21 Groupe orthogonal d`un espace vectoriel euclidien de dimension

2,3
E n ≥1,
EB0,
BdetB0(B) = 1
u∈ O(E) det (u) = ±1
O+(E) = {u∈ O(E)|det (u) = 1}
O(E)
O−(E) = O(E)\ O+(E) = {u∈ O(E)|det (u) = −1}
O(E)
u∈ L(E)A
E A ∈ On(R).
O+
n(R) = {A∈ On(R)|det (A) = 1}
On(R)
O−
n(R) = On(R)\ O+ n(R) = {A∈ On(R)|det (A) = −1}
On(R)
λ u ∈ O(E), λ =±1.
x∈E λ,
1 = ∥x∥=∥u(x)∥=∥λx∥=|λ|∥x∥=|λ|
F E u ∈ O(E), F ⊥
u u (F)⊂F, u (F) = F
u E x ∈F⊥, z ∈F, y ∈F
z=u(y)
⟨u(x)|z⟩=⟨u(x)|u(y)⟩=⟨x|y⟩= 0
u(x)∈F⊥.

2,3
2
E2
B0= (e1, e2).
θ,
Rθ=cos (θ)−sin (θ)
sin (θ) cos (θ)Sθ=cos (θ) sin (θ) sin (θ)−cos (θ)
θ θ′,
RθRθ′=Rθ′Rθ=Rθ+θ′, SθSθ′=Rθ−θ′, RθSθ′=Sθ+θ′, Sθ′Rθ=Sθ′−θ
Rθ, Sθ
R−1
θ=R−θS−1 θ=Sθ
RθRθ′=cos (θ)−sin (θ)
sin (θ) cos (θ)cos (θ′)−sin (θ′) sin (θ′) cos (θ′)
=cos (θ) cos (θ′)−sin (θ) sin (θ′)−(cos (θ) sin (θ′) + sin (θ) cos (θ′))
cos (θ) sin (θ′) + sin (θ) cos (θ′) cos (θ) cos (θ′)−sin (θ) sin (θ′)
=cos (θ+θ′)−sin (θ+θ′)
sin (θ+θ′) cos (θ+θ′)=Rθ+θ′
SθSθ′=cos (θ) sin (θ)
sin (θ)−cos (θ)cos (θ′) sin (θ′) sin (θ′)−cos (θ′)
=cos (θ) cos (θ′) + sin (θ) sin (θ′)−(sin (θ) cos (θ′)−cos (θ) sin (θ′))
sin (θ) cos (θ′)−cos (θ) sin (θ′) cos (θ) cos (θ′) + sin (θ) sin (θ′)
=cos (θ−θ′)−sin (θ−θ′)
sin (θ−θ′) cos (θ−θ′)=Rθ−θ′
RθR−θ=R0=InSθSθ=R0=In, R−1
θ=R−θ
S−1
θ=Sθ.
Sθ+θ′Sθ′=Rθ=Sθ′Sθ′−θ, RθSθ′=Sθ+θ′, Sθ′Rθ=Sθ′−θ.
S−1
θ=SθS2
θ=I2Sθ2.
O+
2(R) = {Rθ|θ∈R} O− 2(R) = {Sθ|θ∈R}
θ,
R−1
θ=R−θ=tRθ,det (Rθ) = 1
Rθ∈ O+
2(R)
S−1
θ=Sθ=tSθ,det (Sθ) = −1
Sθ∈ O−
2(R).

2
A=a b
c d ∈ O2(R)C=d−c −b a 
A∈ O+
2(R),tA=A−1=1
det (A)
tC=tC, A =C,
a=d b =−c, A =a−c
c a det (A) = a2+c2= 1 θ
a= cos (θ)c= sin (θ), A =Rθ.
A∈ O−
2(R), A =−C, d =−a b =c,
A=a c
c−adet (A) = −(a2+c2) = −1θ a = cos (θ)
c= sin (θ), A =Sθ.
O2(R) = A=cos (θ)−εsin (θ)
sin (θ)εcos (θ)|θ∈Rε= det (A)∈ {−1,1}
θ A ∈ O2(R),
]−π, π].
R→ O+
2(R)
θ7→ Rθ
(R,+) O+
2(R) 2πZ
R
2πZO+
2(R).
O+
2(R)
Γ
1,R→Γ
θ7→ eiθ
R
2πZ,+(Γ,·)O+
2(R),◦
(Γ,·).
n≥3,O+
n(R)
O+
2(R) Γ Γ
[0,2π]t7→ eit
O−
2(R)O+ 2(R)
cos (θ)−sin (θ)
sin (θ) cos (θ)7→ cos (θ)−sin (θ) sin (θ) cos (θ)1 0 0−1=cos θsin θ sin θ−cos θ
O2(R)O+
2(R)O− 2(R)

2,3
u E
θ u
B
Rθ=cos (θ)−sin (θ)
sin (θ) cos (θ)
u E θ
uB0
Sθ=cos (θ) sin (θ)
sin (θ)−cos (θ)
u∈ O+(E)B0O+
2(R),
Rθθ
BE, u ∈ O(E)
P−1RθP, P ∈ O+(E)B0BP−1RθP=Rθ
O+
2(R)
u∈ O−(E)B0O−
2(R),
Sθθ.
u∈ O+(E)
BtRθ=R−1
θ=R−θ.
B B−
0= (e1,−e2),
B0B−
0Q=1 0
0−1uB−
0
Q−1RθQ=1 0
0−1cos (θ)−sin (θ) sin (θ) cos (θ)1 0 0−1
=cos θsin θ
−sin θcos θ=R−θ
uB.
u∈ O−(E),
B0BP=Rθ′
P−1SθP=R−θ′SθRθ′=Sθ−θ′Rθ′=Sθ−2θ′
u∈ O−(E)S0=1 0
0−1B0R2
B=1
√2(e1+e2),1 √2(−e1+e2), P =1 √21−1
1 1 uB
S′
0=0−1
−1 0 
u∈ O+(E)v∈ O−(E), v ◦u◦v=u−1.

2
B0
θ θ′,
Sθ′RθSθ′=Sθ′−θSθ′=R−θ=R−1
θ
u◦v∈ O−(E)O(E)−1
u◦v= (u◦v)−1=v−1◦u−1
v, v ◦u◦v=u−1.
2
u∈ O+(E)Rθ
B, θ
θ]−π, π],
−θ.
θ={θ+ 2kπ |k∈Z} ∈ R
2πZ
u∈ O+(E)E.
θ
θ2π.
u∈ O+(E)x=x1e1+x2e2∈E,
u(x) = (cos (θ)x1−sin (θ)x2)e1+ (sin (θ)x1+ cos (θ)x2)e2
⟨u(x)|x⟩= cos (θ)x2
1+x2 2= cos (θ)∥x∥2= cos (θ)∥x∥∥u(x)∥
detB0(x, u (x)) = x1cos (θ)x1−sin (θ)x2
x2sin (θ)x1+ cos (θ)x2= sin (θ)x2
1+x2 2= sin (θ)∥x∥2
θ2π u
cos (θ) = ⟨u(x)|x⟩
sin (θ) = detB0(x, u (x))
x
x̸= 0, θ x u (x).
θ]−π, π],±θ.
uπ
2(f1, f2)
u(f1) = f2u(f2) = −f1.
Id −Id.
0,−Id π
u θ /∈0, π
χu(λ) = cos (θ)−λ−sin (θ)
sin (θ) cos (θ)−λ
= (cos (θ)−λ)2+ sin2(θ)≥sin2(θ)>0
−Id −1
Id 1