SECTION 5.3 DIAGONALIZATION
EXAMPLE.
Use the factorization
A
=
PDP

1
shown below to find a simple formula for
A
k
, where
k
is an arbitrary positive integer. Then discuss what happens to
A
k
as
k
→ ∞
.
"

2
1

15
/
2
7
/
2
#
=
"
2
1
5
3
# "
1
/
2
0
0
1
# "
3

1

5
2
#
A square matrix
A
is said to be
diagonalizable
if
A
is similar to a diagonal matrix, that
is, if
A
=
PDP

1
for some invertible matrix
P
and some diagonal matrix
D
.
Which matrices are diagonalizable?
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A
p
×
p
matrix
A
is diagonalizable exactly when
A
has
p
linearly independent eigenvectors.
It turns out that the dimension of the eigenspace for an eigenvalue is less than or equal to
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 Spring '08
 PAVLOVIC
 Linear Algebra, Matrices, arbitrary positive integer, diagonal matrix D.

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