Agrégation externe de Mathématiques 1 Existence et Calcul

u∈ L(E)χu=Qk
i=1(X−λi)αimu=Qk i=1(X−λi)βi
16βi6αii6k Ei= ker(u−λiId)αiu
λi
E=⊕k
i=1EiEi= ker(u−λiId)βi
u−λiId Eiβi
πiEi⊕j6=iEjd=Piλiπin=u−d
dn =nd
E u d
ndim Ei=αi
det u= det d
A∈ Mn(K)A
D+N D N
u χu=Qk
i=1(X−λi)αi
1
χuQi1 = Pk
i=1 QiPiPi=
Qk
j=1,j6=i(X−λj)αjπi=PiQi(u)
λid=Piλiπin=u−d
1 1 2
0 2 1
0 0 2

u
d n
d P =Qk
i=1(X−λi)k0
P=χu/(χu∧χ0
u)P χu
d K[u]P(d) = 0
u0=u un+1 =un−P(un)◦(P0(un))−1
k>2Q∈K[X, Y ]
P(Y) = P(X)+(Y−X)P0(X)+(Y−X)2Q(X, Y )
a∈k[u]k[u]b∈k[u]a+b
k[u]
P0(u)−1u P (u)u1
u P (u1)P0(u1)−1u
uii∈Nu
P(uk) = P(u)2kQk(u)k∈N(uk)k∈N
u∞u−u∞d=u∞
n=u−u∞
A∈Gln(C)eAA
A eA
eA=In
etA t+∞
A
etA t+∞
N∈ Mn(C)Mt=−Pn−1
k=1
(−tN)k
k
In+tN =eMtt∈[0,1] A∈ Gln(C)P∈C[X]
A=eP(A)AC R
f(z) = Pk∈NakzkR A ∈ Mn(C)
kAk< R k·k Pk∈NakAkMn(C)
A ρ(A) = sup{|λ|, λ A}
ρ(A) = inf{kAk,k · k Mn(C)}
k·k Mn(C)lk=kAkklk+p≤lklp
(k, p)∈Nl
1
k
k→inf{l
1
k
k, k ∈N}ρ(A) = limk→∞ kAkk1
k
Mn(C)
Pk∈NakAkρ(A)< R ρ(A)> R
(A, B)∈ Mn(C)ρ(AB) = ρ(BA)