082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

8.11 Eigenvalues and eigenvectors 303
which produces:
ans =
7
-1
which isavectorcontaining the eigenvalues.
UseMATLAB ® orasimilar language to confirm the
restofthe eigenvalues given in Exercises8.11.2.
8.11.3 Eigenvectors
We have studied the system
AX=λX
anddeterminedthevaluesof λforwhichnon-trivialsolutionsexist.Thesevaluesof λare
calledeigenvaluesofthesystem,or,moresimply,eigenvaluesofA.Foreacheigenvalue
thereisanon-trivialsolution of the system.This solution iscalled aneigenvector.
Example8.46 Find the eigenvectors of
where
AX=λX
A =
( ) ( 4 1 x
andX=
3 2 y)
Solution We seek solutions ofAX = λX which may be written as
(A−λI)X=0
The eigenvalues werefound inExample 8.44 tobe λ = 1,5.
Firstlywe consider λ = 1.The systemequation becomes
(A−λI)X =0
(A−I)X =0
[( ) ( ( ) ( 4 1 1 0 x 0
− =
3 2 0 1)]
y 0)
( ) ( 3 1 x 0
=
3 1)(
y 0)
Writtenas individual equations wehave
3x+y =0
3x+y =0
Clearlythereisonlyoneequationwhichisrepeated.Aslongasy = −3xtheequationis
satisfied.Thusthereareaninfinitenumberofsolutionssuchasx = 1,y = −3;x = −5,
y = 15;and so on. Generally we write
x=t,y=−3t
for any numbert. Thus the eigenvector corresponding to λ = 1 is
( ( ) ( ) x t 1
X = = =t
y)
−3t −3