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

Itfollows that
so that
(2−λ)(4−λ)−3 =0
λ 2 −6λ+5 =0
(λ−1)(λ−5) =0
λ=1 or 5
8.11 Eigenvalues and eigenvectors 299
Thesearethevaluesof λwhichcausethesystemAX = λX tohavenon-trivialsolutions.
They arecalledeigenvalues.
The equation
|A−λI|=0
which when written out explicitly is the quadratic equation in λ, is called the characteristicequation.
Example8.42 Find values of λfor which
x+4y = λx
2x+3y = λy
has non-trivial solutions.
Solution We write the systemas
where
AX=λX
A =
( ) ( 1 4 x
, X =
2 3 y)
To have non-trivialsolutions we require
|A−λI|=0
Now
( ) ( ) 1 4 1 0
A−λI = − λ
2 3 0 1
( ) ( ) 1 4 λ 0
= −
2 3 0 λ
( ) 1 − λ 4
=
2 3−λ
Hence
|A−λI| = (1−λ)(3−λ)−8
= λ 2 −4λ−5
To have non-trivialsolutions we require
λ 2 −4λ−5 =0
(λ+1)(λ−5) =0