Densité des matrices diagonalisables dans Mn(C).

Mn(C)
Dn(C)Mn(C)
D0
n(C)Mn(C)n
C
D0
n(C)Mn(C)
Int(Dn(C)) = D0
n(C)
D0
n(C)Dn(C)Mn(C)
Dn(R)Mn(R)
M∈ Mn(C)M∈ D0
n(C)χM
Cϕ(M) = Res(χM, χ0
M)6= 0
D0
n(C)Mn(C)C∗
ϕ ϕ M
nD0
n(C)⊂ Dn(C)
D0
n(C)Mn(C)D0 n(C)⊂Int(Dn(C))
A∈Int(Dn(C)) λ2
P A =P DP −1
D=
λ0
λ
λ3
0λn
k > 0
∆k=Ãλ1
k
0λ!, Dk=
∆k0
λ3
0λn
Dk∆k(x−λ)2P(Dk) = 0
P(∆k) = P π∆kDkAk=
P DkP−1
A= lim
k∞AkA∈Int(Dn(C))
Int(Dn(C)) n Int(Dn(C)) ⊂ D0
n(C)
Int(Dn(C)) = D0
n(C)
A∈ Mn(C)
P
T=
t11 t12 ··· t1n
t22 ··· t2n
0tnn
A=P T P −1
α=½1tii =tjj i6=j{1, . . . , n}
inf{|tii −tjj |, , 1≤i, j ≤n, tii 6=tjj }

(Tk)k≥1Tk=T+ ∆k
∆k=
α
k0
α
2k
0α
nk
k > 0Tktii +α
ik =tjj +α jk
tii 6=tjj
|tii −tjj |=α
k¯¯¯¯
1
i−1 j¯¯¯¯
<α
k≤α
α Tk∈ D0
n(C)
A= lim
k∞Akk > 0
Ak=P TkP−1D0
n(C)D0 n(C)Dn(C)
Mn(C)
Mn(R)n= 2 ϕ:M2(R)→R
M=µa b
c d ¶
ϕ(M) = (a−d)2+ 4bc
χMχ0
M
A= lim
k∞Ak=⇒ϕ(A) = lim k∞ϕ(Ak)
AkD0
2(R)D2(R)ϕ(Ak)≥0A
ϕ(A)<0A
D0
2(R)D2(R)
f E Cχf
f
χf(f) = 0
A∈ Mn(C)χA
A∈ Mn(C)Q D =
diag(λ1, λ2, . . . , λn)A=QDQ−1
χA(λ) =
n
Y
k=1
(λk−λ), χA(A) = QχA(D)Q−1= 0
A∈ Mn(C)A= lim
k∞Ak(Ak)k
M7→ χM(M)Mn(C)Mn(C)
mij χA(A) = lim
k∞χAk(Ak) = 0