eigenvalues eigenvectors handout

EIGENVALUES AND EIGENVECTORS
ELECTRONIC VERSION OF LECTURE

Dr. Lê Xuân Đại
HoChiMinh City University of Technology
Faculty of Applied Science, Department of Applied Mathematics
Email:

HCMC — 2018.
Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

1 / 56


OUTLINE

1

THE REAL WORLD PROBLEMS

2

EIGENVALUES AND EIGENVECTORS OF A MATRIX

3

DIAGONALIZATION

4

MATL AB

Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

2 / 56


The Real World Problems

MODELLING MOTION

PQR →

P Q R is

the reflection over the x−axis.
Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

3 / 56

The Real World Problems

A=

1 0
0 −1

is the reflection matrix.

Therefore, for every point in the plane
(x1 , x2 ), the matrix that results in a reflection
over the x−axis and then we obtain a new
point in the plane (y1, y2)
x1
−x2
Question: For every point (x1, x2), find
y1
x1
= Ak .
, (k ∈ N).
y2
x2
y1
y2

=

1 0

x1
.
0 −1
x2

Dr. Lê Xuân Đại (HCMUT-OISP)

=

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

4 / 56


Eigenvalues and Eigenvectors of a Matrix

A=

Definition

1 0
−1
0
, u=
,v=
. We have
0 −1
−1

1

A

−1
−1
=
−1
1

A

0
0
0
=
= −1.
1
−1
1

Dr. Lê Xuân Đại (HCMUT-OISP)

and

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

5 / 56

Eigenvalues and Eigenvectors of a Matrix

Definition

DEFINITION 2.1
If A is an n × n matrix, then a nonzero vector
X ∈ Rn , X = 0 is called an eigenvector of A if
AX = λ.X for some scalar λ. The scalar λ is
called an eigenvalue of A and X is said to be
an eigenvector corresponding to λ.
EXAMPLE 2.1
Find eigenvalues and eigenvectors of
A=

1 0
0 −1

Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

6 / 56


Eigenvalues and Eigenvectors of a Matrix

Definition

The equation AX = λX can be rewritten as
(A − λI)X = 0
1 0
x1
0 −1
x2

=

1−λ
0
0 −1 − λ

λx1
λx2
x1
x2

⇔
=

0
.
0

This homogeneous linear system has
non-zero solution X = 0, thus
1−λ

0
= 0 ⇔ λ2 − 1 = 0
0 −1 − λ
⇔ λ1 = −1, λ2 = 1.
Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

7 / 56


Eigenvalues and Eigenvectors of a Matrix

Definition

In the case where λ1 = −1, we have
2x1 + 0x2 = 0
⇔ x1 = 0, x2 = α.
0x1 + 0x2 = 0

Therefore, the eigenvectors corresponding
to λ1 = −1 are α(0, 1), α = 0.
In the case where λ2 = 1. We have
0x1 + 0x2 = 0
⇔ x1 = β, x2 = 0.
0x1 − 2x2 = 0

Therefore, the eigenvectors corresponding

to λ2 = 1 are β(1, 0), β = 0.
Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

8 / 56


Eigenvalues and Eigenvectors of a Matrix

Definition

EXAMPLE 2.2
Find eigenvalues and eigenvectors of
A=

1 2
−2 1

Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

9 / 56

Eigenvalues and Eigenvectors of a Matrix

Definition

AX = λX can be rewritten
λx1
λx2

⇔

1−λ 2
−2 1 − λ

x1
x2

1 2
−2 1
=

x1
x2

=

0
.
0

This homogeneous linear system has

non-zero solution X = 0, thus
1−λ 2
= 0 ⇔ (1 − λ)2 + 4 = 0
−2 1 − λ
⇔ λ1,2 = 1 ± 2i.
Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

10 / 56


Eigenvalues and Eigenvectors of a Matrix

Definition

In the case where λ1 = 1 + 2i. We have
−2ix1 + 2x2 = 0
⇔ x1 = α, x2 = αi.
−2x1 − 2ix2 = 0

Therefore, the eigenvectors corresponding
to λ1 are α(1, i), α = 0.
In the case where λ2 = 1 − 2i. We have
2ix1 + 2x2 = 0
⇔ x1 = β, x2 = −βi.
−2x1 + 2ix2 = 0

Therefore, the eigenvectors corresponding
to λ2 are β(1, −i), β = 0.
Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

11 / 56


Eigenvalues and Eigenvectors of a Matrix

Characteristic equation

If λ is an eigen value of A ⇔ ∃X = 0 : AX = λ.X
⇔ AX − λX = 0 ⇔ (A − λI).X = 0.

This homogeneous linear system has
non-zero solution X = 0, thus det(A − λI) = 0
DEFINITION 2.2
If A is an n×n matrix, then λ is an eigenvalue
of A if and only if χA(λ) = det(A − λI) = 0. This
is called the characteristic equation of A. The
polynomial χA(λ) = det(A − λI) is called the
characteristic polynomial.
Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

12 / 56


Eigenvalues and Eigenvectors of a Matrix

Characteristic equation

FINDING EIGENVALUES AND EIGENVECTORS OF A
SQUARE MATRIX

STEP 1. Finding the characteristic equation
det(A − λI) = 0.
STEP 2. Solving this equation to find
eigenvalues.
STEP 3. For every eigenvalue λi , solve the
homogeneous system (A − λi I)X = 0
to find eigenvectors X
corresponding to the eigenvalue λi .
Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

13 / 56


Eigenvalues and Eigenvectors of a Matrix

Characteristic equation

THEOREM
2.1



a11 a12 a13


If A =  a21 a22 a23  , then
a31 a32 a33
χA (λ) = |A − λI| = −λ3 + tr(A)λ2 −
−

a11 a13
a11 a12
a22 a23
λ + det(A)
+
+
a31 a33
a32 a33
a21 a22

where tr(A) = a11 + a22 + a33 is called the trace
of A.
Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

14 / 56


Eigenvalues and Eigenvectors of a Matrix

Eigenspace corresponding to the eigenvalue

DEFINITION 2.3
The eigenvetors corresponding to the
eigenvalue λ, together with the zero vector,
form the null space of the matrix (A − λI).
This subspace is called the eigenspace
corresponding to the eigenvalue λ.

Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

15 / 56


Eigenvalues and Eigenvectors of a Matrix

Geometric and Algebraic Multiplicity

DEFINITION 2.4
If λ0 is an eigenvalue of an n × n matrix A,
then the dimension of the eigenspace
corresponding to λ0 is called the geometric
multiplicity of λ0, and the number of times
that λ − λ0 appears as a factor in the
characteristic polynomial of A is called the
algebraic multiplicity of λ0.

Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

16 / 56


Eigenvalues and Eigenvectors of a Matrix

EXAMPLE
2.3


Geometric and Algebraic Multiplicity



3 1 1



Let A =  2 4 2 
1 1 3
1

2

3

Find the characteristic polynomial of A
Evaluate det(A − 2013.I)
Find eigenvalues and eigenvectors of A

Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

17 / 56


Eigenvalues and Eigenvectors of a Matrix

Geometric and Algebraic Multiplicity

1. The characteristic polynomial of A
3−λ 1
1

2 4−λ 2
=
χA (λ) = |A − λI| =
1
1 3−λ
= −λ3 + 10λ2 − 28λ + 24 = −(λ − 2)2 (λ − 6)
2. det(A − 2013.I) = −(2013 − 2)2(2013 − 6)
3. The characteristic equation of A
3−λ 1
1
2 4−λ 2
χA (λ) = |A − λI| =
=0
1
1 3−λ
⇔ −(λ − 2)2 (λ − 6) = 0 ⇔ λ1 = 2, λ2 = 6.
Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

18 / 56


Eigenvalues and Eigenvectors of a Matrix

Geometric and Algebraic Multiplicity

In the case where λ1 = 2, we have



x1 + x2 + x3 = 0

2x1 + 2x2 + 2x3 = 0


x1 + x2 + x3 = 0




−1
−1




⇒ X1 = α  1  + β  0  , α2 + β2 = 0.
0
1
The algebraic multiplicity of λ1 = 2 is 2 and
the geometric multiplicity of λ1 = 2 is 2.
Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

19 / 56

Eigenvalues and Eigenvectors of a Matrix

Geometric and Algebraic Multiplicity

In the case where λ2 = 6, we 
have



1
 −3x1 + x2 + x3 = 0
 
2x1 − 2x2 + 2x3 = 0 ⇒ X2 = γ  2  , γ = 0.

 x + x − 3x = 0
1
1
2
3
The algebraic multiplicity of λ2 = 6 is 1 and
the geometric multiplicity of λ2 = 6 is 1.

Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

20 / 56


Diagonalization

Definition

DEFINITION 3.1
If A and B are square matrices, then we say
that B is similar to A if there is an invertible
matrix S such that B = S−1AS.
DEFINITION 3.2
A square matrix A is said to be
diagonalizable if it is similar to some
diagonal matrix D, that is, if there exists an
invertible matrix S such that S−1AS = D. In
this case the matrix S is said to diagonalize A.
Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

21 / 56


Diagonalization

Definition

We have S−1AS = D = dig(λ1, λ2, . . . , λn). It
follows that AS = SD


a11 a12
a a
 21 22
A=
 ... ...
an1 an2

s11 s12
s s
 21 22
S=
 ... ...
sn1 sn2
Dr. Lê Xuân Đại (HCMUT-OISP)

...
...
...
...

a1n
a2n
...
ann

...

...
...
...

s1n
s2n
...
snn

λ1 0

 0 λ


2
,D = 

 ... ...
0 0





...
...
...
...

0

0
...
λn









 = S∗1 S∗2 . . . S∗n


EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

22 / 56


Diagonalization



a11
a
 21
AS = 

 ...
an1

a12
a22
...
an2

...
...
...
...

Definition

a1n
a2n
...
ann

 

s11
 s
  21
.
  ...
sn1

s12

s22
...
sn2

...
...
...
...

s1n
s2n
...
snn







= A S∗1 S∗2 . . . S∗n = AS∗1 AS∗2 . . . AS∗n


s11
s
 21
SD = 
 ...
sn1
Dr. Lê Xuân Đại (HCMUT-OISP)

s12
s22
...
sn2

...
...
...
...

s1n
s2n
...
snn

λ1 0
 0 λ

2

 ... ...
0 0


EIGENVALUES AND EIGENVECTORS



... 0


...


... ... 
. . . λn
HCMC — 2018.

23 / 56


Diagonalization

Definition

Therefore,
(AS)∗i = AS∗i = (SD)∗i = λi S∗i , (i = 1, 2, . . . , n).

So, S∗i is the eigenvector corresponding to
eigenvalue λi (i = 1, 2, . . . , n) of A.
Form the matrix S whose column vectors
are the n basis eigenvectors of A.

Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

24 / 56

Diagonalization

EXAMPLE
3.1


Definition



15 −18 −16


Let A =  9 −12 −8  . Find a matrix S that
4 −4 −6
diagonalizes A.

Step 1. Find eigenvalues, eigenvectors of A.
15 − λ −18
−16
9
−12 − λ −8
=0
χA (λ) = |A − λI| =
4
−4
−6 − λ
⇔ −(λ + 3)(λ + 2)(λ − 2) = 0 ⇔ λ1 = −3 (AM=1),

λ2 = −2 (AM=1), λ3 = 2 (AM=1).
Dr. Lê Xuân Đại (HCMUT-OISP)

EIGENVALUES AND EIGENVECTORS

HCMC — 2018.

25 / 56