22 Réduction et classi cation des formes quadratiques sur un

EK2.
E q E K
∀x∈E, q (x) = φ(x, x)
φ E.
Q(E)EK
R2, φ ψ
φ(x, y) = x1y1+x2y2
ψ(x, y) = x1y1+x1y2−x2y1+x2y2
q(x) = φ(x, x) = x2
1+x2 2
=ψ(x, x) = x2
1+x1x2−x2x1+x2 2
φ
q E,
φ q (x) = φ(x, x)x∈E.
q q (x) = φ0(x, x)x∈E,
φ0E. φ E ×E
φ(x, y) = 1
2(φ0(x, y) + φ0(y, x))

K2φ(x, x) = q(x)
x∈E, φ.
φ x, y E
q(x+y) = φ(x+y, x +y) = φ(x, x)+2φ(x, y) + φ(y, y)
=q(x) + 2φ(x, y) + q(y)
φ(x, y) = 1
2(q(x+y)−q(x)−q(y))
φ.
φ
q.
∀(x, y)∈E2, φ (x, y) = 1
2(q(x+y)−q(x)−q(y)) = 1 4(q(x+y)−q(x−y))
λ x,
q(λx) = φ(λx, λx) = λ2φ(x, x) = λ2q(x)
q2.
q φ
Q(E)Bil s(E)
E. E n, Q (E)n(n+ 1)
2.
φ1φ2E φ1(x, x) = φ2(x, x)x∈E.
E n B= (ei)1≤i≤nE.
q E φ, q φ
BA= ((φ(ei, ej)))1≤i,j≤n
qB
Q(E)Sn(K)
n.
aij =φ(ei, ej)i, j 1n,
q∈Q(E)B= (ei)1≤i≤n
q(x) = 
1≤i,j≤n
aijxixj=tXAX
XKnxB.

q E
A= ((aij))1≤i,j≤nB.
E
Q(E) 2 n
aij =aji,
q(x) =
n
i=1
aiix2
i+ 2 
1≤i<j≤n
aijxixj
φB
φ(x, y) = 1
2(q(x+y)−q(x)−q(y)) = ⟨AX |Y⟩
A∈ Sn(K)
E=Knq
∀X∈Kn, q (X) = ⟨AX |X⟩
p ℓ1,··· , ℓpE
λ1,··· , λpq E
∀x∈E, q (x) =
p
j=1
λjℓ2
j(x)
φ E2
∀x∈E2, φ (x, y) =
p
j=1
λjℓj(x)ℓj(y)
q(x) = φ(x, x)x∈E.
qRn, φ
φ(x, y) = 1
2
n
j=1
∂q
∂xj
(x)yj=1
2
n
i=1
∂q
∂yi
(y)xi
A= ((aij))1≤i,j≤nq
q(x) = 
1≤i,j≤n
aijxixj
q
φ(x, y) = 
1≤i,j≤n
aijxiyj.

k1n,
∂q
∂xk
(x) = ∂
∂xkn
i=1
xi
n
j=1
aijxj=
n
i=1
∂
∂xkxi
n
j=1
aijxj
=∂
∂xkxk
n
j=1
akj xj+
n
i=1
i̸=k
∂
∂xkxi
n
j=1
aijxj
=
n
j=1
akj xj+xkakk +
n
i=1
i̸=k
xiaik =
n
j=1
akj xj+
n
i=1
aikxi
=
n
j=1
ajkxj+
n
i=1
akixi= 2
n
i=1
aikxi
akj =ajk A
φ(x, y) = 
1≤i,j≤n
aijxiyj=
n
j=1 n i=1
aijxiyj
=1
2
n
j=1
∂q
∂xj
(x)yj
qR3
q(x) = x2
1+ 3x2 2+ 5x2 3+ 4x1x2+ 6x1x3+ 2x2x3.
φ(x, y) = 1
2((2x1+ 4x2+ 6x3)y1+ (4x1+ 6x2+ 2x3)y2+ (10x3+ 6x1+ 2x2)y3)
= (x1+ 2x2+ 3x3)y1+ (2x1+ 3x2+x3)y2+ (5x3+ 3x1+x2)y3
EK
2q E φ.
x, y E φ φ (x, y) = 0.
XRn, X φ
E X.
X⊥X φ
X⊥={y∈E| ∀x∈X, φ (x, y) = 0}.
X={0}, X⊥=E.

X, Y E.
X⊥E.
X⊂X⊥⊥.
X⊂Y, Y ⊥⊂X⊥.
X E, X⊥E,
X⊂X⊥⊥X
E=R2, E⊥={0}. y ∈E⊥,
y·y=y2
1+y2 2= 0 y1=y2= 0, y = 0.
φR2
φ(x, y) = x1y1−x2y2
x x2
1−x2 2= 0,
x2=±x1.
x E φ
E, φ,
φ.
φ E
Cφ=q−1{0}={x∈E|q(x) = φ(x, x) = 0}.
Cφq.
φ E.
ker (φ)φ,
ker (φ) = E⊥={y∈E| ∀x∈E, φ (x, y) = 0}
E.
ker (φ)qker (q).
φ
ker (φ)⊂Cφ.
x∈ker (φ), E
φ.
E A ∈ Sn(K)φ
Bu∈ L(E)E A
φ u.