Dm

NOM : Abid
Prénom : Youcef
Section : M1 RO2MIR
1◦
eA+B = eAB
Si A ,B MN (k) commutent alors exp (A+B)= exp A ∗ exp B
on a (A+B)k =
Pk
eA+B = limn→∞
Pn
A+B
k
i=0
k=0
Ak
k≥0 k!
P
!
Ai Bk-i
i
(A+B)
k!
k
=
P
k≥0
car A ,B commutent
k
Pk
i=0
Bk
k≥0 k!
P
e
=(
(
) = ( limn→∞
◦
2
M
Calculer e!
01
M=
−1 0
Calculer MK Par
! récurrence on trouve :
−1
0
M2 =
0 −1
M
i
Pn
!
Ai B k−i
k!
Bk
k=0 k!
M4 =
M2K =
01
!
; on obtient
!
(−1)k 0
0 (−1)k
P
(−1)k
M2K+1 =
0 (−1)k
M
eM =
cos (1) sin (1)
!
− sin (1) cos (1)
!
(−1)k+1 0
!
P
(−1)k
0
k≥0 (2k)! 0
k≥0 (2k+1)!
e =
+ P
P
(−1)k
(−1)k+1
0
k≥0 (2k)!
k≥0 (2k+1)! 0
k
P
∗ x2k =cos(x)
On a ∀ x R, k=0 (−1)
(2k)!
P
(−1)k
Et k=0 (2k+1)!
∗ x2k+1 = sin(x)
!
.
1
P
) = eA eB
3= 0 1−0 1
10
=
k≥0
Ai B k−i
i=0 i! (k−i)!
Pk