1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 357

first equation.) Consequently, system (5) reduces to a single condition- either

the first equation or the second equation of (5). Hence, one variable x 11 or x 21

is arbitrary and the other is determined by one equation from (5). Choosing

X21 = 1 and solving the second equation of (5) for x 11 , we get x 11 = -1 + i.

Hence, an eigenvector of the matrix A associated wit h the eigenvalue > 11 = 2+i

is

Let

be an eigenvector of the matrix A associated with the eigenvalue >. 2 = 2 - i.

The vector x 2 must satisfy

-2 ) (X12)

3 - (2 - i) X22

• 2 ) (X12) (Q)
  1 + i X22 - 0.

Thus, X12 and X22 must simultaneously satisfy

( -1 + i)x12 - 2x22 = 0

(6)

X12 + (1 + i)X22 = Q.

Since det (A - .A 2 1) = 0 these two equ ations must also be multiples of one

another. Therefore, either x 12 or x 22 may be selected arbitrarily and the other

variable then determined from either equation of (6). Choosing x 22 = 1 and

solving the second equation of (6) for x 12 , we get x 12 = -1 - i. Consequently,

a n eigenvector of t he matrix A associated with the eigenvalue >. 2 = 2 - i is

!Comments on Computer Software! Computer algebra systems (CAS) of-
ten include a routine to numerically compute all eigenvalues and eigenvectors
of a n n x n matrix with real entries (elements). The input for such pro-
grams is the size, n, of the matrix and the matrix itself. The output from the
program is a set of n eigenvalues and associated eigenvectors. The software