Réduction des endomorphismes

E k f :E→E
E f E f
E
f
E
E k f E
F⊂E f f f(F)⊂F
x∈E f
λ∈k f −λId
x∈E\ {0}f(x) = λx
fSp(f)f
λ∈Sp(f)f λ
ker(f−λId) Eλ
λ∈k x ∈E\ {0}λ
x x λ
f−λId f P (f)P(X) = X−λ
f
P=a0+a1X+· · · +adXdk P (f)
E a0Id +a1f+· · · +fdfif i
f◦f◦ · · · ◦ f f P (f)
f E P, Q k

P(f) + Q(f)=(P+Q)(f)
P(f)◦Q(f) = Q(f)◦P(f)=(P Q)(f)
ker P(f) im P(f)f E
P=a0+a1X+· · · +XdQ=b0+b1X+· · · +beXe
P(f)+Q(f) = a0Id +a1f+· · ·+fd+b0Id +b1f+· · ·+bede= (a0+b0) Id +(a1+b1)f+· · · = (P+Q)(f)
XlP Q Pi+j=laibj
P(f)◦Q(f) = d
X
i=0
aifi◦e
X
j=0
bjfj=
d
X
i=0
aifi◦e
X
j=0
bjfj Xaifi,
=
d
X
i=0
ai
e
X
j=0
bj(fi◦fj)fi,
=
max(d,e)
X
l=0 X i+j=l
aibjfk= (P Q)(f).
(P Q)(f)=(QP )(f) ker(P(f))
im(P(f))
f P (f)(x)=0
P(f)(f(x)) = f(P(f)(x)) = f(0) = 0 ker P(f)x=P(f)(y)f(x) =
f(P(f)(y)) = P(f)(f(y)) im P(f)
evf:k[X]→L(E)
P P (f)
k[X]
L(E)
P(f)Q(f)P(f)◦Q(f)
P Q P Q ker P(f)⊂ker Q(f)
im Q(f)⊂im P(f)
E k f
E P1, . . . , PsP Pi
ker(P(f)) = ker(P1(f)) ⊕ · · · ⊕ ker(Ps(f))
s= 2
ker P(f) = ker P1(f)⊕ker P2(f) ker Pi(f)⊂ker P(f)i= 1,2
P1P2
U1U2U1P1+U2P2= 1
f
U1(f)P1(f) + U2(f)P2(f) = Id .

x∈ker P(f)x1= (U2(f)P2(f))(x)x2= (U1(f)P1(f))(x)P1(f)P2(f) = P(f)
x1∈ker P1(f)x2∈ker P2(f)x=x1+x2ker P1(f)
ker P2(f) ker P(f)x∈ker P1(f)∩ker P2(f)x1=x2= 0
x= 0 ker P1(f) ker P2(f)
f∈L(E)λ1, . . . , λs
Eλis6n= dim(E)
s=n EλiE
Pi=X−λis6n
1
s
n6dim(Eλ1) + · · · + dim(Eλn)6n1
EλiE
evf:k[X]→L(E) evf(P) = P(f)
k[X]L(E)
ker(evf)6={0}
f P P (f)=0
ker(evf)
evfker(evf)k[X]
k[X]
f∈L(E)f µf
f
f
Xdd0
X2−X
f
f χf
χf(X) = det(XId −f).
X
E A f χf(X)
XId −A k0=k(X)

n= dim(E)n
Xn
χf(X) = det(f−XId) (−1)n
χff
λ∈k χfdet(λId −f)=0
λId −f
E k f ∈L(E)
f F f|Ff
µf|Fµf
χf|Fχf
P P (f)|F=P(f|F)µf(f|F) =
µf(f)|F= 0 µf|Ff|Fµf|F
µf|Fµf
{e1, . . . , em}F G F {em+1, . . . , en}
G{e1, . . . , en}E
f A =A11 A12
0A22 A11 f|F
f
XId −A=XId −A11 −A12
0XId −A22 .
χf(X) = det(XId −A11) det(XId −A22) = χf|F(X) det(XId −A22)
χf|Fχf
f∈L(E)
χ χ(f)=0
χ f
µ χ
χ(f) = 0 (χ(f))(x) = 0 x x
m
fm(x) + am−1fm−1(x) + · · · +a1f(x) + a0x= 0.
mB={x, f(x), . . . , fm−1(x)}F=
Vect(B)f m Bf|F
0−a0
1−a1
0
1−am−1
.

χ0=χf|F
χ0(X) = Xm+am−1Xm−1+· · · +a1X+a0
χ0(f)(x)=0 P χ =P χ0χ(f) = P(f)χ0(f)
(χ(f))(x) = P(f)(χ0(f)(x)) = P(f)(0) = 0,
deg(µf)6n
µfχfdeg(µf)6deg(χf) = n
E k f ∈L(E)f
E f
BE A f
fB0P−1AP P B B0f A
P P −1AP
f k
f
χ f
1
⇒f
f f
f
χ(X) = det 
XId −
a11 ∗ ∗
∗
0ann
= det 
X−a11 ∗ ∗
∗
0X−ann
,
χ(X)=(X−a11). . . (X−ann)
⇒n χ f n = 1
n−1λ1, . . . , λn
χ e1f λ1
{e1, e0
2, . . . , e0
n}E f
A=λ1B
0C.