Lecture 12 Eigenanalysis

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Lecture 12 - Eigen-analysis

• CVEN 302
• June 22, 2001

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Lectures Goals

• What is an eigenvalue and eigenvector?
• Direct computation of the ls and eigenvectors.
• Power Method
• Shift technique

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Eigenvalue Problems

• In solving homogeneous linear differential
  equations, an acceptable form of solution is
  determined by an eigenvalue problem.
• To solve ODE, one must find an eigenvalue/eigenfun
  ction for which in general are infinite in
  numbers. These are different than eigen problems.

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Eigen-Analysis

• Matrix eigenvalues arise from discrete models of
  physical systems
• Discrete models
• Finite number of degrees of freedom result in a
  finite number of eigenvalues and eigenvectors.

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Eigenvalues

• Computing eigenvalues of a matrix is important in
  numerous applications
• In numerical analysis, the convergence of an
  iterative sequence involving matrices is
  determined by the size of the eigenvalues of the
  iterative matrix.
• In dynamic systems, the eigenvalues indicate
  whether a system is oscillatory, stable (decaying
  oscillations) or unstable(growing oscillation)

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Eigenvalues

• Oscillator system, the eigenvalues of
  differential equations or the coefficient matrix
  of a finite element model are directly related to
  natural frequencies of the system
• Regression analysis, eigenvectors of correlation
  matrix are used to select new predictor variables
  that are linear combinations of the original
  predictor variables.

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Physical Examples

• Natural vibration of systems of mass springs
• Flutter of the airplane wings
• vibration of membranes
• oscillation of a suspension bridge
• torsional vibration of multi-cylindrical engine
• structural response of earthquakes

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General form of the equations

• The general form of the equations

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Example

• For a set of equations

Rewrite the equations
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Example

• The equations can be

The determinant of the matrix is
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Example
Determinant

• Expand the equation

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Examples

• The equation can be factored

Eigenvalues are
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Example

• The eigenvector for l 3, can be determined by
  plug-in the equation

The matrix is singular so there are infinite
number of results.
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Example
Assume that one value of the x values is 1.
Therefore, x2 is 1. So the eigenvector for l 3
is 1, 1T.
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Example
For second eigenvalue, l -1, the equation
becomes
Assume x11 therefore x2 is -1. So the
eigenvector for l -1 is 1, -1T.
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Eigen-analysis
Unfortunately, we can not find the eigenvalues of
A general matrix by simply reducing it to a
triangular form by Gaussian elimination as we
might hope.
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Eigen-analysis
We can find the largest eigenvalue by using an
iterative procedure called the power method. Any
x vector can be represented by a combination of
the systems eigenvectors.
Multiply the equation by A for each Af is
equal to lf.
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Power Method
The equation can be rewritten as
Multiply the equation by A again.
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Power Method
In general continue the multiplication
where,
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Power Method
Factor the large l value term
As you continue to multiply the vector by A
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Power Method
The basic computation of the power method is
summarized as
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Power Method
The basic computation of the power method is
summarized as
The equation can be written as
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The Power Method Algorithm
Initialize u0 x0 vector for k 1,2,n w
Auk-1 l w uk (1/l) w end
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Example of Power Method
Consider the follow matrix A
Assume an arbitrary vector x0 1 1 1T
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Example of Power Method
Multiply the matrix by the matrix A by x
Normalize the result of the product
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Example of Power Method
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Example of Power Method
As you continue to multiple each successive
vector l 4 and the vector uk1 0 0T
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Power method
The special advantage of the power method is that
the eigenvector corresponds to the dominant
eigenvalue and is generated at the same
time. The disadvantage is that the method only
supplies obtains one eigenvalue
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Shift method
It is possible to obtain another eigenvalue from
the set equations by using a technique known as
shifting the matrix.
Subtract the a vector from each side, thereby
changing the maximum eigenvalue
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Shift method
The eigenvalue, s, is the maximum value of the
matrix A. The matrix is rewritten in a form.
Use the Power method to obtain the largest
eigenvalue of B.
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Example of Power Method
Consider the follow matrix A
Assume an arbitrary vector x0 1 1 1T
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Example of Power Method
Multiply the matrix by the matrix A by x
Normalize the result of the product
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Example of Power Method
Continue with the iteration and the final value
is l -5. However, to get the true you need to
shift back by
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Inverse Power Method
The inverse method is similar to the power
method, except that it finds the smallest
eigenvalue. Using the following technique.
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Inverse Power Method
The algorithm is the same as the Power method and
the eigenvector is not the eigenvector for the
smallest eigenvalue. To obtain the smallest
eigenvalue from the power method.
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Matlab Program

• There are set of programs Power and InversePower.
  
• The Power program does the power method
  Power(A,x0,iter).
• The InversePower(A, x0,iter,tol) does the inverse
  method.

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Summary

• Eigen-analysis of the set of equations
• Finding an eigenvalue.
• Power Method
• Shifting technique
• Inverse Power Method

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Homework

• Check the Homework webpage