1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 351

Then x 1 must satisfy

-3 ) (Xu)

-2 - 1 X21

= (1 -3) (Xu) = (0).

1 - 3 X21 0

Performing the matrix multiplication, we find that xu and x 21 must simul-

taneously satisfy the system of equations

xu - 3x21 = 0

xu - 3x21 = 0.

Since these equations are identical, we conclude xu = 3x2 1 and x 21 is arbi-

trary. Choosing x2 1 = 1, we find an eigenvector associated with the eigenvalue

>11 = 1 is

X1 = G~~) = G).

As we noted earlier the vector

where c =/=- 0 is any arbitrary constant is also an eigenvector of A associated

with the eigenvalue >-1 = 1.

An eigenvector x2 of A associated with >- 2 = -1 must satisfy the equation

Ax 2 = >-2x 2 or Ax2 - >-2x 2 = (A - >- 2 I)x 2 = 0. Letting x 2 = ( x^12 ) , we see

X22
that x 2 must satisfy

(

2+1

(A - >. 2 I)x 2 = (A+ I)x2 = l -3 ) (X12)

-2+1 X22

= (3 -3) (X12) = (0).

1 -1 X22 0

Performing the required matrix multiplication, we see x 12 and x22 must satisfy
the system of equ ations
3x12 - 3x22 = 0

X12 - X22 = 0.

Notice that the first equation of this system is three times the last equation.
So actually there is only one equation- say, X21 - X22 = 0- to be satisfied by