1550078481-Ordinary_Differential_Equations__Roberts_

506 Ordinary Differential Equations

By clicking on the button labeled CALCULATE THE EIGENVALUES

AND EIGENVECTORS OF A NEW MATRIX, we can find the eigenvalues

and eigenvectors of another matrix. By clicking on the box in the upper right-

hand corner of the monitor with an X in it, we can exit to the selection screen

(Figure A.l) which will allow us to run a different program in CSODE or to

exit CSODE.

EXAMPLE Finding the Eigenvalues and Eigenvectors of a

3 x 3 Matrix

Find the eigenvalues and eigenvectors of the matrix

(2) A = ( ~ ~ -~)

-2 0 -1

SOLUTION
In order to use EIGEN to solve this problem, we double click the CSODE
icon and when the program selection screen appears on the monitor, we single
click on the EIGEN button. Since the size of the matrix A is three, we enter
3 in the highlighted box after "N =" and press the Enter key. We input the
nonzero entries of A as integers just as they appear in (2). After making
sure we have entered the values correctly, we click on the VERIFY MATRIX
ENTRIES AND CALCULATE button. When the Please Confirm yes/no box
appears, we click yes, since the entries are entered correctly. This causes the
screen shown in Figure A.8 to appear on the monitor. The matrix A with
entries written in exponential form appears at the top of the output list box.
We used the bar at the right of the box to scroll down so we could see the
three eigenvalues and their associated eigenvectors. From Figure A.8 we see

that one eigenvalue is .A 1 = 1 and the associated eigenvector is

A second eigenvalue is .A2 = i and the associated eigenvector is

(

.5-.5i)

X2 = -.5 7 .5i ·

And a third eigenvalue is .A3 = -i and the associated eigenvector is

(

.5+.5i)

X3 = -.5_~ .5i.