2019학년도 1학기

반도체 공학과 공학수학1

주교재: Erwin Kreyszig, Engineering Mathematics, 10th Edition

부교재: 이상구 외 4인, 최신공학수학 I,  1st Edition

강의시간:  공학수학1, (화09:00-10:15)   (목10:30-11:45)

담당교수:  김응기 박사

Week 14

14주차

8.1: The Matrix Eigenvalue Problem

1.6 고유값과 고유벡터

*8.2: Some Applications of Eigenvalue Problems

1.7 닮음, 행렬의 대각화, 이차형식

8.1 Eigenvalues, Eigenvectors

From the view point of engineering application, eigenvalue problems are among the most important problems in connection with matrices.

Let be a given matrix and consider the vector equation

(1)            .

Here is an unknown vector and is an unknown scalar.

Our task is to determine ’s and ’s that satisfy .

should be proportional to .

The zero vectors is a solution of for any value of , because .

A value of for which has a solution is an eigenvalue ( characteristic value) of the matrix .

The corresponding solutions of are an eigenvectors (characteristic vectors) of corresponding to that eigenvalue .

The set of all the eigenvalues of is called the spectrum of .

The spectrum consist of at least one eigenvalue and at most numerical different values.

The largest of the absolute values of eigenvalues of is called the spectral radius of .

How to Fine Eigenvalues and Eigenvectors

is homogeneous linear system

has a non-trivial solution is zero.

: Characteristic determinant or characteristic polynomial

: characteristic equation of

The eigenvalues of are the solutions of the characteristic equation of .

Example 1  Determination of Eigenvalues and Eigenvectors

The eigenvalues and eigenvectors of the matrix .

Solution

Eigenvalues

: characteristic determinant, characteristic polynomial

:  characteristic equation of

Solutions of this quadratic equation are , .

Eigenvector of corresponding .

Solution is .

This determines an eigenvector corresponding to .

We choose ,

.

Eigenvector is

.

.

Eigenvector of corresponding .

Solution is .

This determines an eigenvector corresponding to .

We choose ,

.

Eigenvector

.

.

Sage Coding

① Characteristic polynomial of

 A = matrix([[-5, 2], [2, -2]])    # input A print A.charpoly()            # characteristic equation of A

Evaluate

x^2 + 7*x + 6

② Hence the eigenvalues are as follows.

 solve(x^2 + 7*x + 6 == 0, x)

Evaluate

[x == -6, x == -1]

③ We can find the eigenvalues directly by using the built in command.

 A.eigenvalues()     # eigenvalues of A

Evaluate

[-1, -6]

④ In order to find eigenvector for , solve

 (A + identity_matrix(2)).echelon_form() # consider only coefficient matrix

Evaluate

[ 2 -1]

[ 0  0]

We obtain the eigenvector

.

⑤ In order to find eigenvector for , solve .

 (A + 6*identity_matrix(2)).echelon_form() # consider only coefficient matrix

Evaluate

[1 2]

[0 0]

We obtain the eigenvector

.

⑥ We can find the eigenvectors directly by using the built in command.

 A.eigenvectors_right()

Evaluate

[(-1, [(1, 2)], 1), (-6, [(1, -1/2)], 1)]

Equation written in components is

Transferring the terms on the right side to the left, we have

In matrix notation

(3)

This homogeneous linear system of equations has a nontrivial solution if and only if the corresponding determinant of the coefficients is zero :

(4)

is the characteristic matrix.

is the characteristic determinant of .

is the characteristic equation of .

By developing we obtain a polynomial of th degree in .

This is the characteristic polynomial of .

Theorem 1  Eigenvalues

The eigenvalues of a square matrix are the roots of the characteristic equation (4) of . Hence an matrix has at least one eigenvalue and at most numerically different eigenvalues.

Theorem 2  Eigenvectors, Eigenspace

If and are eigenvectors of a matrix corresponding to the same eigenvalue ,

so are and for any .

Hence the eigenvectors corresponding to one and the same eigenvalue of , together with , from a vector space called the eigenspace of corresponding to that .

Proof

and imply

and

hence

In particular, an eigenvector is determined only up to a constant factor.

Hence, we can normalize , that is, multiply it by a scalar to get a unit vector.

For instance

Let

Length of :

Normalized of :

Sage Coding

 x1 = vector([1, 2]) x1.norm() x1/(x1.norm())

Evaluate

(1/5*sqrt(5), 2/5*sqrt(5))

Example 2  Multiple Eigenvalues

Find the eigenvalues and eigenvectors of

.

Solution

The characteristic determinant gives the characteristic equation

.

The roots (eigenvalues of ) are , (Double root).

For the characteristic matrix is

Hence it rank .

.

Choosing we obtain form .

Taking and , we obtain from .

Hence an eigenvector of corresponding to is .

For the characteristic matrix is

.

Hence it rank .

.

Choosing ,       .

Choosing ,       .

We obtain two linearly independent eigenvectors of corresponding to ,

and .

Sage Coding

 A = matrix([[-2, 2, -3], [2, 1, -6], [-1, -2, 0]]) A.eigenvectors_right()

Evaluate

[(5, [(1, 2, -1)], 1), (-3, [(1, 0, 1/3),(0, 1, 2/3)], 2)]

Algebraic multiplicity of

The order of an eigenvalue as a root of the characteristic polynomial is called the algebraic multiplicity of .

Geometric multiplicity of

The number of linearly independent eigenvectors corresponding to is called the geometric multiplicity of .

Thus is the dimension of the eigenspace corresponding to this . Since the characteristic polynomial has degree , the sum of all the algebraic multiplicities must equal .

In general .

Defect of

The difference is called the defect of .

In example 2

For : and .

Example 3  Algebraic multiplicity, Geometric multiplicity, Positive defect

The characteristic equation of the matrix

is .

(double root)

Eigenvector of corresponding .

Solution is .

This determines an eigenvector corresponding to .

If we choose ,

.

We obtain the eigenvector

.

Algebraic multiplicity is .

Geometric multiplicity is ,

Defect is .

Since eigenvectors result from , hence , in the from .

Hence for the defect is .

Sage Coding

 A = matrix([[0, 1], [0, 0]]) A.eigenvectors_right()

Evaluate

[(0, [(1, 0)], 2)]

The characteristic equation of the matrix

is .

(double root)

Eigenvector of corresponding .

Solution is .

This determines an eigenvector corresponding to .

We choose ,

.

Eigenvector is

.

An eigenvalue of algebraic multiplicity . but its geometric multiplicity is only , since eigenvectors result from in the from .

For , the defect is .

Sage Coding

 A = matrix([[3, 2], [0, 3]]) A.eigenvectors_right()

Evaluate

[(3, [(1, 0)], 2)]

Example 4  Real Matrices with Complex Eigenvalues and Eigenvectors

Since real polynomials may have complex roots, and real matrix may have complex eigenvalues and eigenvectors.

The characteristic equation of the skew-symmetric matrix

is .

Two eigenvalues , .

Eigenvector of corresponding .

Solution is .

This determines an eigenvector corresponding to .

We choose ,

, .

Eigenvector is

.

Eigenvector of corresponding .

Solution is .

This determines an eigenvector corresponding to .

We choose ,

, .

Eigenvector is

.

Sage Coding

 A = matrix([[0, 1], [-1, 0]]) A.eigenvectors_right()

Evaluate

[(-1*I, [(1, -1*I)], 1), (1*I, [(1, 1*I)], 1)]

Theorem 3  Eigenvalues of the Transpose

The transpose of a square matrix has the same eigenvalues as .

Proof

Transposition does not change the value of the characteristic determinant.

*8.2 Some Applications of Eigenvalue Problems

Example 1  Stretching of an Elastic Membrane

An elastic membrane in plane with boundary circle is stretched so that a point goes over into the point given by

(1)          in components

Find the principal directions, that is, the directions of the position vector of for which the direction of the position vector of is the same or exactly opposite. What shape does the boundary circle take under this deformation?

Solution

We are looking for vectors such that .

Since , this gives , the equation of an eigenvalue problem.

is

(2)          or

The characteristic equation is

(3)

The roots (eigenvalues of ) are .

Eigenvector of corresponding .

A solution is .

This determines an eigenvector corresponding to .

If we choose ,

.

We obtain the eigenvector

.

Eigenvector of corresponding .

A solution is .

This determines an eigenvector corresponding to .

If we choose ,

.

We obtain the eigenvector

.

Sage Coding

 A = matrix([[5, 3], [3, 5]])    # input A print A.eigenvectors_right()   # [(eigenvalue, [eigenvectors], multiplicity)]

Evaluate

[(8, [(1, 1)], 1), (2, [(1, -1)], 1)]

The vectors make angle with the positive direction.

Principal directions.

The vectors make angle with the positive direction.

Principal directions.

They give the principal directions.

The eigenvalues show that in the principal directions the membrane is stretched by

factor and , respectively.

Accordingly, if we choose the principal directions as directions of a new Cartesian

coordination system, say, with the positive semi-axis in the first quadrant

and semi-axis in the second quadrant of the system.

system    coordination system

axis    semi-axis

axis    semi-axis

If we set , then a boundary point of the unstretched circular

membrane has coordinate , . Hence, after the stretch we have

,

Since , this show that the deformed boundary is an ellipse

(4)        .

Example 2  Eigenvalue Problems Arising from Markov Processes

Markov processes as considered in Example 13 of Sec. 7.2 lead to eigenvalue problems if we ask for the limit state of the process in which the state vector is reproduced under the multiplication by the stochastic matrix governing the process, that is,

.

1. Hence should have the eigenvalue , and should be a corresponding eigenvector.

This is of practical interest because it shows the long-term tendency of the development modeled by the process. In that example,

.    For the transpose  .

Hence has the eigenvalue has the eigenvalue

An eigenvector of for is obtained from

,  row-reduced to

Taking , we get from and then from .

An eigenvalue of for is .

It means that in the long ren, the ratio Commercial : Industrial : Residential will approach , provided that probabilities given by remain (about) the same.

Sage Coding

 A = matrix(QQ, [[0.7, 0.1, 0], [0.2, 0.9, 0.2], [0.1, 0, 0.8]])    # input A print A.eigenvectors_right()   # [(eigenvalue, [eigenvectors], multiplicity)]

Evaluate

[(1, [(1, 3, 1/2)], 1), (7/10, [(1, 0, -1)], 2)]

Example 3  Eigenvalue Problems Arising from Population Models. Leslie Model

The Leslie model describes age-specified population growth, as follows.

Let the oldest age attained by the females in some animal population be years.

Divide the population into three age classes of years each.

Let the “Leslie matrix” be

(5)

where is the average number of daughters born to a single female during the time she is in age class , and is the fraction of females in age class that will survive and pass into class .

(a) What is the number of females in each class after years if each class initially consists of females?

(b) For what initial distribution will the number of females in each class change by the same proportion? What is this rate of change?

Solution

(a) Initially, .

.

.

(b) Proportional change means that we are looking for a distribution vector such that , where is the rate of change (growth if , decrease if ).

The characteristic equation is

.

A positive root is equation to be .

A corresponding eigenvector can be determined from the characteristic matrix

,    say

where is chosen, then follows from and from

.

To get an initial population of as before, we multiply by .

Proportional growth of the numbers of female in the three classes will occur if the initial values are in classes respectively. The growth rate will be per years.

Sage Coding

 A = matrix(QQ, [[0, 2.3, 0.4], [0.6, 0, 0], [0, 0.3, 0]])    # input A print A.eigenvectors_right()   # [(eigenvalue, [eigenvectors], multiplicity)]

Evaluate

[(6/5, [(1, 1/2, 1/8)], 1),

(-1.147722557505166?, [(1, -0.5227744249483389?, 0.1366465496900332?)], 1),

(-0.05227744249483389?, [(1, -11.47722557505167?, 65.8633534503100?)], 1)]

Example 4  Vibrating System of Two Masses on Two Springs

Mass-Spring systems involving several masses and springs can be treated as eigenvalue problems.

For instance, the mechanical system is governed by the system of ODEs

(6)

where and are the displacements of the masses from the rest and primes denote derivatives with respect to time .

Vector from of the system

(7)      .

Try a vector solution of the form

(8)

Substitution into gives

.

Dividing by and writing , we see that our mechanical system leads to the eigenvalue problem

(9)         where

From Example 1 in Sec. 8.1, A has the eigenvalues and .

and

Corresponding eigenvectors are

(10)        and    .

From , obtain the four complex solutions

,

By addition and subtraction, we get the four real solutions

.

A general solution is obtained by taking a linear combination of these

with arbitrary constants (to which values can be assigned by prescribing initial displacement and initial velocity of each of the two masses).

By , the components of are

These functions describe harmonic oscillations of the two masses.

Physically, this had to be expected because we have neglected damping.

Sage Coding

 A = matrix([[-5, 2], [2, -2]])    # input A print A.eigenvectors_right()   # [(eigenvalue, [eigenvectors], multiplicity)]

Evaluate

[(-1, [(1, 2)], 1), (-6, [(1, -1/2)], 1)]

[한빛 아카데미] Engineering Mathematics with Sage:

[저자] 이상 구, 김영 록, 박준 현, 김응 기, 이재 화

Contents

A. 공학수학 1 – 선형대수, 상미분방정식+ Lab

Chapter 01 벡터와 선형대수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-1.html

Chapter 02 미분방정식의 이해 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-2.html

Chapter 03 1계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-3.html

Chapter 04 2계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-4.html

Chapter 05 고계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-5.html

Chapter 06 연립미분방정식, 비선형미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-6.html

Chapter 07 상미분방정식의 급수해법, 특수함수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-7.html

Chapter 08 라플라스 변환 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-8.html

B. 공학수학 2 - 벡터미적분, 복소해석 + Lab

Chapter 09 벡터미분, 기울기, 발산, 회전 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-9.html

Chapter 10 벡터적분, 적분정리 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-10.html

Chapter 11 푸리에 급수, 적분 및 변환 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-11.html

Chapter 12 편미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-12.html

Chapter 13 복소수와 복소함수, 복소미분 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-13.html

Chapter 14 복소적분 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-14.html

Chapter 15 급수, 유수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-15.html

Chapter 16 등각사상 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-16.html

Made by Prof. Sang-Gu LEE  sglee at skku.edu

http://matrix.skku.ac.kr/sglee/   with Dr. Jae Hwa LEE