1 Vecteurs propres et valeurs propres - ANMC

A
A f spec(A) spec(f)
A0/∈spec(A)
A
λ∈K→det(λI −A)
A pA= det(λI −A)
A∈Mn×n(K)pAA
n
pA=a0+a1t+. . . +an−1tn−1+tn,
a0= (−1)ndet(A)an−1=−(a11 +. . . +ann) = −trace(A)
Atrace(A)
p∈K(t)p=a0+a1t+. . . +an−1tn−1+antnp
A∈Mn×n(K)
p∈C(t)n≥1
pC
A∈Mn×n(K)
A A
V K f ∈L(V, V )λ∈K
λ f
ker(λid −f)6={0}
λid −f
A∈Mn×n(K)P∈Mn×n(K)
A P −1AP
x∈KnA P −1x
P−1AP

f
f
V K finie f ∈L(V, V )
B V λ ∈spec(f)v∈V λ ∈
spec([f]B,B ) [v]B∈Knλ∈spec([f]B,B )
x∈Knλ∈spec(f) [x]−1
B∈V
λ Eλ(A)A Eλ(f)
f
A∈Mn×n(K)λ1, . . . , λrA
v1, . . . , vr{v1, . . . , vr}Kn
A∈Mn×n(K)f∈L(V, V )V K
λ1, . . . , λrA f
Eλ1(A) + . . . Eλr(A)Eλ1(f) + . . . Eλr(f)
A∈Mn×n(K)A
KnA
A∈Mn×n(K)λ1, . . . , λrA
f∈L(V, V )A f
Eλ1(A)⊕. . . ⊕Eλr(A) = Kn, Eλi(f)).
A∈Mn×n(K)A n
f∈L(V, V )A f
n
In
malg(λ)
mgom
A∈Mn×n(K)λi∈spec(A)f λispec(f)
mgom(λi)≤malg(λi)
A∈Mn×n(K)A
pA(λ)pA(λ) = Πr
i=1(λ−λi)miPr
i=1 mi=n
λi∈spec(A)mgom(λi) = malg(λi), i = 1, . . . , r