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MATH2071: LAB #12: The Eigenvalue Problem
TABLE OF CONTENTS
•
Introduction
•
The Rayleigh Quotient
•
The Power Method
•
The Inverse Power Method
•
Using Shifts
•
The QR Method
•
ASSIGNMENT
Introduction
For any square matrix
A
, consider the equation
det(Alambda*I)=0
. This is a polynomial equation of degree
N
in the variable
lambda
. Hence there are exactly
N
complex roots that satisfy the equation. If the matrix
A
is real, then the complex roots occur in conjugate pairs. The roots are known as
eigenvalues
of
A
. Interesting
questions include:
•
how can we find one or more of these roots?
•
when are the roots distinct?
•
when are the roots real?
In textbook examples, the determinant is computed explicitly, the (usually cubic) equation is factored
exactly, and the roots "fall out". This is not how a real problem will be solved.
If
lambda is an eigenvalue of A, then Alambda*I is a singular matrix, and therefore there is at least
one nonzero vector x with the property that (Alambda*I)*x=0. This equation is usually written
A * x = lambda * x
Such a vector is called an
eigenvector
for the given eigenvalue. There may be as many as N linearly
independent eigenvectors. In some cases, the eigenvectors are, or can be made, into a set of pairwise
orthogonal vectors. Interesting questions about eigenvectors include:
•
how do we compute an eigenvector?
•
when will there be a full set of N independent eigenvectors?
•
when will the eigenvectors be orthogonal?
In textbook examples, the singular system (Alambda*I)*x=0 is examined, and by inspection, an
eigenvector is determined. This is not how a real problem is solved.
Some useful facts about the eigenvalues lambda of a matrix A:
•
A
1
has the same eigenvectors as A, and eigenvalues 1/lambda;
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A
2
has the same eigenvectors as A, and eigenvalues lambda
2
;
•
A+mu*I has the same eigenvectors as A, and eigenvalues lambda+mu;
•
if A is symmetric, all its eigenvalues are real;
•
if B is invertible, then inv(B)*A*B has the same eigenvalues as A;
The Rayleigh Quotient
If a vector x is an exact eigenvector of a matrix A, then it is easy to determine the value of the
associated eigenvalue: simply take the ratio of the Ith components of A*x and x, for any index I that
you like.
But suppose x is only an approximate eigenvector, or worse yet, just a wild guess. We could still
compute the ratio of corresponding components for some index, but the value we get will vary from
component to component. We might even try computing all the ratios, and averaging them, or taking
the ratios of the norms. However, the preferred method for estimating the value of an eigenvalue uses
the Rayleigh quotient.
The
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This note was uploaded on 04/26/2010 for the course CEG 616 taught by Professor Taylor during the Winter '10 term at Wright State.
 Winter '10
 Taylor

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