Réduction d`un endomorphisme en dimension finie

K E K
n>1u E
λ∈K u x 6= 0
u(x) = λx u λ
u(u)
λ∈(u) ker(u−λ)
u λ
(λ1, . . . , λk)
uker(u−λi)
u χu
n
χu(X) = det(X· − u) = Xn−(u)Xn−1+· · · + (−1)ndet(u).
u K χu
λ χuλ u
m(λ)
F u χu|F|χu
λ u
16dim ker(u−λ)6m(λ).
K[X]
I(u) = {P∈K[X]|P(u) = 0}
K[X] 0
u µu
µuχu
χu(u) = 0
u K
u
E u
E u
E=Lλ∈(u)ker(u−λ)

χuK λ ∈(u)
dim ker(u−λ) = m(λ).
u n u
A=2 1
1 1(A) = 3±√5
2
u u
µuK
F E u u u|F
u
E u
χuu K
u K
R2ϑ6= 0[π]R
u F E u u|F
K L(E)
Mn(C)Mn(C)
G GLn(C)
u v
u v
u v
u v
♦
u∈L(E)χuK
χu(X) = (X−λ1)m1· · · (X−λp)mp
λi∈(u)

E=Lp
i=1 ker(u−λi)miker(u−
λi)miu u
∀i∈[1, p],dim ker(u−λi)mi=mi
u u =d+n
d u dn =nd d n u
k
Mn(C)
a∈E µu,a K[X]
{P∈K[X]|P(u)(a) = 0}.
< x >u:= {P(u)(x)|P∈K[X]}
F u E F u x ∈F
F=< x >u
u x ∈E
E=< x >u
a∈E µu=µu,a
u
χu=µudeg(µu) = dim(E)
E u χu(X) =
Xn+an−1Xn−1+· · · +a0
C(χu) =
0· · · · · · 0−a0
1 0 −a1
0 1
0−an−2
0· · · 0 1 −an−1
F1, . . . , FrE u
E=F1⊕F2⊕ · · · ⊕ Fr
Fiu i ∈[1, r]
Piu|FiPi+1|Pii∈[1, r −1]

P1, . . . , Pru
u
P1, . . . , Pr
u E u
C(P1)
C(Pr)
.
M u E
u1M−XI ∈ Mn(K[X])
u v E
M∈ Mn(K)MtM
K⊂L A B L K
χuK
u K
E u
J1
Js
Ji
Ji=
λi1
λi1
λi1
λi
λiu
Mn(C)