Télécharger

TSI2
no3
M2+M=A
A=5 3
1 3 M2(R)
A
A
χA(x) = x−5−3
−1x−3=x2−8x+ 12
∆ = 64 −48 = 16 = 42
λ1=8+4
2= 6 λ2=8−4 2= 2
Spec(A) = {6,2}
x
y∈E6⇐⇒ (A−6I)x y=−1 3 1−3 x y=0 0
⇐⇒ x−3y= 0 ⇐⇒ x= 3y
x
y∈E2⇐⇒ (A−2I)x y=3 3 1 1  x y=0 0
⇐⇒ x+y= 0 ⇐⇒ x=−y
E6= Vect  3
1 E2= Vect  −1 1
D=6 0
0 2 =P−1AP P =3−1 1 1 
N=x y
z t M2(R)
N2+N=D
ND =N(N2+N) = N3+N2= (N2+N)N=DN

N D
ND =x y
z t  6 0 0 2 =6x2y 6z2t
DN =6 0
0 2  x y z t =6x6y 2z2t
6y= 2y6z= 2z
y=z= 0
N2+N=D
N2+N=x20
0y2+x0 0y=x2+x0 0y2+y=6 0 0 2 
t2+t= 2
x2+x= 6
t2+t−2 1 −2x2+x−6
2−3N
2 0
0 1 ;−3 0 0 1 ;2 0 0−2;−3 0 0−2
N2+N N
N2+N=D
MM2(R)N=x y
z t x, y, z, t R
M2(R)N=P−1MP
P−1P X =X0⇐⇒ X=P−1X0
Px
y=x0 y0⇐⇒ 3x−y=x0 x+y=y0
⇐⇒ 4x=x0+y0(L1+L2)
4y= 3y0−x0(3L2−L1)
P−1=1
41 1
−1 3 
M=P NP −1
M=1
43x−3y−z+t3x+ 9y−z−3t
x−y+z−t x + 3y+z+ 3t

TSI2
P−1M2P=P−1MP ×P−1MP =N2
M2+M=A⇐⇒ P−1(M2+M)P=P−1AP
⇐⇒ P−1M2P+P−1MP =D
⇐⇒ N2+N=D
M2+M=A⇐⇒ N2+N=D
MM2(R)M2+M=A
M=P NP −1N2+N=D
y=z= 0 x∈ {−3,2}t∈ {−2,1}
S=1
4−11 −3
−1−9,1
4−8−12
−4 0 ,1
44 12
4−4,1
47 3
1 5 
a
Ma=a−1−1
2a+ 2 
Madet Ma6= 0
det Ma=a−1−1
2a+ 2 = (a−1)(a+ 2) + 2 = a2+a=a(a+ 1)
Maa6= 0 a6=−1
χMa(x) = x−a+ 1 1
−2x−a−2=(a−x)−1−1 2 (a−x)+2 = (a−x)(a−x+ 1)
Spec(Ma) = {a, a + 1}
Ma
x
y∈Ea⇐⇒ (A−aI)x y=−1−1 2 2  x y=0 0
⇐⇒ −x−y= 0 ⇐⇒ y=−x
x
y∈Ea+1 ⇐⇒ (A−(a+ 1)I)x y=−2−1 2 1  x y=0 0
⇐⇒ 2x+y= 0 ⇐⇒ y=−2x

Ea= Vect  1
−1 E1+a= Vect  1 −2
Da=a0
0 1 + a=P−1MaP P =1 1 −1−2
P−1P X =X0⇐⇒ X=P−1X0
Px
y=x0 y0⇐⇒ x+y=x0 −x−2y=y0
⇐⇒ x= 2x0+y0(2L1+L2)
y=−x0−y0(−L1−L2)
P−1=2 1
−1−1
a b DaDb
MaMb=P DaP−1P DbP−1=P DaDbP−1=P DbDaP−1=P DbP−1P DaP−1=MbMa
MaMb=MbMa
n An
An=M1M2M3. . . Mn
An=P D1D2. . . DnP−1n∈N
? A1=M1=P D1P−1
? n An=P D1D2. . . DnP−1
An+1 =An×Mn+1 =P D1D2. . . DnP−1×P Dn+1P−1=P D1D2. . . DnDn+1P−1
n+ 1
? n >1
D1D2. . . Dn=1 0
0 2  2 0 0 3 . . . n0 0n+ 1 =n! 0 0 (n+ 1)! 
An=1 1
−1−2 n! 0 0 (n+ 1)!  2 1 −1−1
An=(1 −n)n!−n.n!
2n.n! (2n+ 1)n!

TSI2
(un)n>1(vn)n>1u1=−2v1= 4
∀n∈N, un+1 = (n−1)un−vnvn+1 = 2un+ (n+ 2)vn.
un+1
vn+1 =n−1−1 2n+ 2  un vn=Mnun vn
n>1
un
vn=Mn−1. . . M2M1u1 v1=M1M2. . . Mn−1u1 v1=An−1−2 4
∀n∈N∗, un=−2n!vn= 4n!
E=R3[X]
f E P ∈E f(P)
f(P)(X) = −3XP (X) + X2P0(X), P 0P.
E(1, X, X2, X3)
dim E= 4
f E f
E
P, Q ∈E λ, µ ∈R
f(λP +µQ)(X) = −3X(λP +µQ)(X) + X2(λP +µQ)0(X)
=−3X(λP (X) + µQ(X)) + X2(λP 0(X) + µQ0(X))
=λ(−3XP (X) + X2P0(X)) + µ(−3XQ(X) + X2Q0(X))
f(λP +µQ)(X) = λf(P)(X) + µf(Q)(X)
f
f(1)(X) = −3X
f(X)(X) = −3X2+X2=−2X2
f(X2)(X) = −3X3+ 2X3=−X3
f(X3)(X) = −3X4+ 3X4= 0
E
f E