Réduction des matrices symétriques réelles

(E, < ., . >)f
E E
f
∀S∈ Sn(R),∃(Ω, D)∈On(R)×Dn(R)/ S = ΩDΩ−1
(E, < ., . >)f
E f
λ µ f λ 6=µ x λ y µ < x, y >= 0
λ, µ ∈R(f)λ6=µ x ∈SEP (f, λ)y∈SEP (f, µ)
x6= 0, y 6= 0, f(x) = λx, f(y) = µy
hλx, yi=hf(x), yi=hx, f(y)i=hx, µyi(λ−µ)hx, yi= 0 hx, yi= 0
f E
F E f F ⊥f
F E f x ∈F⊥∀y∈F, hf(x), yi=hx, f(y)i= 0
x∈F⊥f(y)∈F f(x)∈F⊥
n= dim(E)
n= 1
n≥1E n + 1
f E B1E A =B1(f)A
A=µαtC
C B ¶, α ∈R, C ∈ Mn,1(R), B ∈ Sn(R)
Ω∈On(R), D ∈Dn(R)B= ΩDΩ−1
U=µ1 0
0 Ω ¶U U−1=µ1 0 0 Ω−1¶
U−1AU =µαtCΩ
Ω−1C D ¶
G= Ω−1C A0=U−1AU =µαtG
G D ¶
•A0
G=
g1
gn
D= (d1, . . . , dn)A0=
α g1· · · gn
g1d10
gn0dn
k∈ {1, . . . , n}gk= 0 dkA0
Ek+1 ∀k∈ {1, . . . , n}, gk6= 0
λ∈R, V ∈ Mn+1,1(R)
V V =µx
X¶x∈RX∈ Mn,1(R)
A0V=λV ⇐⇒ µαtG
G D ¶µ x X¶=λµx X¶⇐⇒ ½αx +tGX =λx xG +DX =λX
d1≥. . . dnλ > d1D−λIn
A0V=λV ⇐⇒ ½X=−x(D−λIn)−1G
αx −xtG(D−λIn)−1G=λx

tG(D−λIn)−1G+λ=α⇐⇒
n
X
i=1
g2
i
di−λ+λ=α
ϕ:
]d1,+∞[→R
λ7→
n
X
i=1
g2
i
di−λ+λ]d1,+∞[−∞ d+
1
+∞+∞λ∈]d1,+∞[
ϕ(λ) = α
f
•x0fRx0f(Rx0)⊥
f E 1
kx0kx0
fµλ00
0S¶λ0∈RS∈ Sn(R)
Ω1∈On(R), D1∈Dn(R)S= Ω1D1Ω−1
1Ω2=µ1 0
0 Ω1¶
D2=µλ00
0D1¶
Ω2∈On+1(R), D2∈Dn+1 (R),µλ00
0S¶= Ω2D2Ω−1
2
E f
≥2µ2i
i0¶
S={x∈E / kxk= 1}ϕ:S→R
x7→ hx, f(x)i
•ϕ E x0∈S
ϕ(x0) = sup
x∈S
ϕ(x)
•x1∈E(x0, x1)hx1, f(x0)i= 0
cos(θ)x0+ sin(θ)x1θ∈Rkcos(θ)x0+ sin(θ)x1k2= cos2(θ) + sin2(θ) = 1
ϕ(cos(θ)x0+ sin(θ)x1)≤ϕ(x0)
2 sin(θ) cos(θ)hx1, f(x0)i ≤ sin2(θ) (hx0, f(x0)i−hx1, f(x1)i)
½∀θ∈]0, π[,2 cos(θ)hx1, f(x0)i ≤ sin(θ) (hx0, f(x0)i−hx1, f(x1)i)
∀θ∈]−π, 0[,2 cos(θ)hx1, f(x0)i ≥ sin(θ) (hx0, f(x0)i−hx1, f(x1)i)
θ0+0−hx1, f(x0)i= 0
•x0f
x∈x⊥
0\{0}¿1
kxkx, f(x0)À= 0 hx, f(x0)i= 0
x⊥
0⊂(f(x0))⊥
Rx0=x⊥
0⊃(f(x0))⊥⊥ =Rf(x0)
f(x0)∈Rx0x0f