The document discusses the eigenvalue-eigenvector problem, which has applications in solving differential equations, modeling population growth, and calculating matrix powers. It provides mathematical background on homogeneous systems of equations where the eigenvalues are the roots of the characteristic polynomial. Iterative methods like the power method are presented for finding the dominant or lowest eigenvalue of a matrix. Physical examples of mass-spring systems are given where the eigenvalues correspond to vibration frequencies and the eigenvectors to mode shapes.

1. Eigenvalue – Eigenvector Problem

2. EIGENVALUES & EIGENVECTORS
The eigenvalue problem is a problem of considerable
theoretical interest and wide-ranging application.
For example, this problem is crucial in solving systems of
differential equations, analyzing population growth models,
and calculating powers of matrices (in order to define the
exponential matrix (A100)).
Other areas such as physics, sociology, biology, economics
and statistics have focused considerable attention on
“eigenvalues” and “eigenvectors”-their applications and their
computations

4. Eigenvalue Problems
(Mathematical Background)
 
 
  0xaxaxa
0xaxaxa
0xaxaxλa
nnn2n21n1
n1n222121
n1n212111









   0XA
The roots of polynomial D(λ) are the eigenvalues of the eigen system
A solution {X} to [A]{X} = λ{X} is an eigen vector
(homogeneous system)
      0 XIA 
       IA   tDeterminanD
(eigen system)

6. Example 1

8. CW

9. Example 2

15. 15
The Power Method
(An iterative approach for determining the largest eigenvalue)
Example (3):
Iteration 1: initialization [x1, x2, x3]T = [1 1 1]T
Iteration 2: A [1 0 1]T
Iteration 3: A [1 -1 1]T
Iteration 4: A [-0.75 1 -0.75]T
Iteration 4: A [-0.714 1 -0.714]T
(Exact solution = 6.070)

16. Class Work: Use the power method to find the dominant eigenvalue and
eigenvector for the matrix

19. Example 4

21. 21
Power Method for Lowest Eigenvalue
(An iterative approach for determining the lowest eigenvalue)
 












422028101410
281056202810
141028104220
1
...
...
...
A










































7510
1
7510
1241
8840
1241
8840
1
1
1
422028101410
281056202810
141028104220
.
.
.
.
.
.
...
...
...
Example (5):
Iteration 1:
Iteration 3:
Exact solution is 0.955 which is the reciprocal of the smallest eigenvalue,
1.0472 of [A].
Iteration 2:










































7150
1
7150
9840
7040
9840
7040
7510
1
7510
422028101410
281056202810
141028104220
.
.
.
.
.
.
.
.
...
...
...










































7090
1
7090
9640
6840
9640
6840
7150
1
7150
422028101410
281056202810
141028104220
.
.
.
.
.
.
.
.
...
...
...
Idea: The largest eigenvalue of [A]-1 is the reciprocal of the lowest
eigenvalue of [A]
 













 5563
422077810
778155632810
077815563
7781 .
..
...
..
.A

22. Faddeev-Leverrier Method for Eigenvalues

27. Use Faddeev’s Method to find the eigenvalues of the following matrix

28. Eigenvalue Problems
(Physical Background 1)
 1212
1
2
1 xxkkx
dt
xd
m 
 tsinAx ii 
Mass-spring system
Analytical solution
(vibration theory)
  2122
2
2
2 kxxxk
dt
xd
m 
k k
Ai = the amplitude of the vibration of mass i
and ω = the frequency of the vibration, which is
equal to ω = 2π/Tp, where Tp is the period.
1
2
3
 tsin-Ax 2
ii  4
To Find A1, A2, and :
Substitute equations 3 and 4 into 1 and 2
0AA
0AA
21
21














2
22
1
2
1
2
2


m
k
m
k
m
k
m
k
The eigenvalues to this system are the
correct frequency , and the eigenvectors
are the correct A1 and A2.
take 2 as λ

29. 0AA
0AA
21
21














2
22
1
2
1
2
2


m
k
m
k
m
k
m
k
The eigenvalues to this system are the
correct frequency , and the
eigenvectors are the correct A1 and A2.
take 2 as λ
 
 
  0xaxaxa
0xaxaxa
0xaxaxλa
nnn2n21n1
n1n222121
n1n212111









   0XA (homogeneous system)
      0 XIA  (eigen system)

30. 30
Eigenvalue Problems
(An instance of mass-spring problem)
0AA
0AA
21
21














2
22
1
2
1
2
2


m
k
m
k
m
k
m
k
 
 




modeseconds
modefirsts
-
-
1
1
2362
8733
.
.

k=200 N/m
First mode: A1 = -A2
Second mode: A1 = A2
 
  0AA
0AA
21
21


2
2
105
510

m1=m2=40 kg

31. A unique set of values cannot be obtained
for the unknowns. However, their ratios can
be specified by substituting the eigenvalues
back into the equations. For example, for
the first mode (ω2 = 15 s−2), Al=−A2. For the
second mode (ω2 = 5 s−2), A1 = A2.
Note:

32. Eigenvalue Problems
(Physical Background 2)

35. Eigenvalue Problems
(Physical Background 3)

37. Special Problem 5

38. Home Work
1.
2.
3.