Chapter 8.   Eigenvalues and Eigenvectors


2018학년도 2학기

       반도체 공학과 공학수학2 (GEDB005) 강의교안  (2018학년도용)

    주교재: Erwin Kreyszig, Engineering Mathematics10th Edition

    부교재: 최신공학수학 I 과 II, 한빛출판사 및 (영문) 다변량 Calculus (이상구, 김응기 et al)   (http://www.hanbit.co.kr/EM/sage/)

    강의시간: E12 (금 12:00 - 14:45), 반도체관 400126호 담당교수: 김응기 박사 

 

주차

주교재

부교재

6

8.1: Eigenvalues and Eigenvectors

8.4: Eigenbases, Diagonalization , Quadratic Forms

1.6 고유값과 고유벡터

1.7 닮음, 행렬의 대각화, 이차형식

web

http://www.hanbit.co.kr/EM/sage/1_chap1.html 

 


   Week 6



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 determinantcharacteristic 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

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/ 

① Characteristic polynomial of 




Evaluate

x^2 + 7*x + 6


② Hence the eigenvalues are as follows.




Evaluate

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


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




Evaluate

[-1, -6]


④ In order to find eigenvector for , solve 




Evaluate

[ 2 -1]

[ 0  0]


We obtain the eigenvector

                                       .


⑤ In order to find eigenvector for , solve .




Evaluate

[1 2]

[0 0]


We obtain the eigenvector

                                      .


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




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




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

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




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

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




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

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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

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




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.




8.4 Eigenbases. Diagonalization. Quadratic Forms


General properties of eigenvectors

Eigenvectors of an  matrix  may form a basis for . If we are interested in a transformation , such an “eigenbasis”(basis of eigenvectors).

We can represent any  in  uniquely as a linear combination of the eigenvectors ,

                            .

And denoting the corresponding eigenvalues of the matrix  by , we obtain

                    

(1)                                  

                           



Theorem 1  Basis of Eigenvectors

If an  matrix  has distinct eigenvalues, then  has a basis of eigenvectors  for .



Example 1  Eigenbases. Nondistinct Eigenvalues. Nonexistence

The eigenvalues and eigenvectors of the matrix .

Solution

Eigenvalues

                       

                        

                        

    

Characteristic equation is

   

Roots (eigenvalues of ) 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 is

                     .

Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

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



Even if not all  eigenvalues are difference, a matrix  may still provide an eigenbasis for .


A may not have enough linearly independent eigenvectors to make up a basis. For instance

  and has only one eigenvector  ( arbitrary).



Theorem 2  Symmetric Matrices

A symmetric matrix has an orthonormal basis of eigenvectors for .



Example 2  Orthonormal Basis of Eigenvectors

The matrix  is  symmetric, and an orthonormal basis of eigenvectors is

                             .

Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[ 1  1]

[ 1 -1]


[ 0.7071067811865475?  0.7071067811865475?]

[ 0.7071067811865475? -0.7071067811865475?]



Similarity of Matrices, Diagonalization


Definition  Similar Matrices Similarity Transformation

An  matrix  is similar  to an  matrix  if

                                   

for some  matrix .

This transformation, which gives  from  is a similarity transformation.

                         .



Theorem 3  Eigenvalues and Eigenvectors of Similar Matrices

If  is similar to , then  has the same eigenvalues as . Furthermore, if  is an eigenvector of , then  is an eigenvector of  corresponding to the same eigenvalue.

Proof

 ( an eigenvalue, ) we get .

Now .

By this “identity trick” the previous equation gives

.

Hence  is an eigenvalue of  and  a corresponding eigenvector.

Indeed,  would give , contradicting .



Example 3  Eigenvalues and Eigenvectors of Similar Matrices

Let  and .

Then .

Here  was obtained with . We see that  has the eigenvalues .

Characteristic equation of  is .

The roots (the characteristic equation of ) is .

From the first component of  we have .

For  this gives , say, .

For  this gives , say, .

We have

,  

These are eigenvectors of the diagonal matrix . We see that  and  are the column of . This suggests the general method of transforming a matrix  to diagonal from  by using , the matrix with eigenvectors as column.

Sage Coding

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Evaluate

([3 0]  [  1   1] ,[0 2], [  1 4/3] )


Theorem 4  Diagonalization of Matrix

If an  matrix  has a basis of eigenvectors, then

(5)                                   

is diagonal, with the eigenvalues of  as the entries on the main diagonal. Here  is the matrix these eigenvectors as column vectors. Also

                                  ()



Example 4  Diagonalization

Diagonalize .

Solution

The characteristic equation is .

The roots (eigenvalues of ) are .

By the Gauss elimination applied to  with .

We find eigenvectors and then  by the Gauss-Jordan elimination.

The results are

        ,   ,   ,   ,   

Calculating  and multiplying by  from the left, we thus obtain

             .

Sage Coding

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Evaluate

True

[(3, [(1, -3, 1)], 1), (0, [(1, 1/2, 2)], 1), (-4, [(1, -1, 3)], 1)]




Evaluate

[ 3  0  0]

[ 0  0  0]

[ 0  0 -4]



Quadratic Forms. Transformation to Principal Axes

By definition a quadratic forms  in the components  of a vector  is a sum of  terns, namely,

                 

                                         (7)

 is called the coefficient matrix of the from. The matrix  is symmetric.

We can take off-diagonal terns together in pairs in pairs and write the result as a sum of two equal terns.



Example 5  Quadratic Form. Symmetric Coefficient Matrix

Let .

Here .

From the corresponding symmetric matrix , where ,

thus , we get the same result,

.



By Theorem 2 the symmetric coefficient matrix  of  has an orthonormal basis of eigenvectors. Hence if we take these as column vectors, we obtain a matrix  that is orthogonal.

                      ().

          .                              (8)

If we get , then, since , we get

                        .                                  (9)

Furthermore, in (8) we have  and , so that  becomes simply

                 .            (10)



Theorem 5  Principal Axes Theorem

The substitution  transforms a quadratic form

                                ()

to the principal axes form or canonical form , where  are the (not necessarily distinct) eigenvalues of the matrix , and  is an orthogonal matrix with

corresponding eigenvectors , respectively, as column vectors.


Example 6  Transformation to Principal Axes. Conic Sections

Find out what type of conic section the following quadratic form represents and transform it to principal axes :

                         .

Solution

We have , we have

                         ,  .

The characteristic equation .

The roots (eigenvalues of ) are .

Hence (10) becomes

                                .

 represents the ellipse , that is

                                  .

The direction of the principal axes in the -coordinates,

we have determine normalized eigenvectors from  with  and 

and  then (9). We get

                              and 

hence

             ,        .

This is  rotation.

Sage Coding

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① Computing eigenvalues of 




Evaluate

[32, 2]


② Computing eigenvectors of 




Evaluate

[(32, [(1, -1)], 1), (2, [(1, 1)], 1)]


③ Computing  diagonalizing 




Evaluate

[ 1/2*sqrt(2)  1/2*sqrt(2)]

[-1/2*sqrt(2)  1/2*sqrt(2)]


④ Sketching two ellipses simultaneously




Evaluate

그림입니다.
원본 그림의 이름: mem000025100001.png
원본 그림의 크기: 가로 594pixel, 세로 584pixel

 

   http://matrix.skku.ac.kr/sglee/ 


[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

 





Copyright @ 2018 SKKU Matrix Lab. All rights reserved.
Made by Manager: Prof. Sang-Gu Lee and Dr. Jae Hwa Lee
*This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03035865).