1550078481-Ordinary_Differential_Equations__Roberts_

358 Ordinary Differential Equations

which accompanies this text contains a program named EIGEN, which com-
putes the eigenvalues and eigenvectors of an n x n matrix with real entries
where 2 :::; n :::; 6. Complete instructions for running this program appear in
Appendix A. The next example shows the typical output of EIGEN. When

an eigenvalue has multiplicity m > 1, EIGEN generates m associated vectors.

When the m associated eigenvectors are linearly independent the vectors pro-
duced by the program are also linearly independent, as they should be. When

an eigenvalue of multiplicity m > 1 does not have m associated linearly in-

dependent eigenvectors, the associated m vectors produced by EIGEN will

not be linearly independent. In general, if there are k < m linearly indepen-

dent eigenvectors associated with a particular eigenvalue, k of the m vectors
produced by EIGEN will be linearly independent eigenvectors.

EXAMPLE 5 Using Computer Software to Compute

Eigenvalues and Eigenvectors

Use EI GEN to compute the eigenvalues and associated eigenvectors of the
following matrices. Compare the results with the results obtained in examples
1, 2, 3, and 4.

(

2 -3)

l. 1 -2 2. (-~ ~)

SOLUTION

3 (-:

-1

1

1

(

4 1 -2)

· 1 3

1. We used the computer program EIGEN to calculate the eigenvalues and
   associated eigenvectors of the given matrix by setting the size of the
   matrix equal to two and then entering the values for the elements of the
   matrix. From Figure 8.1 we see that one eigenvalue is >. 1 = 1 + Oi = 1
   and the associated eigenvector is

= ( 1.66410 + Oi) = ( 1.64410)

Xi .55470+0i .55470.

We noted in example 1 that any vector of the form

where c-:/= 0 is an eigenvector associated with >. 1 = l. Observe that the

first component of x 1 is 3 times the second component of x 1. So x 1 is

clearly an eigenvector of the given matrix associated with the eigenvalue

>-1 = l. When we computed an eigenvector associated with >. 1 = 1