Exercices de Maths MP* - Lycée Montaigne

EKn>1u1, . . . , uk∈ L(E)
k>n u1◦ ··· ◦ uk
n
n= 1
n>2< n
EKn u1, . . . , uk
uiF:= Im(uk)ui
i∈J1, k −1KviuiF ui
viv1,...vk−1k−1F
k−1>dim F F ukuk
v1◦ ··· ◦ vk−1= 0L(F)
∀x∈E, (u1◦ ··· ◦ uk−1)(uk(x)
|{z}
∈F
)=(v1◦ ··· ◦ vk−1)(uk(x)) = 0
M:= 0 1
0 0N:= 0 0 1 0
M∈ Mn(R)
tr(M)=0 M
Mn(R)
M
tr(M)=0 M= 0
n>2M∈ Mn(R)
M= 0
M
u
RnM u

x∈Rn(x, u(x))
Rn
B= (x, u(x), e3, . . . , en).
u
MatB(u) = 
0∗ ∗ ∗ ∗
1
0N
0
tr(u) = 0 N
N P ∈GLn−1(R)P−1NP
Q:= 
1 0 . . . 0
0
P
0
∈GLn(R)
Q−1MatB(u)Q
0∗. . . . . . ∗
∗
∗P−1NP
∗
=: M0
MMatB(u)
M0M
tr(M) = 0 M
Ei,j i6=j
M P −1Ei,jP i 6=j
KEK
E E I ⊕N=E
G:= {f∈ L(E)|Im(f) = Iker(f) = N}
G◦
L(E)
◦
GL(E)G N {0}
◦
G(G, ◦)

◦
◦
◦f, g ∈G
f g L(E)f◦gL(E)
I= Im(f) = f(I⊕N) = f(I) = f(Im(g)) = Im(f◦g)
ker(f◦g) = g−1(ker(f)) = g−1(N) = g−1(N∩I) = ker(g) = N
f◦g∈G◦G
(G, ◦)◦ L(E)
e∈G f ∈G f ◦e=f=e◦f
e
f=e◦f e|I=id e ∈G e N
e∈ L(E)e N e I e I N
(G, ◦)
f∈G g ∈G f ◦g=g◦f=e f ∈G I
ker(f) = N E f I f
I fII f g
g|N= 0 g G
g|I=f−1
I
E E =I⊕N f ◦g N
I f ◦g=e g ◦f=e
N={0}I=E(GL(E),◦)
n∈N∗A, B ∈ Mn(R)
(A2+B2=√3(AB −BA)
AB −BA ∈GLn(R)
n
A2+B2AB −BA
A2+B2= (A−iB)(A+iB) + i(BA −AB)
Mn(C)n= 1
N⊆ker(f)

(A−iB)(A+iB) = (√3 + i)(AB −BA) = 2eiπ/6(AB −BA)
det(A−iB) det(A+iB)
| {z }
=|det(A−iB)|2
= 2neinπ
6det(AB −BA)
M∈ Mn(C) det(M) = det M
AB −BA ∈GLn(R) det(AB −BA)
einπ/6n
f:Mn(C)→ Mn(C)A, B ∈
Mn(C)t
det(f(A) + tf(B)) = det(A+tB).
f
f
A, B ∈ Mn(C)f(A) = f(B)
M∈ Mn(C)t∈C
det(A+tM) = det(f(A) + tf(M)) = det(f(B) + tf(M)) = det(B+tM).
N∈ Mn(C)M=N−B t = 1
det(A−B+N) = det(N).
A−B r A −B
P, Q ∈GLn(C) (A−B) = P JrQ
Jr:= Ir0
0 0.
N=P Q det(P(Jr+In)Q) = det(P Q) det(P Q)∈C×
det(Jr+In) = 1 det(Jr+In)=2rr= 0 A−B
A=B
f
M∈ Mn(C)M
det(A+XM )AMn(C)
r M P, Q ∈GLn(C)M=P JrQ
A∈ Mn(C)
det(A+XM ) = det(A+XP JrQ) = det(P(P−1AQ−1+XJr)Q) = det(P Q) det(P−1AQ−1+XJr)
r A =P Q det(A+XM ) = det(P Q) det(In+XJr) =