MTH 309 Y
LECTURE 24.1
Motivating example: Fibonacci numbers
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1� 1� 2� 3� 5� 8� 13� 21� 44� 65� 111� 176� 287� 463���
MTH 309 Y
LECTURE 24.2
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Problem. Find a formula for the �-th Fibonacci number F�.
MTH 309 Y
LECTURE 24.3
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General problem. If A is a square matrix how to compute A� quickly?
Easy case:
Definition. An � × � matrix D is a diagonal matrix if all its
entries outside the main diagonal are zero.
⎡
⎢
⎢
D=⎢
⎢
⎣
λ1
0
...
0
0
λ2
...
0
···
···
...
···
0
0
...
λ�
Proposition. If D is a diagonal matrix
⎡
λ�1 0 · · ·
⎢
�
⎢
0
λ
···
2
�
D =⎢
⎢ ... ... . . .
⎣
0 0 ···
⎤
⎥
⎥
⎥
⎥
⎦
as above then
⎤
0
⎥
0 ⎥
⎥
... ⎥
⎦
�
MTH 309 Y
LECTURE 24.4
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Definition. An � × � matrix A is a diagonalizable matrix if A is
of the form
A = PDP −1
where D is a diagonal matrix and P is an invertible matrix.
Example.
⎡
⎤
1 0 1
A = ⎣0 1 1⎦ is a diagonalizable matrix:
1 1 0
⎡
⎤ ⎡
⎤ ⎡
⎤−1
1 1 −1
2 0 0
1 1 −1
A = ⎣ 1 1 1 ⎦ · ⎣ 0 −1 0 ⎦ · ⎣ 1 1 1 ⎦
1 −2 0
0 0 1
1 −2 0
Proposition. If A is a diagonalizable matrix A = PDP −1 then
A� = PD � P −1
MTH 309 Y
LECTURE 24.5
Example.
⎡
⎤
1 0 1
Let A = ⎣0 1 1⎦. Compute A10.
1 1 0
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MTH 309 Y
LECTURE 24.6
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How to diagonalize matrices?
Diagonalization Theorem.
1) An � × � matrix A is diagonalizable iff it has � linearly
independent eigenvectors v1� v2� � � � � v�.
2) In such case A = PDP −1 where
P=
�
⎡
⎢
⎢
D=⎢
⎢
⎣
v1 v2 � � � v�
λ1
0
...
0
0
λ2
...
0
···
···
...
···
0
0
...
λ�
�
⎤
λ1 = eigenvalue corresponding to v1
⎥
⎥ λ2 = eigenvalue corresponding to v2
⎥
...
...
...
⎥
⎦
λ� = eigenvalue corresponding to v�
MTH 309 Y
LECTURE 24.7
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Example. Diagonalize the following matrix if possible:
⎡
⎤
4 0 0
A = ⎣1 3 −1⎦
1 −1 3
MTH 309 Y
LECTURE 24.8
Note. Not every square matrix is diagonalizable.
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Example. Check if the following matrix is diagonalizable:
�
�
2 1
A=
0 2
MTH 309 Y
LECTURE 24.9
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Proposition. If A is an � × � matrix with � distinct eigenvalues
then A is diagonalizable.
MTH 309 Y
LECTURE 24.10
Back to Fibonacci numbers:
�
F�
F�+1
�
=
�
��−1 � �
0 1
1
·
1 1
1
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