Algèbre bilinéaire

R
Q
q q(ei)>0 (ei)
Q(x, y, z) = 2x2−2y2−6z2+ 3xy −4xz + 7yz.
Q(x1, x2, x3) = |x1|2+|x2|2+|x3|2−ix1x2+ix1x2+ix1x3−ix1x3+x2x3+x2x3
n E R[X]6n
Φ : P7→ R1
0P2−(R1 0P)2
n×n
"0 1
...
1 0#
0 1 ... 1
1.......
.
.
.
.
.......1
1... 1 0
ϕ E ϕ
Φ : E→E∗
F⊂E F ⊥ϕ⊂E ϕ F ⊥∗⊂E∗
ΦF⊥ϕF⊥∗
Rn
Cn
k2h0 1
1 0i
h0 1
1 1ia b a+b6= 0
ha0
0bi ha+b0
0ab
a+bi.
Fq6= 2
FqF∗
q/(F∗ q)2
u, v ∈F∗
qa, b ∈Fqua2+vb2= 1
{ua2, a ∈Fq} {1−vb2, b ∈Fq}
u∈FqS∈ Sn(Fq)
idnhidn−10
0uiFq

H∈Mn(C) 1 6k6n
Hkk k
16k6n Hk
det Hk>0>0
kdet Hk6= 0 H
det Hk
M∈Mn(C)
P(PtP)M(PtP)−1=tM
Mn(R)→GLn(R)
P∈R[X]n
S χS= (−1)nP
S
Sn(ai, bi) := 
a1b10
b1a2b2
.........
0bn−1an
ai, bi
SnχSnχSn−1χSn−2
P
c, d ∈RR∈R[X]n−2P= (X−d)P0
n−c2R
P P 0P0P
P0R P 0
R+∞R P 0
P
P
P∈R[X]nAk k[X]/(P)
α∈ A A(α) := (mα)mα:A → A
α
ϕP:A × A → k(α, β)7→ A(αβ)
ϕPP
P
k[X]/(Qi)Qα
ϕP(r1+r2, r2)r1
P r2P
a1, . . . , an"0a1
0.
.
.
a1. . . an#
M, N 7→ (tMN) Mn(R)
Rn

x= (x1, x2, x3)∈C3
q(x) = |x1|2+ 3|x2|2+ 6|x3|2+ix1x2−ix1x2+ 2ix2x3−2ix2x3.
f
f
C3
A=4i−i
−i4 1
i1 4 . U D
D=U−1AU
(E, k·k)Rk·k
∀(x, y)∈E2,kx+yk2+kx−yk2= 2(kxk2+kyk2).
ϕ:E2→R
ϕ(x, y) = 1
4(kx+yk2− kx−yk2)
ϕ
Rn
S1, S2∈ Sn(R)S1
S1S−1
1
S1S2S−1
1
S1S2
(E, h·i)n u ∈ L(E)λ16. . . λn
u{e1, . . . , en}
x∈E q(x) := hx, u(x)i
min
kxk=1 q(x) = λ1max kxk=1 q(x) = λn
16p6n Fp:= Vect{e1, . . . , ep}Gp= Vect{ep, . . . , en}
p
λp= min
x∈Gp
kxk= 1
q(x) = max
x∈Fp
kxk= 1
q(x).
16p6nFpGpE p
n−p+ 1 p
λp= max
G∈Gp
min
x∈G
kxk= 1
q(x) = min
F∈Fp
max
x∈F
kxk= 1
q(x).
G∈ GpG∩Fp6= 0
M∈ Snλ16. . . , 6λn
N∈ Sn−1M
µ16··· 6µnN
λ16µ16λ26··· 6µn−16λn.

A B ∈ Sn(R)C:= A+B A, B
C α1>··· >αn
A β1>··· >βnB γ1>··· >γnC
(fk)16k6n(gk) (hk)A B
C αkβkγk
E n F, G, H
dim F+ dim G+ dim H>2n+ 1 F∩G∩H6={0}
x∈Rnkxk= 1 αn6hx, Axi6α1
i, j i +j−16n
γi+j−16αi+βj.
(i, j)
F:= Vect{fi, fi+1, . . . , fn}G:= Vect{gj, gj+1, . . . , gn}H:= Vect{h1, . . . , hi+j−1}
x∈F∩G∩Hkxk= 1
x
hx, Axi6αi,hx, Bxi6βj,hx, Cxi>γi+j−1.
C3F
x1−x2+ix3= 0 F
EL(E)
E p ∈ L(E)p
∀x∈E, kp(x)k6kxk
E f ∈ L(E)x∈Ehf(x)|xi=
0. f
E u E
x E ku(x)k6kxk
∀x∈E, ku∗(x)k6kxk
∀x∈E, (u(x) = x)⇐⇒ (u∗(x) = x)
Ker(u−idE) Im(u−idE)E
n vn=1
nPn−1
k=0 uk(vn)
Ker(u−idE)
E
x, y E θ ∈[0, π]
hx, yi=kxkkykcos(θ)θ x y
f∈ L(E)E
α > 0f∗◦f=λid
λ > 0∀x∈E, kf(x)k=λkxk
f λ > 0
f

f
f
f λ
E u ∈ L(E)
u u∗
λ u Eλ
u∗E⊥
λu u
u X
u u∗
P={Re(X),Im(X)}u u∗X X
P u ha−b
b a iu
O(n)SO(n)U(n)
SO(n)U(n)
SO(n)
On(R)
On(R)
A= (aij )16i,j6n
P16i,j6n|ai,j |6n√n
P16i,j6naij 6n
O(n)
n= 3 πSO(3)
2O(n)
nO(n)∼
=SO(n)×Z/2
E u ∈O(E)
u u
O(n)Sn−1:= {x∈
Rn,kxk= 1}(0; . . . ; 0; 1)
A∈Mn(R)
A
idn−AC(A) := (idn+A)(idn−A)−1SO(n)
SO(n)−1
C(A)A
n= 2