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TSI1
no3
t E(t)
E(t) = I+tA +t2
2A2.
A3= 0 A3
E(s)E(t) = I+sA +s2
2A2I+tA +t2 2A2
=I+tA +t2
2A2+sA +stA2+s2 2A2=I+ (s+t)A+s2+ 2st +t2 2A2
=I+ (s+t)A+(s+t)2
2A2=E(s+t)
∀(s, t)∈R2E(s)E(t) = E(s+t).
t∈RnE(t)n=E(nt)
? n = 0
? n ∈Nn
E(t)n+1 =E(t)n×E(t)
=E(nt)×E(t)
=E(nt +t) = E((n+ 1)t)
? n
∀t∈R,∀n∈N, E(nt) = E(t)n.
E(t−t) = E(t+ (−t)) = E(t)E(−t) = E(−t)E(t) = E(0) = I
E(t) (E(t))−1=E(−t)
λ1, λ2λ3∈R
λ1I+λ2A+λ3A2= 0
A3= 0
A
λ1A+λ2A2= 0
λ1A2= 0
A26= 0 λ1=0=λ2=λ3

(I, A, A2)Mp(R)
t7→ E(t)
E(s) = E(t) =⇒s=t
(s, t)∈R2E(s) = E(t)
0 = E(s)−E(t)=(s−t)A+s2−t2
2A2
(I, A, A2)s=t A
E:t7→ E(t)RMp(R)
A=
011
001
000
A2=
001
000
000
A3= 0
A3E(t) = 
1t t +t2
2
0 1 t
0 0 1 
B0= (−→ e1,−→ e2)R2A=4−6
1−1∈ M2(R)
fR2A
f
χf(x) = x−4 6
−1x+ 1 =x2−3x+ 2 = (x−1)(x−2)
Spec(f) = {1,2}
f
f
(x, y)E1
(A−I)x
y= 0 ⇔3x−6y= 0
x−2y= 0 ⇔x= 2y
(x, y)E2
(A−2I)x
y= 0 ⇔2x−6y= 0 x−3y= 0 ⇔x= 3y
B= (−→ u , −→ v) = (2,1),(3,1)
A=P DP −1P=2 3
1 1 D=1 0 0 2 

TSI1
P−1
Px
y=x0 y0⇐⇒ 2x+ 3y=x0 x+y=y0⇐⇒ y=x0−2y0(L1−2L2) x=−x0+ 3y0(3L2−L1)
P−1=−1 3
1−2
Dn=1 0
0 2n
A0=P D0P−1k Ak=P DkP−1
Ak+1 =A×Ak=P DP −1×P DkP−1=P DDkP−1=P Dk+1P−1
∀n∈N, An=P DnP−1
P DkP−1=2 3
1 1  1 0 0 2n −1 3 1−2
=2 3 ×2n
1 2n −1 3 1−2=−2+3×2n6−3×2n+1 −1+2n3−2n+1 
An=−2+3×2n6−3×2n+1
−1+2n3−2n+1 
t et= lim
n→+∞ n
X
k=0
tk
k!!
t n
En(t) =
n
X
k=0
tk
k!Ak=
n
X
k=0
tk
k!−2+3×2k6−3×2k+1
−1+2k3−2k+1 
En(t) = 
−2Pn
k=0 tk
k!+ 3 ×Pn
k=0
(2t)k
k!6Pn
k=0 tk
k!−6×Pn
k=0
(2t)k
k!
−Pn
k=0 tk
k!+Pn
k=0
(2t)k
k!3Pn
k=0 tk
k!−2Pn
k=0
(2t)k
k!
t∈RE(t)E(t) = a(t)b(t)
c(t)d(t)
a(t) = lim
n→+∞an(t) ; b(t) = lim n→+∞bn(t) ; c(t) = lim n→+∞cn(t) ; d(t) = lim n→+∞dn(t).
E(t) = 3e2t−2et6et−6e2t
−et+e2t3et−2e2t

E(t) = 3−6
1−2e2t+−2 6 −1 3 et
∀t∈R, E(t) = e2tQ+etR Q =3−6
1−2R=−2 6 −1 3 
Q2=Q, R2=R, QR =RQ = 0
q◦q=q r ◦r=r q r
Im q= Vect(3,1) Im r= Vect(2,1)
Ker q= Vect(2,1) = Im rKer r= Vect(3,1) = Im q q ◦r= 0 r◦q= 0
qR−→ vR−→ u r R−→ uR−→ v
(s, t)∈R2
E(t)E(s)=(e2tQ+etR)(e2sQ+esR) = e2t+2sQ2+e2tesQR +ete2sRQ +et+sR2
=e2(s+t)Q+et+sR=E(t+s)
∀(s, t)∈R2E(s)E(t) = E(s+t).
E(t)n=E(nt)E(t)−1=E(−t)
E(t) = E(s)
(e2t−e2s)Q+ (et−es)R= 0
(Q, R)M2(K)et=est=s
E:R→ M2(R)
t7→ E(t)
M.MT=MT.M (1)
M2(R)
I=1 0
0 1 A=0 1 1 0 C=0−1
1 0 
AAT=ATA=A2=I CCT=CTC=I
A C

TSI1
A2=I
n
? n = 2k A2k= (A2)k=Ik=I
? n = 2k+ 1 A2k+1 = (A2)k×A=A
n An
A2=I
A A−1=A
uR2B= (−→ i , −→ j)
A
u(−→ i) = −→ j u(−→ j) = −→ i
A2=I u ◦u=IdR2u
(A−I)x
y=0 0⇐⇒ −x+y= 0 x−y= 0 ⇐⇒ x=y
(A−I)x
y=0 0⇐⇒ x+y= 0 x+y= 0 ⇐⇒ x=−y
uR(−→ i+−→ j)R(−→ i−−→ j)
U=A+I
UTU= (A+I)TU= (AT+IT)U= (A+I)U=U2=UUT
U
U2= (A+I)2=A2+ 2AI +I= 2AI + 2A= 2(A+I) = 2U
Un= 2n−1U
? n = 1 n= 2
? n
Un+1 =Un×U= 2n−1U×U= 2n−1U2= 2n−1×2U= 2nU
∀n∈N, Un= 2n−1U
(Un)TUn= (2n−1U)T2n−1U= 4n−1UTU= 4n−1UUT=Un(Un)T
Unn∈N∗
E2M2(R)
A+C=0 0
2 0 
(A+C)(A+C)T=0 0
2 0  0 2 0 0 =0 0 0 4 
(A+C)T(A+C) = 0 2
0 0  0 0 2 0 =4 0 0 0 
(A+C)(A+C)T6= (A+C)T(A+C)A C E2