Linear Systems of First-Order Differential Equations 359
by hand, we selected c = 1 and got a different eigenvector than the
eigenvector selected by the computer. But this is to be exp ected, si n ce
eigenvectors are not unique. They are unique only up to an arbitrary
scalar multiple, as we proved earli er. From Figure 8.1 we see that the
second eigenvalue is .A 2 = -1 and a n associated eigenvector is
(
l.80278)
X^2 = 1.80278.
In example 1 the associated eigenvector which we computed manually
was x = ( ~). Since the first and second components of x 2 are identical,
t h e vectors x 2 and x are both eigenvectors associated with .A 2 = -1.
Clearly, x 2 = l.80278x.
THE MATRIX WHOSE EIGENVALUES .AND EIGENVECTORS ARE TO BE CALCULATED IS
2.0000EtOO -3.0000EtOO
l .OOOOEtOO -2.0000EtOO
AN EIGENVALUE IS l.OOOOOEtOO t O.OOOOOEtOO I
THE ASSOCIATED EIGENVECTOR IS
l .66410Et00 t O.OOOOOEtOO I
5.54700E-Ol t O.OOOOOEtOO I
AN EIGENVALUE IS -1.00000EtOO t O.OOOOOEtOO I
THE ASSOCIATED EIGENVECTOR IS
l.80278Et00 t O.OOOOOEtOO I
l.80278Et00 t O.OOOOOEtOO I
Figure 8.1 Eigenvalues and Eigenvectors for Example 5.1.
- We used EIGEN to calculate the eigenvalues and associated eigenvectors
of t h e given matrix by setting the size of the matrix equ al to two and
then entering the values for the elements of t he matrix. The two eigen-
values and associated eigenvectors computed by EIGEN a re displayed
in Figure 8.2. In this case, .A 1 = .A 2 = 2 as we found in example 2. From
Figure 8 .2 we see that the computer has generated the following two
associated vectors
(
.707107)
Xl = - .707107 and (
.318453 x 1016 )
X^2 = -. 318453 X 1016.
These vectors are linearly dependent, since x 2 is a multiple of x 1 - that
is, x 2 = kx 1. In example 2, we found the matrix under consideration
here has only one eigenvector associated with the eigenvalue .A = 2. The
