materi vektor eigen
TRANSCRIPT
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NILAI EIGEN DANVEKTOR EIGEN
EIGEN VALUE AND EIGEN VEKTOR
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Definisi
Diberikan matriks A nxn, maka vektortak nol x∈Rn disebt vektorkarakteristik !eigen vector " darimatriks A#
$ika berlak Ax % λx ntk satskalar λ, maka λ disebt nilaikarakteristik !eigen value" darimatriks A#
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&en'elesaian
Ax % λxAx ( λx % )!A ( λI"x % )
Vektor karakteristik mer*akan solsi nontrivial !solsi 'an+ tidak seman'a nol" dari!A ( λI"x % )
A+ar di*erole solsi non trivial maka
-A ( λI- % )-A ( λI- % ) disebt *olinomial karakteristik
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.onto
Tentkan nilai ei+en dan vektor ei+en darimatriks A%
000
010
001
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&en'elesaian
0
00
010
001
=
−
−−
λ
λ
λ
!/(λ" !/(λ" !(λ" % )
−
−−
=
−
λ
λ
λ
λ
λ
λ
00
010
001
00
00
00
000
010
001
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&en'elesaian
$adi *olinomial karakteristik
!/(λ" !/(λ" !(λ" % )
Akar(akar *olinomial karakteristik λ/%), λ0%/, λ1%/
$adi nilai ei+en matriks A adala )
dan /#
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&en'elesaian
Vektor ei+en ntk λ%)A(λI %
!A(λI"x % )
=
−
−−
000
010
001
00
010
001
λ
λ
λ
=
0
0
0
000
010
001
3
2
1
x
x
x
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&en'elesaian
$adi x/%), x0%), x1%t, t≠), t∈R $adi x% mer*akan vektor ei+en 'an+
berkores*ondensi den+an
λ%)
t
0
0
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&en'elesaian
Vektor ei+en ntk λ%/A(λI %
!A(λI"x % )
−
=
−
−−
100
000
000
00
010
001
λ
λ
λ
=
− 00
0
100
000
000
3
2
1
x
x
x
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&en'elesaian
$adi x/%a, x0%b, x1%), a,b≠), a,b∈R $adi x% mer*akan vektor ei+en 'an+
berkores*ondensi den+an
λ%/
0
b
a
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Latian
.arila nilai ei+en dan vektor ei+en dari2
/#
0#