127 - Exponentielle de matrices. Applications. - IMJ-PRG

K=R C n∈N∗Mn(K)
u∈L(E)πuk
d∈L(E)n∈L(E)
u=d+n d n
Pn≥0An/n!Mn(K)
A
A, B ∈Mn(K)P∈Gln(K)
exp(0) = In(λ1, . . . , λn) exp(diag(λ1, . . . , λn)) = diag(exp λ1,...,exp λn)
exp(trig(λ1, . . . , λn)) = trig(exp λ1,...,exp λn)
A=P BP −1exp(A) = Pexp(B)p−1
det(exp A) = etr(A)
exp Mn(K)
SpC(exp(A)) = eSpC(A)
N p exp(N) = Pp−1
n=0 Nn/n!
exp(A)A A
A∈Mn(K),∃PA∈K[X] : exp(A) = PA(A)
P∈K[X]A
A A exp A
A, B ∈Mn(K) [A, B] = 0 exp(A+
B) = exp(A) exp(B)
A=0 1
0 0 
B=0 0
1 0 exp(A+B)6= exp(A) exp(B)
Aexp A

A∈Mn(K),exp(A)∈Gln(K) exp(A)−1=
exp(−A)
A
AK[X]/(πA)
A A =D+N D
N[N, D] = 0 exp(A) = exp(D) exp(N)
A∈Mn(K)K
Aexp(A)
A∈Mn(C) exp(A) = InAC
SpC(A)⊂2πZ
A B exp A= exp B
A=B
exp : Mn(K)→Gln(K)C∞
dexp(0) = Id
Gln(K)
V Gln(K)Gln(K)
V
X:R→Mn(K)C1t
X(t)X0(t)t7→ exp(X(t)) C1
td
dt [exp(X(t))] = X0(t) exp(x(t))
X(t) = 1t
0 0 
(X, H)∈Mn(K)2
dexp(X).H = exp(x)
∞
X
k=0
1
(k+ 1)!(−adX)k(H)
adX:Mn(K)→Mn(K), H 7→ [X, H]

P∞
k=1(−1)k+1 (M−In)k
k
B(In,1)
M
log : B(In,1) →Mn(R)
log : B(In,1) →B(0, ln(2)) exp : B(0, ln(2)) →B(In,1)
A B log(AB) = log(A) + log(B)
A B Mn(K)
(Ak)kA
exp(A) = lim
K→+∞(In+1
kAk)k
exp(A+B) = limk→+∞(exp(A/k) exp(B/k))k
exp(AB −BA) = limk→+∞(exp(A/k) exp(B/k) exp(−A/k) exp(−B/k))k
exp : Mn(C)→Gln(C)
Gln(C)
Gln(C)→Gln(C), A 7→ App≥1
p
exp(Mn(R)) = {M2|M∈Gln(R)}
npp
exp np{In+
N|N∈np}
exp Sn(R)S++
n(R)
exp Hn(R)H++
n(R)
exp(θJ) = Rθ
exp : An→SOn
Gln(R)≈On(R)×S++
n(R)≈On(R)×Rn(n−1)/2
Gln(C)≈Un(C)×H++
n(R)≈Un(C)×Rn2

ϕRGln
ϕ(t) = exp(tX)
SOn
S1Gln(C)
Gln(R)
IRt0∈I B :I→Rn
A∈Mn(R)Y
Y(t) = exp((t−t0)A)Y(t0) + Zt
t0
exp((t−s)A)B(s)ds