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Chapter 04.
Eigenvalues and Eigenvectors
After reading this chapter, you should be able to:
- define eigenvalues and eigenvectors of a square matrix,
- find eigenvalues and eigenvectors of a square matrix,
- relate eigenvalues to the singularity of a square matrix, anduse the power method to numerically find the largest eigenvalue in magnitude of a square matrix and the corresponding eigenvector. What does eigenvalue mean? The characteristic word eigenvalue and Wert means comes (^) value from (^). theHowever, what the word means is not on your mind! German word Eigenwert where Eigen means You want to know why I need to learn about eigenvalues and eigenvectors. Once I give you an example of an application of eigenvalues and eigenvectors, you will want to know how to find these eigenvalues and eigenvectors. Can you give me a physical example application of eigenvalues and eigenvectors? Look at the spring-mass system as shown in the picture below.
Assume each of the two mass-displacements to be denoted by x 1 and x 2 , and let us assume each spring has the same spring constant k. Then by applying Newton’s 2nd^ and 3 rd^ law of motion to develop a force-balance for each mass we have m 1 (^) ddt^2 x 21 kx 1 k ( x 2 x 1 ) m 2 (^) ddt^2 x 22 k ( x 2 x 1 )
x 1 x 2
m 1 m 2
k k
04.10.2 Chapter 04.
Rewriting the equations, we have m 1 ddt^2 x 21 k ( 2 x 1 x 2 ) 0 m 2 ddt^2 x 22 k ( x 1 x 2 ) 0 Let m 1 10 , m 2 20 , k 15
20 ddt^2 x 22 15 ( x 1 x 2 ) 0 From vibration theory, the solutions can be of the form
x i Ai sin t 0
where A i = amplitude of the vibration of mass i , = frequency of vibration, 0 = phase shift. then
ddt^2 x 2 i Aiw 2 Sin ( t 0 )
Substituting xi and d dt^^2 x 2 i^ in equations,
10 A 1 ^2 15 ( 2 A 1 A 2 ) 0
20 A 2 ^2 15 ( A 1 A 2 ) 0
gives
( 10 ^2 30 ) A 1 15 A 2 0
15 A 1 ( 20 ^2 15 ) A 2 0
or
( 2 3 ) A 1 1. 5 A 2 0
0. 75 A 1 ( ^2 0. 75 ) A 2 0
In matrix form, these equations can be rewritten as 0. 75 ^3 (^21) .^50. 75 21 00
2 A
A
03. 75 0.^175.^5 AA^12 ^ ^2 AA 21 00
Let 2
[ A ] 03. 75 0.^175.^5
10 ddt^2 x 21 15 ( 2 x 1 x 2 ) 0
04.10.4 Chapter 04.
Solution
[ A ] [ I ]^30 . 75 0 . 751 .^5
det( A I ) ( 3 )( 0. 75 )( 0. 75 )( 1. 5 ) 0
2. 25 0. 75 3 ^2 1. 125 0
^2 3. 75 1. 125 0
(^ ^3.^75 ) (^3.^75 )^2 ^4 (^1 )(^1.^125 )
^3.^75 ^3.^092
So the eigenvalues are 3.421 and 0.3288. Example 2 Find the eigenvectors of A (^) 03. 75 0.^175.^5 Solution The eigenvalues have already been found in Example 1 as
Let [ X ] xx 21 be the eigenvector corresponding to
Hence
([ A ] 1 [ I ])[ X ] 0
^03.^75 ^0.^175.^5 ^3.^421 ^0110 ^ xx^12 ^0 ^00 ..^42175 21. 671.^5 xx 21 00 If x 1 s then
x s
s x
- 2808
2
2
The eigenvector corresponding to 1 3. 421 then is
Eigenvalues and Eigenvectors 04.10.
[ X ] (^) 0. 2808 s s s 0. 28081 The eigenvector corresponding to
is 0.^12808 Similarly, the eigenvector corresponding to
is 1. 7811
Example 3 Find the eigenvalues and eigenvectors of
[ A ]
Solution The characteristic equation is given by
det([ A ] [ I ]) 0
det
( 1. 5 )[( 0. 5 )()( 0. 5 )( 0 )]( 1 )[( 0. 5 )( 0 )( 0. 5 )( 0. 5 )] 0
^3 2 ^2 1. 25 0
The roots of the above equation are
Note that there are eigenvalues that are repeated. Since there are only two distinct eigenvalues, there are only two eigenspaces. But, corresponding to = 0.5 there should be two eigenvectors that form a basis for the eigenspace.To find the eigenspaces, let
3
2
1 [ ] x
x
x X
Given
[( A I )][ X ] 0
Eigenvalues and Eigenvectors 04.10.
What are some of the theorems of eigenvalues and eigenvectors? Theorem 1: If [ A ] is a n n triangular matrix – upper triangular, lower triangular or diagonal, the eigenvalues of [ A ] are the diagonal entries of [ A ].
Theorem 2: 0 is an eigenvalue of [ A ] if [ A ] is a singular (noninvertible) matrix.
Theorem 3: [ A ] and [ A ] Thave the same eigenvalues. Theorem 4: Eigenvalues of a symmetric matrix are real. Theorem 5: Eigenvectors of a symmetric matrix are orthogonal, but only for distinct eigenvalues. Theorem 6: det( A ) is the product of the absolute values of the eigenvalues of [ A ]^.
Example 4 What are the eigenvalues of
[ A ]
Solution Since the matrix [ A ] is a lower triangular matrix, the eigenvalues of [ A ] are the diagonal elements of [ A ]. The eigenvalues are
Example 5 One of the eigenvalues of
[ A ]
is zero. Is [ A ] invertible? Solution
0 is an eigenvalue of [ A ] , that implies [ A ] is singular and is not invertible.
Example 6 Given the eigenvalues of
[ A ]
are
04.10.8 Chapter 04.
What are the eigenvalues of [ B ] if
[ B ]
Solution Since [ B ] [ A ] T , the eigenvalues of [ A ] and [ B ] are the same. Hence eigenvalues of [ B ] also are
Example 7 Given the eigenvalues of
[ A ]
are
Calculate the magnitude of the determinant of the matrix. Solution Since
det[ A ] 1 2 3
How does one find eigenvalues and eigenvectors numerically? One of the most common methods used for finding eigenvalues and eigenvectors is the power method. It is used to find the largest eigenvalue in an absolute sense. Note that if this largest eigenvalues is repeated, this method will not work. Also this eigenvalue needs to be distinct. The method is as follows:
- Assume a guess [ X (^0 )]for the eigenvector in
[ A ][ X ] [ X ]
equation. One of the entries of [ X (^0 )]needs to be unity.
- Find [ Y (^1 )][ A ][ X (^0 )]
- Scale [ Y (^1 )]so that the chosen unity component remains unity.
[ Y (^1 )] (^1 )[ X (^1 )]
- Repeat steps (2) and (3) with [ X ] [ X (^1 )]to get [ X (^2 )].
- Repeat the steps 2 and 3 until the value of the eigenvalue converges.
04.10.10 Chapter 04.
[ A ][ X (^1 )]
[ X (^2 )] 1. 3
(^2 ) 1. 3
[ X (^2 )]
The absolute relative approximate error in the eigenvalues is
a (^2 )( 2 )(^1 ) 100
^1.^31 . 51.^5 100
Conducting further iterations, the values of ( i )and the corresponding eigenvectors is given
in the table below
i ( i ) [ X ( i )]^ a (%)
_____
Eigenvalues and Eigenvectors 04.10.
The exact value of the eigenvalue is 1
and the corresponding eigenvector is
[ X ]
Key Terms: Eigenvalue Eigenvectors Power method
