1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations

Multiplying we find x 1 and x 2 must simultaneously satisfy

XI+ X2 = 0

-X1 - X2 = 0.

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Since the second equation is -1 times the first, there is actually only one

equation to be satisfied, say x 1 + x2 = 0. Thus, x 1 = -x2 and x 2 is arbitrary.

Choosing x2 = 1, we find

is an eigenvector of A associated with >. = 2. Although for k -:/-0 , the vector

kx is an eigenvector associated with >. = 2, the vector kx is not linearly

independent from x, since l(kx) - k(x) = 0. That is, x and kx are linearly

dependent. Hence, >. = 2 is an eigenvalue of A of multiplicity m = 2 which
has only one associated eigenvector.

EXAMPLE 3 Calculating Eigenvalues and Eigenvectors

of a 3 x 3 Matrix

Find the eigenvalues and associated eigenvectors of the matrix

SOLUTION

-1

1

1

The characteristic equation for the given matrix A is

(

1 - ,\

det (A - >.I)= det -i

-1

1-,\

(^1) 1-D
= (1 - >.)^3 + (-1)(1)(1) + (1)(-1)(1)

• (1->.)(1)(1) - (-1)(-1)(1->.) - (1)(1->.)(1)

= 1 - 3.\ + 3.\^2 - >.^3 - 1 - 1 - 1 + >. - 1 + >. - 1 + >.

=-4+3>.^2 ->.^3 =0.

Since >.^3 - 3.\^2 + 4 = (>. + 1)(>. - 2)^2 , the roots of the characteristic equation
are >. 1 = -1 and >-2 = >. 3 = 2. Thus, -1 is an eigenvalue of the matrix A