AGRÉGATION INTERNE DE MATHÉMATIQUES YjY Algèbre

YjY YjY
On(R)Un(C)
A∈Mn(R)t A=A
A∈SnhAx, xi ≥ 0x∈Rn
A∈S+
nhAx, xi>0
x∈Rn\ {0}A∈S++
nE n
u∈L(E)kuxk=kxkx∈E u ∈On(E)
A At A=InA
A∈On(R)
A∈Mn(C)AtA=InA∈Un(R)
ERn≥2ϕ
E O
O2(R)
ϕ
ϕ
U∈On(R)U−1OU
(Ir,−Is, O1, . . . , Ot)Oi=cos θisin θi
−sin θicos θi
A∈S+
n(R)
B∈S+
n(R)A=B2B A
A∈GLn(R) (O, S)∈On(R)×S++
n(R)
A=OS S t AA
On(R)GLn(R)
A∈Mn(R) (O, S)∈On(R)×S++
n(R)
A=OS
GLn(R)3A7→ (O, S)∈On(R)×S++
n(R)
GLn(R)
1
On(R)GLn(R)
A∈On(R) limk→∞
1
nPn−1
k=0 Ak
Rn= ker(A−In)
⊥
⊕(A−In)
On(R)
On(R)
Un(R)