ENGINEERING MATHEMATICS II

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ENGINEERING MATHEMATICS II

010.141

Byeng Dong Youn (윤병동윤병동윤병동)윤병동

CHAP. 8

Linear Algebra: Matrix Eigenvalue

Problem

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2

B.D. Youn Engineering Mathematics II CHAPTER 8

ABSTRACT OF CHAP. 8

Linear algebra in Chaps. 7 and 8 discusses the theory and application of vectors and matrices, mainly related to

linear systems of equations, eigenvalue problems, and linear transformation.

Chapter 8 concerns the solutions of vector equations Ax = λ x

where A is a given square matrix and vector x and scalar λ .

Eigenvalue problems are of greatest practical interest to the engineer, physicist, and mathematician.

Eigenvectors are transformed to themselves multiplied

with a characteristic constant (eigenvalues).

(3)

3

CHAP. 8.1

EIGENVALUES, EIGENVECTORS

Eigenvalues and eigenvectors imply characteristic

values and vectors of a matrix A.

(4)

4

SOME DEFINITIONS

Let A be an n × n matrix and consider

Ax = λ x (1)

λ , such that (1) has solution x ≠ 0 is an eigenvalue or characteristic value of A.

x are eigenvectors or characteristic vectors of A.

The spectrum of A is the set of eigenvalues of A;

max| λ | is the spectral radius of A.

The set of eigenvectors corresponding to λ (including 0) is

the eigenspace of A for λ _.

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5

SOME DEFINITIONS (cont)

Homogeneous linear system in x ₁ , x ₂

→ Cramer's theorem D( λ _{) = det(A-} λ _{I) = 0}

D( λ ) is the characteristic determinant, and

D( λ ) = 0 is the characteristic equation

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6

EXAMPLE 1

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7

EXAMPLE 1

(8)

8

THEOREMS

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9

EXAMPLE 2

(10)

10

EXAMPLE 2

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11

MULTIPLICITY

The algebraic multiplicity is the order M _λ of λ _{in the}

characteristic polynomial.

The geometric multiplicity (m λ _{) of} λ is the number of linearly independent vectors corresponding to λ .

( ^λ )

−

=

∆

≤

=

λ λ

λ

λ λ

∑ λ

of defect

m M

M m

n M

12

(12)

12

Example 3

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13

Example 4

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14

HOMEWORK IN 8.1

HW1. Problem 6

HW2. Problem 10

HW3. Problem 30

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15

CHAP. 8.2

SOME APPLICATIONS OF EIGENVALUE PROBLEMS

Range of applications of matrix eigenvalue problems.

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16

EXAMPLE 1

(17)

17

EXAMPLE 1

(18)

18

EXAMPLE 2 – Markov Process

The eigenvalue problem can be used to identify the limit state of the process, in which the state vector x is reproduced under the multiplication by the

stochastic matrix A governing the process, that is, Ax=x.

the limit state of the process

(19)

19

EXAMPLE 4

(20)

20

EXAMPLE 4

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21

EXAMPLE 4

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22

HOMEWORK IN 8.2

HW1. Problem 8

HW2. Problem 19

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23

CHAP. 8.3

SYMMETRIC, SKEW-SYMMETRIC, AND ORTHOGONAL MATRICES

Three classes of real square matrices frequently

occurring in engineering applications.

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24

SOME DEFINITIONS

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25

SOME DEFINITIONS

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26

ORTHOGONAL TRANSFORMATIONS

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27

RELATED THEOREMS

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28

RELATED THEOREMS

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29

RELATED THEOREMS

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HOMEWORK IN 8.3

HW1. Problem 10

HW2. Problem 12

HW3. Problem 14

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31

CHAP. 8.4

EIGENBASES. DIAGONALIZATION.

QUADRATIC FORMS

General properties of eigenvectors.

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BASIS OF EIGENVECTORS

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BASIS OF EIGENVECTORS

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DIAGONALIZATION

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35

DIAGONALIZATION

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36

DIAGONALIZATION

Example: Diagonalize

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37

QUADRATIC FORMS

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38

QUADRATIC FORMS

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39

HOMEWORK IN 8.4

HW1. Problem 3

HW2. Problem 7

HW3. Problem 17

HW4. Problem 22

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40

CHAP. 8.5

COMPLEX MATRICES AND FORMS.

Encountered in some applications in quantum

mechanics and wave propagations.

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41

Definitions:

A square matrix A = [a _jk ] is

•

• COMPLEX MATRICES:

HERMITIAN, SKEW-HERMITIAN, UNITARY

1 T

T T

A A

if Unitary

A A

if Hermitian Skew

A A

if

Hermitian

= −

−

=

−

= Symmetric

Skew-symmetric

Orthogonal

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42

•

• COMPLEX MATRICES: HERMITIAN, SKEW-HERMITIAN, UNITARY (cont)

orthogonal

A A

A , nitary u

and real is

matrix a

If

symmetric -

skew

A A

A real is

matrix hermitian

- skew a

If

symmetric

A A

A real is

matrix hermitian

a If

imaginary pure

are elements diagonal

a

a hermitian -

skew is

A If

real are

elements diagonal

a

a ermitian h

is A If

1 T

T

T T

jj jj

→

=

→

−

=

→

=

→

−

=

→

=

−

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43

EIGENVALUES

Theorem :

• The eigenvalues of a Hermitian matrix are real.

• The eigenvalues of a skew-Hermitian matrix are pure imaginary or 0.

• The eigenvalues of a unitary matrix have absolute

value of "1".

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EIGENVALUES (cont)

Proof: Let λ be an eigenvalue of A, x be a corresponding eigenvector.

Ax = λ λλλx (a) Assume A is Hermitian

[ ]

1

2

1 2 n

1 1

2 2

1

x x x

0 since x 0

T T T

T

n n n

n

x Ax x x x x

x x x x

x

x x x x

x x

λ λ

= =

   

=  

   

 

= + +

≠ ≠

L M

L

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EIGENVALUES (cont)

( )

T T

T T number

T T

A A

or

A A

: Hermitian

Ax x

x A x

Ax x

real is

Ax x

if real is

x x

Ax x

=

 

 



= 

=

= λ

3

2

1

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EIGENVALUES (cont)

( )

( ) ^T ( ) ^T ^T

T

T T

T T T

T T

x x

x A and x Ax

A A

Ax x

x A x

Ax x

propertry.

of use no

made we

since

x x

Ax x

λ

= λ

=

−

=

 

 



− 

=

−

=

= λ

(b) If A is skew-Hermitian

(c) If A is unitary

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EIGENVALUES (cont)

( )

1 x x Ax

A x

Ax A

x Ax

x A

x x

Ax x

A

2 T 1

T

T T T

2 T T T T

= λ

=

= λ

=

λ λ

=

λ λ

=

−

Multiplying :

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QUADRATIC FORM (cont)

If A is Hermitian or skew-Hermitian; the form is called Hermitian or skew-Hermitian form.

Theorem:

For every choice of x, the value of an Hermitian form is real, and the value of a skew-Hermitian form is pure imaginary or 0 . Inner Product:

b a b

a ⋅ = ^T

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Length or Norm

Theorem :

A unitary transformation, y = Ax, A unitary, preserves the value of the inner product and the norm.

Proof:

QUADRATIC FORM (cont)

2 n 2

1 n n 2

2 1

1 T

a a

a a a

a a

+ +

=

+ +

+

=

⋅

=

⋅

=

L

( ) ^{( )} ( ) ^{( )}

b a b a Ab A

a

Ab A

a Ab

a A Ab

a A v

u v

u

T 1

T

T T T

T T

⋅

=

⋅

−

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50

ENGINEERING MATHEMATICS II

ENGINEERING MATHEMATICS II

010.141

CHAP. 8

Linear Algebra: Matrix Eigenvalue

Problem

ABSTRACT OF CHAP. 8

Linear algebra in Chaps. 7 and 8 discusses the theory and application of vectors and matrices, mainly related to

linear systems of equations, eigenvalue problems, and linear transformation.

Chapter 8 concerns the solutions of vector equations Ax = λ x

where A is a given square matrix and vector x and scalar λ .

Eigenvalue problems are of greatest practical interest to the engineer, physicist, and mathematician.

Eigenvectors are transformed to themselves multiplied

with a characteristic constant (eigenvalues).

CHAP. 8.1

EIGENVALUES, EIGENVECTORS

Eigenvalues and eigenvectors imply characteristic

values and vectors of a matrix A.

SOME DEFINITIONS

Let A be an n × n matrix and consider

Ax = λ x (1)

λ , such that (1) has solution x ≠ 0 is an eigenvalue or characteristic value of A.

x are eigenvectors or characteristic vectors of A.

The spectrum of A is the set of eigenvalues of A;

max| λ | is the spectral radius of A.

The set of eigenvectors corresponding to λ (including 0) is

the eigenspace of A for λ .

SOME DEFINITIONS (cont)

Homogeneous linear system in x 1 , x 2

→ Cramer's theorem D( λ ) = det(A- λ I) = 0

D( λ ) is the characteristic determinant, and

D( λ ) = 0 is the characteristic equation

EXAMPLE 1

EXAMPLE 1

THEOREMS

EXAMPLE 2

EXAMPLE 2

MULTIPLICITY

The algebraic multiplicity is the order M λ of λ in the

characteristic polynomial.

The geometric multiplicity (m λ ) of λ is the number of linearly independent vectors corresponding to λ .

( λ )

−

=

∆

≤

=

λ λ

λ

λ λ

∑ λ

of defect

m M

M m

n M

12

Example 3

Example 4

HOMEWORK IN 8.1

HW1. Problem 6

HW2. Problem 10

HW3. Problem 30

CHAP. 8.2

SOME APPLICATIONS OF EIGENVALUE PROBLEMS

Range of applications of matrix eigenvalue problems.

EXAMPLE 1

EXAMPLE 1

EXAMPLE 2 – Markov Process

The eigenvalue problem can be used to identify the limit state of the process, in which the state vector x is reproduced under the multiplication by the

stochastic matrix A governing the process, that is, Ax=x.

the limit state of the process

EXAMPLE 4

EXAMPLE 4

EXAMPLE 4

HOMEWORK IN 8.2

HW1. Problem 8

HW2. Problem 19

CHAP. 8.3

SYMMETRIC, SKEW-SYMMETRIC, AND ORTHOGONAL MATRICES

Three classes of real square matrices frequently

the eigenspace of A for λ _.

Homogeneous linear system in x ₁ , x ₂

→ Cramer's theorem D( λ _{) = det(A-} λ _{I) = 0}

The algebraic multiplicity is the order M _λ of λ _{in the}

The geometric multiplicity (m λ _{) of} λ is the number of linearly independent vectors corresponding to λ .

( ^λ )

A square matrix A = [a _jk ] is