130: matrice symétriques réelles et matrices

R C A∗=A
A∗A
R−ev
n(n+ 1)/2n2
A=−A∗
(u(x)|u) =
(x|u(y))
E
¯
f(y, x)
E
E×E
q E
q q(Pxiei) = Pλi|xi|2=Pλiei∗(x)λi
e∗
i

q ϕ2
1+. . . +ϕ2
r−ψ2
1−. . . ϕ2
p=ϕ02
1+. . . +
ϕ02
r0−ψ02 1−. . . ϕ02 p0r=r0p=p0
A1−1 0 r1p−1
p+r
p+r=dim(E) =: n p +r=dim(E) =: n
C
q=Paiix2
i+P
i<j
aij xixj
aii a11q a11x2
1+
x1B(x2, . . . , xn) + C(x2, . . . , xn) = a11(x1+B(x2, . . . , xn)
2a)2+ (C(x2, . . . , xn)−B(x2, . . . , xn)2 4a)
aii aij aij
a12 q(x) = ax1x2+x1B+x2C+D=a(x1+C/a)(x2+B/a) + (D−BC/a) =
a/4((x1+x2+ (B+C)/a)2−(x1−x2+ (C−B)/a)2+ (D−BC/a)
Pλiϕ2
i
λi>0<0
ϕ(x, y, z) = x2−2y2+xz +yz = (x+z/2)2−z2/4−2y2+yz = (x+z/2)2−2(y−z/4)2+z2/8−z2/4
(1,2) 1 + 2 = 3
ϕ(x, y, z) = ¯xx + ¯yy + ¯zz + ¯xy + ¯yx −¯yz −¯zy = (x+y)(¯x+ ¯y) + z¯z−¯yz −y¯z= (x+y)(¯x+ ¯y) +
(z−y)(¯z−¯y)−y¯y(1,2) 1 + 2 = 3
(Ax, x)>0>0x6= 0
S+
n, S++ n, H+ n, H++ n
q
p
|ϕ(x, y)|26q(x)q(y)
Q(x+y)6Q(x) + Q(y)

F uF
A=0 1
−1 0C
λ1, µ1λ1µ1
λnµn
1
1 + |i−j|
SnS++
n
Hn
Sp(A)
C
A ϕA
F(Sp(A),C) (Hn)ϕA(Id)A ϕAϕA(f)∗=ϕ(¯ f)
ϕA(f)A ϕA(f)g
ϕA(g◦f) = ϕϕA(f)(g)
R+∗
exp(A) = B
R+∗B
B2=A
RA
B3=A
GLn(mathbbC)→Un×H++
nA(U, H)