Algèbre Linéaire et Bilinéaire

◦2
kR C n E
k n
•
R C
A=
1−1 0
−111
−1−1 0
;B=
1 2 −2
2 1 −2
2 2 −3
;C=
1. . . 1 0 . . . 0
1. . . 1 0 . . . 0
0. . . . . . 0
0. . . . . . 0
,
C”1” p q q ≤p<n
uR3
A=011
100
001R3
u
E
E=C∞(R,R)φ:f∈E7→ f0
E=k[X] Φ : P7→ P0,Ψ : P7→ XP, Θ : P7→ P(2X).
ECu E
1
(u)⊂(u)u2= 0 u
(a) (u)∩(u) 0 (u)⊂(u)
(b) (u) (u) (u)⊕(u) = E
u
u
2×2
2×2R
n≥3A= (ai,j )n ai,j = 1 i
j n 0
A=
0. . . 0 1
0. . . 0 1
1. . . 1 0
.
A0A

A
A λ+λ−
E+E−
A
A
•
A=3 1 −1
0 2 0
1 1 1 A
A
Φ : A∈ Mn(k)7→ tAΦ
Φ
A=
1 1 0 0
0 2 0 0
0 0 2 2
0 0 0 2
;B=
1 0 0 0
0 2 1 0
0 0 2 0
0 0 0 1
;C=
2 1 0 0
0 2 1 0
0 0 2 0
0 0 0 1
u, v ∈ L(E)u◦v=v◦u
u v
F E v v
v F
u v
u∈ L(E)E
u r ∈N\{0}F1, F2, . . . , Fr
u E =F1⊕. . . ⊕Fr
u
u Fii= 1, . . . , r
u r
u Fii= 1, . . . , r
u u Fi
•
λ n
Jλ=
λ1 0 . . . . . . 0
0
0
1
0. . . 0λ
J0
J0Jλλ∈R
JλtJλ
u∈ L(E)ECn
u un= 0

(a)u(b)Cu=Xn,(c)u(u) = {0},
Cuu Mu
A∈ Mn(C)
A(Ap) = 0 p∈N\{0}
(Ap)=0 p∈N\{0}
(a)A
CA=
r
Y
k=1
(λk−X)mk,
r∈N\{0}λkmk∈N\{0}
(b)p∈N\{0}(Ap) = Pr
k=1 mkλp
k
(c)M(λ1, . . . , λr)=(λi−1
j)1≤i,j≤rV= (mkλk)k=1,...r
M(λ1, . . . , λr)V
(d)λk= 0 k= 1, . . . , r
•
uR5
CuMuu
Cu(X)=(X−1)5(X+ 5), Mu(X) = (X−1)2(X+ 5)
Cu(X)=(X−1)3(X+ 2)2, Mu(X)=(X−1)2(X+ 2)2
ER−7u∈ L(E)
Cu(X) = (X−2)3(X+ 1)4
( (u−2Id)) = 3 ( (u+Id)) = 2
u
u
u
A=
−4 1 0 1
−2−1 0 1
−12630
−2 1 0 −1
;B=
1 0 0 0
−1001
2 2 1 −2
−1−1 0 2
;C=
2−4 0 −12
1−2 0 −3
000−2
000−2
.
E k 9u∈ L(E)
α β k
(u−αid)=3,(u−αid)2= 5,(u−αid)3= 6.
(u−βid)=2,(u−βid)2= 3.
u
u(u−γid)p
p∈N\{0}γ∈k
u