03 term at SUNY Stony Brook.
 Spring '03
 Andant

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