Exponentielles de matrices. Applications 1 Définitions

Mn(k)k
A
PAk/k!Mn(R)exp(A)
|||exp(A)||| 6e|||A|||
exp(−A)
A A A B exp(A+B) =
exp(A)exp(B)exp(A)exp(B)exp(P−1AP ) = P−1exp(A)P
(Id +A/n)n→exp(A)An→A(Id +A/n)n→exp(A)
Sp(Exp(A)) = Exp(Sp(A))
A A exp(A)
det(exp(A)) = eT r(A)
A A =D+N exp(A)
exp(D) + exp(D)(exp(N)−1)
Pλipiλi
pi
Q(X−λi)ri
1/Q(X−λi)ri1/χA=
PiPjxij /(X−λi)jUi:= Pri
j(X−λi)ri−j1 = PUiQiQi:= Qj6=i(X−
λj)rjUiQi(A)

Mn(C)Gln(C)
Mn(R)GLn(R)∩A|∃B, B2=A
Gln(C)
Mn(R)
exp(A) = Id
21πZL
exp(A) = B exp(A0)
A0+l l ∈L A0
B(0,1)
P∞
n=1(−1)n+1(A−Id)n/n
exp(log(A)) = A
C1C∞
0 0
Id
GLn(K)
GLn(K)
t7→ exp(tA)Aexp(tA)t7→
exp(tA)x exp(tA)x
X0=AX A A0
X0(t) = A(t)X(t)
t exp(tA)exp(tB)A B
ad :A7→ (X7→ AX −XA)
Ad :A7→ (M7→ AMA−1
exp(ad(M)) = Ad(exp(M))
d(exp(M))(X) = exp(M)∗Pm= 0∞(−1)mMm/(m+ 1)!)X
d(exp(M)) Ad(M) 2iπZ
|Im(λi)|< π
Idd GLn(R)

RGLn(C)t7→ etA
RSL2(R)t7→ etA A
C1RMn(C)s7→ P exp(sA)
U Gln(C)z7→ P−1Diag(zk1, . . . , zkn)P
U Gln(R)eiθ 7→ Qf(θ)Q−1f(θ)
kiθ
Gln(K)
Exp(X+Y) = lim(exp(X/n)exp(Y/n))n
G Gln(K)g={X∈Mn(k)|exp(tX)∈G, ∀t∈R}
g
G
G g
G Gln(k) 0 g
U Id G V U
V
G Mn(k)
dim(g)Id g
G exp(g)G
On(R)SOn(R)Mn(k)n(n−1)/2
Id
SLn(R)Mn(k)n2−1
Id
Gln(R)On(R)×S++
n(R)Gln(C)
Un(C)×H++
n(C)
G GLn(C)M∈
G G G
G(G∩On)×(G∩HDP )Gln(R)
G M∗JM =J J2=Id
G
(G∩Un)×Rdd
U(p, q)Up×Uq×Rpq
O(p, q)Op×Oq×Rpq
p q
SnS++
nHnH++ n
exp(A/p)
Gln(R)On(R)×Rn(n+1)/2
Gln(C)Un(C)×Rn2

O(n, C)On(R)×Rn(n−1)/2
R C
etM 0 +∞M
etM
1
X0=AX 0
+∞
y0=f(y), y(0) = x f
C1A f 0A