PSI* 2016/17 - Révision 3B - matrices de Frobenius I. Propriétés

KR C n
u K E u0=idE∀n∈N, un+1 =un◦u
Kn[X]K n Mn(K)K
n K InGLn(K)Mn(K)
Mn(K)M= (mi,j )
AMn(K)tA A rg (A)χA(x) = det (xIn−A)
Sp A
P=Xn+an−1Xn−1+. . . +a1X+a0Kn[X]
CP=
0 0 . . 0−a0
1 0 . . 0−a1
010.0−a2
. . . . . .
0.010−an−2
0. . 0 1 −an−1
∈ Mn(K)
CP= (ci,j )ci,j = 1 i−j= 1 ci,n =−ai−1ci,j = 0
P=Xn+an−1Xn−1+. . .+a1X+a0Kn[X]CP
CPP(0)6=0
CPk χCp=kP
Q Kn[X]A
Mn(K)χA=Q
tCPCP
Sp CP= Sp tCP
λSp tCPtCPλ
tCPP K
P n λ1λ2λntCP
1 1 . . 1
λ1λ2. . λn
λ2
1λ2 2. . λ2
n
. . . . .
λn−1
1λn−1 2..λn−1
n
rg (U−V) = 1
A CAχA
A CA
(U, V )GLn(K)
rg (U−V)=1
P U =P−1CUP V =P−1CVP
(U, V )GLn(K)
(U, V )GL2(K)n= 2
χUχV
(U, V )GLn(K)χUχV
R C
E K n B u v E
U V u v B
H= ker (u−v)

H E
F6={0}E u v
u(F)⊂F v(F)⊂F
uFvFu v F
χuFχu
F H
F6=E F +H=E BFF
H B0E u v B0
u v
j∈NGj={x∈E, uj(x)∈H}
GjE
n−2
T
j=0
Gj6={0}
y
n−2
T
j=0
Gj0≤j≤n−1ej=uj(y)
B00 = (e0, e1, . . ., en−1)E
F= Vect y, u(y), . . ., up−1(y)p
y, u(y), . . ., up−1(y)
u v B00 CUCV