1550078481-Ordinary_Differential_Equations__Roberts_

356 Ordinary Differential Equations

EXAMPLE 4 A 2 x 2 Matrix with Complex Eigenvalues

and Eigenvectors

Find the eigenvalues and associated eigenvectors of the matrix

-2) 3.

SOLUTION

The characteristic equation of the matrix A is

Solving the quadratic equation >-^2 - 4>-+ 5 = 0 , we find the eigenvalues of the
matrix A are >- 1 = 2 + i and >- 2 = 2 - i.

An eigenvector

X1-_ (X11)

X21

of the matrix A corresponding to the eigenvalue >- 1 = 2 + i must satisfy

-2 ) (X11)

3 - (2 + i) X21

1 --2 i ) (X11X21 ) - (0) 0.

Multiplying, we see that x 11 and x 21 must simultaneously satisfy the system
of equations

( -1 - i)x11 -2x21 = 0
(5)
X11 + (1 - i)X21 = 0.

Since det (A->- 1 1) = 0, these two equations must be multiples of one another.

(To check this fact, multiply the second equation by ( -1 - i) and obtain the