Maatriks eksponentsiaali harjutused: Lineaarse algebra ülesanded

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
i=0 k
i!AiBk-i car A ,B commutent
eA+B=limn→∞ Pn
k=0
(A+B)k
k!=Pk≥0Pk
i=0 k
i!AiBk−i
k!=Pk≥0Pk
i=0
Ai
i!
Bk−i
(k−i)!
eA+B= ( Pk≥0
Ak
k!(Pk≥0
Bk
k!) = ( limn→∞ Pn
k=0
Bk
k!) = eAeB
2◦
Calculer eM
M= 0 1
−1 0!
Calculer MKPar récurrence on trouve :
M2= −1 0
0−1!
M3= 0−1
1 0 
M4= 1 0
0 1!; on obtient
M2K= (−1)k0
0 (−1)k!M2K+1= 0 (−1)k (−1)k+1 0!
eM= Pk≥0
(−1)k
(2k)! 0
0Pk≥0
(−1)k
(2k)! !+ 0Pk≥0
(−1)k
(2k+1)!
Pk≥0
(−1)k+1
(2k+1)! 0!
On a ∀xR, Pk=0
(−1)k
(2k)! ∗x2k =cos(x)
Et Pk=0
(−1)k
(2k+1)! ∗x2k+1 = sin(x)
eM= cos (1) sin (1)
−sin (1) cos (1)!.
1