Formes quadratiques 1 Formes bilinéaires symétriques

=
E n
{e1,· · · , en}E xinE x =x1e1+· · · +xnen
q:E→
q(x) =
n
X
i=1
aiix2
i+X
i6=j
aij xixj
xi
E
f:E×E→
f(x+x0, y) = f(x, y) + f(x0, y)
f(λx, y) = λf(x, y)
f(x, y) = f(y, x)
S(E)E
=
E=2f(x, y) = x1y1−x2y2

f∈ S(E){e1,· · · , en}E
f
n
X
i=1
xiei,
n
X
j=1
yjej
=
n
X
i=1
n
X
j=1
xiyjf(ei;ej)
E={e1,· · · , en}E f ∈ S(E)
f e Mf= (aij )∈ Mn( )
ai,j =f(ei, ej)
f
A f aj,i =f(ej, ei) =
f(ei, ej) = ai,j tA=A
E={e1,· · · , en}E M
f
EM
f M Ef(ei, ej) = mij i
j
x=X
i
xieiy=Pjyjejf(x, y) = Pi,j xiyjf(ei, ej)
x=X
i
xieiy=PjyjejX=
x1
xn
Y=
y1
yn
M f
f(x, y) =tXMY
M= (mi,j )tXM = · · · ,
n
X
i=1
ximij ,· · ·!
n
X
i=1
ximij j

tXMY =
n
X
j=1 n
X
i=1
ximij !yj=
n
X
j=1 n
X
i=1
xiyjf(ei, ej)!=f(x, y)
E E0E P
E E0M M 0fE E0
M0=tP MP
X Y X0Y0x
yE E0X=P X0Y=P Y 0
f(x, y) = tX0M0Y0
=tXMY
=X0tPM(P Y 0)
=tX0tP MP Y0
M0=tP MP f E0
f E =2
f(x, y) = x2
1−2x1x2+ 3x2 2
f
A=1−1
−1 3 
E0e0
1=1
3, e0
2=2
1
f(e0
1, e0 1) = 22 f(e0 1, e0 2) = 4
f(e0
2, e0 1) = 4 f(e0 2, e0 2) = 3
fE0
A0=22 4
4 3 
P=1 2
3 1 
tP AP =A0
f E
f(x+y, x +y) = f(x, x) + f(x, y) + f(y, x) + f(y, y) = f(x, x)+2f(x, y) +
f(y, y)
f(x, y) = 1
2(f(x+y)−f(x, x)−f(y, y))

E q :E→
∀x∈E, ∀λ∈, q(λx) = λ2q(x)
fq:E×E→: (x, y)7→ 1
2(q(x+y)−q(x)−q(y))
fq
E
f E q :E→q(x) = f(x, x)
fq=f
q fqfq(x, x, ) =
q(x)
fqq
qEfq
(aij )
q
f(x, y) =
n
X
i=1
xiyi+X
1≤i<j≤n
aij (xiyj+xjyi)
q(x) =
n
X
i=1
aiix2
i+ 2 X
1≤i<j≤n
aij xixj(∗)
q
E n
E
E
q(x) =
n
X
i=1
aix2
i+ 2 X
1≤i<j≤n
bij xixj
n
a1· · · an
bij

q(x) = x2
1+ 3x2 2−2x1x2
1−1
−1 2 
f(x, y) = x1x21−1
−1 2 y1 y2=x1y1+ 3x2y2−x1y2−x2y1
q(x) = x2
1+ 3x2 2−4x2 3+ 6x1x2+ 8x1x3
1 3 4
3 3 0
4 0 4
x y E f(x, y) = 0
A⊂E A A⊥={x∈
E;∀a∈A f(x, a) = 0}
fKer(f)
x∈E∀y∈E f(x, y) = 0
Ker f6={0}f
Ker f={0}f
f1 1
1 1
Ker f= Vect(1,−1)
q(x) = x2
1−x2
1 0
0−1{0}
fdim E−dim (Ker f)
f