Algèbre Linéaire : Corrigé du Devoir Maison n

◦1
λ A X = (xk)k=1,...,n (A−λIn)X= 0
−λx1+x2= 0
x1−λx2+x3= 0
x2−λx3+x4= 0
xk−2−λxk−1+xk= 0
xn−2−λxn−1+xn= 0
xn−1−λxn= 0
x0=xn+1 = 0
∀k∈ {2, . . . , n + 1}, xk−λxk−1+xk−2= 0.
(xk)k
r2−λr + 1 = 0 λ= 2 cos(θ) ∆ = 4 sin2(θ)
θ6= 0[π]r1= exp(iθ)r2= exp(−iθ) (xk)
∀k∈ {0, . . . , n}, xk=αrk
1+βrk 2=αexp(ikθ) + βexp(−ikθ),
α, β ∈Rx0= 0 α+β= 0
∀k∈ {0, . . . , n}, xk=α(exp(ikθ)−exp(−ikθ)) = 2αsin(kθ).
xn+1 = 0 2αsin((n+ 1)θ) = 0 sin((n+ 1)θ) = 0 α= 0
X= (xk)θ=lπ/(n+ 1) l∈ {1, . . . , n}
λl= 2 cos lπ
n+ 1, l ∈ {1, . . . , n},
A l
Xl=sin klπ
n+ 1k=1,...,n
λl
n A ∈
Mn(R). A
A
Γ : v∈ L(E)7→ u◦v−v◦u∈ L(E)
C(u) = (Γ) L(E)v
R[u]d(ak)k=0,...,d v=Pd
k=0 akuk
u◦v=u◦ d
X
k=0
akuk!=
d
X
k=0
ak(u◦uk) =
d
X
k=0
ak(uk◦u) = d
X
k=0
akuk!◦u=v◦u,
uku u v ∈ C(u)R[u]⊂ C(u)

{x0, u(x0), . . . , un−1(x0)}n= (E)
E E v(x0)
ai
{x0, u(x0), . . . , un−1(x0)}.
06p6n−1v∈ C(u)v up
v(up(x0)) = up(v(x0)) = up n−1
X
k=0
akuk(x0)!=
n−1
X
k=0
akup(uk(x0)) =
n−1
X
k=0
akuk(up(x0)) .
x E x
v
x
v=Pn−1
k=0 akuk
x E v ∈ C(u)uC(u)⊂R[u]
{x0, u(x0), . . . , un−1(x0)}E
x0n= (E)
a0, . . . , an−1∈RPn−1
k=0 akuk(x0)=0
u
uk(x0) = uk n
X
i=1
xi!=
n
X
i=1
uk(xi) =
n
X
i=1
λk
ixi.
0 =
n−1
X
k=0
akuk(x0) =
n−1
X
k=0
ak n
X
i=1
λk
ixi!=
n
X
i=1 n−1
X
k=0
akλk
i!xi.
{x1, . . . , xn}u
Pn−1
k=0 akλk
i= 0 1 6i6n
n×n
a0+a1λ1+a2λ2
1+· · · +an−1λn−1
1= 0
a0+a1λ2+a2λ2
2+· · · +an−1λn−1
2= 0
a0+a1λn+· · · · · · · · · +an−1λn−1
n= 0
ai
MA = 0 A=
a0
an−1
M=
1λ1λ2
1· · · λn−1
1
1λnλ2
n· · · λn−1 n
.
M
V(λ1, . . . , λn) = Y
1≤i<j≤n
(λj−λi).
λi2 2 M MA = 0
A=M−10=0 A= 0
a0=· · · =an−1= 0
{x0, u(x0), . . . , un−1(x0)}
u XnXn−1
x0∈E un−1(x0)6= 0 uk(x0)6= 0 k= 0, . . . , n −2

{x0, u(x0), . . . , un−1(x0)}
E u n 0
C(u) = R[u]
x∈Eiu(v(x)) = v(u(x)) = v(λix) = λiv(x)v(x)∈EiEi
v
x∈Ei
vi(ui(x)) = vi(u(x)) x∈Ei,
=v(u(x)) Eiu,
=u(v(x)) v∈ C(u),
=u(vi(x)), vi(x)∈Eiv(Ei)⊂Ei,
=ui(vi(x)).
vi◦ui=ui◦vivi∈ C(ui)
Φv∈ C(u)vi
viC(ui)
16i6p α, β ∈Rv, w ∈ C(u) (αv +βw)|Ei=α(v|Ei) + β(w|Ei)
Φ(αv +βw)=(α(v|E1) + β(w|E1), . . . , α(v|Ep) + β(w|Ep)),
=α(v|E1, . . . , v|Ep) + β(w|E1, . . . , w|Ep),
=αΦ(v) + βΦ(w).
Ψ : C(u1)× · · · × C(up)→ C(u)
(v1, . . . , vp)7→ Pp
i=1 evi
evi∈ L(E)evi(x) = vi(x)x∈Eievi= 0 eviviL(E)
E=Lp
i=1 EiΨ
Φ◦Ψ = IdC(u1)×···×C(up)Ψ◦Φ = IdC(u)Φ
Ψ
Eiu λix∈Eiui(x) = u(x) = λix
uiλiui=λiidEui
EiC(ui) = L(Ei)
(C(ui)) = (L(Ei)) = ( Ei)2=n2
i.
C(u)C(u1)× · · · × C(up)
(C(u)) = (C(u1)× · · · × C(up)) =
p
X
i=1
(C(ui)) =
p
X
i=1
n2
i.
u
u p λi
Mu(X) = (X−λ1)· · · (X−λp).
a0, . . . , an−1∈R
p−1
X
k=0
akuk= 0.
akak
p−1Pp−1
k=0 akXk
u Mup
ak{Id, u, . . . , up−1}

k>pR[X]XkMu(X)Xk=Mu(X)Q(X)+R(X)
R(X) (Mu) = p uk=Mu(u)◦Q(u) +
R(u) = R(u)Mu(u) = 0 uku p −1uk∈
(Id, u, . . . , up−1)
P(u)∈R[u]ukk>p P (u)
(Id, u, . . . , up−1){Id, u, . . . , up−1}
R[u]
R[u]
(R[u]) = p(C(u)) = Pp
i=1 n2
i
R[u]⊂ C(u)
R[u] = C(u)⇐⇒ (R[u]) = (C(u)) ⇐⇒ p=
p
X
i=1
n2
i.
ni= 1 1 6i6p ni
1u
u(E) = n p =n
R[u] = C(u)u