2020학년도 1학기
반도체 공학과 공학수학1
주교재: Erwin Kreyszig, Engineering Mathematics, 10th Edition
부교재: 이상구 외 4인, 최신공학수학 I, 1st Edition
실습실: http://www.hanbit.co.kr/EM/sage/ http://matrix.skku.ac.kr/LA/
강의시간: 공학수학1, (월15:00-16:15) (수16:30-17:45)
담당교수: 김응기 박사
Week 14
14주차
8.1: The Matrix Eigenvalue Problem
1.6 고유값과 고유벡터
http://matrix.skku.ac.kr/LA/Ch-4/
*8.2: Some Applications of Eigenvalue Problems
1.7 닮음, 행렬의 대각화, 이차형식
http://matrix.skku.ac.kr/LA/Ch-8/
http://matrix.skku.ac.kr/2018-EM/EM-2-W6-lab.html
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
http://math3.skku.ac.kr/ http://sage.skku.edu/ http://mathlab.knou.ac.kr:8080/
① Characteristic polynomial of 
② Hence the eigenvalues are as follows.
③ We can find the eigenvalues directly by using the built in command.
④ In order to find eigenvector for 
, solve 
We obtain the eigenvector
.
⑤ In order to find eigenvector for 
, solve 
.
We obtain the eigenvector
.
⑥ We can find the eigenvectors directly by using the built in command.
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
http://math3.skku.ac.kr/ http://sage.skku.edu/ http://mathlab.knou.ac.kr:8080/
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/
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/
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
http://math3.skku.ac.kr/ http://sage.skku.edu/ http://mathlab.knou.ac.kr:8080/
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/
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
http://math3.skku.ac.kr/ http://sage.skku.edu/ http://mathlab.knou.ac.kr:8080/
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
http://math3.skku.ac.kr/ http://sage.skku.edu/ http://mathlab.knou.ac.kr:8080/
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
http://math3.skku.ac.kr/ http://sage.skku.edu/ http://mathlab.knou.ac.kr:8080/
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
http://math3.skku.ac.kr/ http://sage.skku.edu/ http://mathlab.knou.ac.kr:8080/
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
Made by Prof. Sang-Gu LEE sglee at skku.edu
http://matrix.skku.ac.kr/sglee/ with Dr. Jae Hwa LEE and DongGeoul Han
Copyright @ 2020 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).
