15 Valeurs propres

K K [X]K.
EKn≥1.
L(E)E, GL (E)
E.
Mn(K)nK, GLn(K)
Mn(K).
Id In
E
L(E)Mn(K).
AMn(K)Kn,
AA:Kn→Kn
x7→ y=Ax.
u E.
λ∈Kuker (u−λId)̸={0}.
λ∈Ku x
E u (x) = λx.
x u λ
E, Eλ= ker (u−λId), λ.
u∈ L(E)uSp (u).
Eker (u−λId)
{0}u−λId E
λ u −λId
E.
A∈ Mn(K)Kn
Kn,

λKA∈ Mn(K)
xKnAx =λx.
x A λ
Kn, Eλ={x∈Kn|Ax =λx}ker (A−λIn)
λ.
A A Sp (A).
E n ≥1, λ ∈K
u∈ L(E)u−λId
u Pu(X) = det (u−XId)X=λ.
dim (E) = n
Sp (u) = P−1
u{0}={λ∈K|det (u−λId) = 0}
Kn
K K =R
A∈ Mn(K)PA(X) = det (A−XId).
u A E, A u
E
F E u,
u F u
B1F E, B=B1∪B2.
u A =A1A2
0A3A1B1,
u F F u u
Pu(X) = det (A1−XIn1) det (A3−XIn3).
Puu
F.
E
λ∈Ku α
u,
1≤dim (ker (u−λId)) ≤α.
Eλ= ker (u−λId)
1.
u, Pλ
u EλPuu. Pλ(X)=(λ−X)δ
δ Eλ,
Pu(X) = (λ−X)δQ(X)
λ Puα, δ ≤α.
A=a b
c d ∈ M2(K)
PA(X) = a−X b
c d −X=X2−Tr (A)X+ det (A).

A=
a1a2a3
a4a5a6
a7a8a9
∈ M3(K)
PA(X) = 
a1−X a2a3
a4a5−X a6
a7a8a9−X
=−X3+aX2−bX +c
a= Tr (A), c = det (A)
b=a5a6
a8a9+a1a3 a7a9+a1a2 a4a5
2A
A∈ Mn(K),
PA(X) = (−1)nXn+ (−1)n−1αn−1Xn−1+ (−1)n−2αn−2Xn−2+··· +α0
αn−1= Tr (A), α0= det (A)
αn−2=
1≤i<j≤n
det (Ai,j)
det (Ai,j),1≤i < j ≤n, 2
A,
det (Ai,j) = aii aij
aji ajj (1 ≤i < j ≤n)
n= 4
A=
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
PA(X) = X4−α3X3+α2X2−α1X+α0
α3=a11 +a22 +a33 +a44
α2=a11 a12
a21 a22 +a11 a13 a31 a33 +a11 a14 a41 a44 +a22 a23 a32 a33 +a22 a24 a42 a44 +a33 a34 a43 a44 

Rθ=cos (θ)−sin (θ)
sin (θ) cos (θ)M2(R),
Pθ(X) = X2−2 cos (θ)X+ 1 = (X−cos (θ))2+ sin2(θ)
θ∈R\πZ.Sp (Rθ) = ∅
Rθ=±I2θ∈πZ
Sθ=cos (θ) sin (θ)
sin (θ)−cos (θ)M2(R),
Pθ(X) = X2−1
Sp (Sθ) = {−1,1}.
E=C∞(R)R R
u u :f7→ f′.
λ,
ker (u−λId) = {f∈ C∞(R)|f′=λf}=Rfλ
fλ
∀x∈R, fλ(x) = eλx
f01
Sp (u) = R
a < b E =C0([a, b])
[a, b]Ru u :f7→ g, g
∀x∈[a, b], g (x) = x
a
f(t)dt
λ,
f∈ker (u−λId)⇔∀x∈[a, b],x
a
f(t)dt =λf (x)
λ= 0,x
a
f(t)dt = 0 x∈[a, b]f= 0
0
λ̸= 0, f ker (u−λId)λf ′=f f (a)=0,
α f (x) = αex
λf(a) = 0 α= 0.
Sp (u) = ∅.
u u C1([a, b]) ̸=E
u0u.
E=K[X]u u :P7→ XP.
λ,
ker (u−λId) = {P∈K[X]|XP =λP }={0}
Sp (u) = ∅.
0u.

C∞(R+,∗)u u :f7→ xf′.
u.
λ,
ker (u−λId) = f∈ C∞R+,∗|xf′=λf=Rfλ
fλ
∀x∈R+,∗, fλ(x) = xλ
f01
Sp (u) = R
A
λ A ∈ Sn(R)⊂ Mn(C)x
Ax =λx,
Ax =λx A
txAx =λtxx =λ
n
k=1 |xk|2
A
txAx =t(Ax)x=t(λx)x=λtxx =λ
n
k=1 |xk|2
λ=λ
n
k=1 |xk|2̸= 0.
R
R
A∈ Sn(R), A A
A=
1 1 ··· 1 1
1 0 ··· 0 1
1 0 ··· 0 1
1 1 ··· 1 1
∈ Mn(R)
n≥3.