Université Montpellier II Fac des Sciences Département de

uR3R3
−2 1 1
8 1 −5
4 3 −3
u−X(X+ 2)2
u E0= Vect(3,1,5) E−2=
Vect(1,−1,1)
u
2(= 1 + 1)
3u
u
µ u
µ=X(X+ 2) X(X+ 2)2u
µ µ =X(X+ 2)2
R3u
u
u(3,1,5) (1,−1,1)
E0E−2
FR3u uFF
uFR3
u
PFuFu
PF=X(X+ 2) (X+ 2)2
PF=X(X+ 2) uFF
uFu F =E0⊕E−2
F u

PF= (X+ 2)2PF(uF) = 0 PF(uF)
PF(u)F
F⊂Ker((u+ 2Id)2).
F
u x ∈R3
uR3
u
e1= (1,0,0)
(e1, u(e1), u2(e1)) e1
P, Q ∈R[X]D P Q
M P Q
M:= P Q
D
D P Q
A, B ∈R[X]M=AP =BQ
A, B ∈R[X]
P=DB Q =DA M M =P Q/D =
DAB =QB =P A
E u ∈ L(E)
Ker P(u) + Ker Q(u) = Ker M(u).
y∈Ker P(u)z∈Ker Q(u)M(u)(y+z) = M(u)(y) + M(u)(z) =
A(u)(P(u)(y))+B(u)(Q(u)(z)) = 0+0 = 0 Ker P(u)+Ker Q(u)⊂
Ker M(u)
U, V ∈K[X]AU +
BV = 1 x∈Ker M(u)x=id(x)=(AU)(u)(x)+(BV )(u)(x)
y:= (AU)(u)(x)z:= (BV )(u)(x)x=y+z
P(u)(y) = P(u)◦A(u)◦U(u)(x) = U(u)◦M(u)(x) = 0 x∈Ker M(u)
Q(u)(z)=0 x∈Ker P(u) + Ker Q(u)
Cn
A B n A
B P B =P AP −1χA
µAA
det(P AP −1−XIn) = det(P AP −1−XP P −1) = det(P(A−XIn)P−1) =
det(A−XIn).det(P).det(P−1) = det(A−XIn)
µ(P AP −1)k=P AkP−1
k
µ(P AP −1) = P µ(A)P−1.

µ(B) = 0 µ(A) = 0 A B
n= 2 µA=µBA B
µ=µA=µB
µ λ λ0
λ λ0A
µAA A
λ0
0λ0.
µ=X20 1
A
A A
0 1
0 0 .
µ λ
λ A
A−λI2
0 1
0 0 .
A
λ1
0λ.
µ A
B A B
µ λ A B λI2
n= 2
0 1
0 0 
X2

Sn1,· · · , n σ ∈Snk
σkσ k
σ0= Id σ−k= (σ−1)k
σ∈Snk l
σk=σl
Snσkk∈N∗
k l σk=σl
k σk= Id
k l k l
l > k σk=σlσ−kId = σl−k
k σ ord(σ)
σk= Id ord(σ)k
σk= Id kord(σ)
k= ord(σ)q+r0≤r < ord(σ).
σk=σord(σ)q+r=σord(σ)q◦σr= (σord(σ))q◦σr=σrσr= Id
r < ord(σ)
ord(σ)r= 0 ord(σ)
c= (a1,· · · , al)l ai
0< k < l ck(a1) = a1+kck6= Id
cl(a1) = a1i cl(ai) = aicl(b) = b b
aicl= Id
l= ord(c)
σ
σ
σ=c1◦ · · · ◦ ckp
ci
ci
l
σl=cl
1◦ · · · ◦ cl k.
ord(ci)p cp
i= Id (1) σp= Id
p
σl= Id l cl
i
(1) cl
i= Id i
ord(ci)l i p l
l≥p