Démonstrations du théorème de Hamilton-Cayley

M∈Mn(A)A PM(X) =
det(XIn−M)PM(M) = 0
M x 7→ Mx An
PM(X) = (X−λ1)···(X−λn)
M−λnIn
M
M(X−λ1)· · · (X−λn−1)
D
λ1M
Q= (X−λ2)· · · (X−λn)Q(M)
D

G= (Ti,j )i,j=1,...,n
R=Z[Ti,j ]
Q(Ti,j )M∈Mn(A)θ:R→A
G M G M θ θ
G
M G
G
G∆∈R
G
MC
∆M
G
Ck,` ∈Z[Ti,j ]k
` PG(G)G
Ck,` Mn(C)
Ck,`
C
M7→ PM(M)
Mn(C)
M
K x KnPx
P P (M)x= 0
PxM
M Exx M
M x, Mx, . . . , Mk−1x Ex
k= dim Ex= deg PxPx
M PM
PM(M)(x) = 0
X=M
PM(X) = det(XIn−M)
A[X]→A[M]
δi,j M−mi,j InA[M]δi,j
0

X=M
Nt (com N) = t (com N)N= det(N)In,
com N N n
N=
XIn−M A[X]
Mn(A[X])
(†) (XIn−M)t (com(XIn−M)) = PM(X)In.
A[M]
(†)
com(XIn−M) (XIn−t M) = PM(X)In.
A ²M:A[X]→
A[M]⊂Mn(A)X M ²M:Mn(A[X]) →
Mn(A[M])
Mn(A[M])
e1
en
(e1, . . . , en)An
²M(PM(X)In)
e1
en
=
PM(M)e1
PM(M)en
.
²M(XIn−t M)
e1
en
=
Me1−Pn
j=1 mj,1ej
Men−Pn
j=1 mj,nej
=
0
0
,
PM(M)e1=. . . =PM(M)(en)=0 PM(M) = 0
²M(XIn−t M)
AnAn
X
Mn(A[X]) XdQd+
· · ·+XQ1+Q0Qi∈Mn(A)Mn(A)
X=
M XdQd+· · · +XQ1+Q0

XIn−M MdQd+···+MQ1+Q0= 0
X(†)PM(M) =
0
(†)
XIn−M PM(X)In−PM(M)U
Mn(A)PM(M)=(XIn−M)U
X U =PM(M) = 0
A[X]
XIn−MMn(A[X])
A[X] (A[X])n→(A[X])n
M AnA[X]adXd+· · · +
a1X+a0∈A[X]x∈An
(adXd+· · · +a1X+a0)x=adMdx+· · · +a1Mx +a0x .
X=M A[X]
ϕ: (A[X])n→An(A[X])n
Xdxd+· · · +Xx1+x0xi∈An
ϕ(Xdxd+· · · +Xx1+x0) = Mdxd+· · · +Mx1+x0X=M
XIn−M ϕ◦(XIn−M) : (A[X])n→An
(†)x∈An
PM(M)x=ϕ(PM(X)x) = (ϕ◦PM(X)In)(x)
= (ϕ◦(XIn−M)◦t (com(XIn−M)))(x) = 0 .
ϕ
(A[X])nXIn−M −−−−−−−−→(A[X])nϕ −−−−−→ An→0
ϕker ϕ
XIn−M AnA[X]
M(A[X])nXIn−M
M M0Mn(A)
XIn−M XIn−M0XIn−M XIn−M0
(A[X])nXIn−M
XIn−M0M M0
A[X]An

A K
XIn−M K[X]
Q1
Qn
,
QiQiQi+1 i= 1, . . . , n −1
XIn−M Qi
XIn−M PM
Qr, . . . , Qn
KnK[X]M
(K[X])nXIn−M
K[X]/Qr⊕ · · · ⊕ K[X]/Qn.
K[X]/QiQi= 1
M Qr, . . . , Qn
PM(X)
K[X]K[X]/QiK[X]Kn