Exercice

uRn λ ∈R
u
λ u det(u−λE)=0
u n
R20 1
A∈n(K)A−1∈K[A]A
A∈n(K)A A−1
f E λ
f m 16dim (Ker (f−λE)) 6m
A=
1111
2222
3333
4444
A A
A=0a
b0∈ Mn(R)A
p∈NM∈ M2p(R)mi,j =λii+j= 2p+ 1 ai,j = 0
M
n∈N∗Φ : P∈E=Rn[X]7→ P(1 −X)
P1−X P 1−X P 1−X
ΦE
Φ◦Φ Φ
exp(Φ)
A, B ∈ M3(R)BA =
011
101
110
=C AB
A=
2 3 1
0−4−2
4 12 5 

A
B∈ M3(C)B2=A A B
Rac(A) = {B∈ M3(C); B2=A}
1
u∈ L(E) 1 u
n∈N∗C∈ Mn,1(K)L∈ M1,n(K)M=CL
M
M M
χM
DnMn(C)
DnC
D∈ Dnn(I, D, D2, ..., Dn−1)
Dn
n>2A∈ Mn(R)ai,j = 1 j=i+ 1 ai,j = 0 B∈ Mn(R)
B2=A
A∈ Mn(C)
(A)⊂
n
[
i=1
Df
ai,i,X
j6=i|ai,j |
M∈n(C)d>2M Md
M∈ Mn(C)M2M
Ker M= Ker M2
n∈N∗
InIn
0 0  0In In0 InIn 0In
H∈ Mn(R) (i, j)∈[[1, n]]2hi,j >0i∈[[1, n]]
n
X
j=1
hi,j = 1

1H
λ∈(H)|λ|61
M∈ Mn(R)mi,j = 1 −δi,j =(0 si i=j
1 si i6=j
M+InM
M
A=
0··· ··· 0 1
2
0··· ··· 0n−1
1 2 ··· n−1n
A
1
(0)
1··· 1
A∈ Mn(K)n(In, A, ..., An−1)
A A
P P (A) = B A =
010
101
010
B=
001
010
100
Q(B) = A Q ∈R[X]
R3
011
101
001
n>2An=1 2
3 6M2(R)
a∈RD=
1··· 1a
(1)
1··· 1a
D
D E1,21 (1,2)
u E n φu
L(E)
∀v∈ L(E), φu(v) = u◦v◦u
u φu

A=
102
010
201
A
A
x0=x+ 2z
y0=y
z0= 2x+z
A=1 2
2 1
B=A A
A A
A=4−4
1 0 Ann∈N∗
M∈ M4(C)M2=
0111
1011
1101
1110
A∈ Mn(C)B=In0
A AB
A In−A
a b
A=
a b b
b a b
b b a
A∈ M3(R)A2=
100
120
123
n
A=
−3−8−6
1/2 3/2 1
3/2 7/2 3 
A=
100
101
010
B=
1 1 0
0 1 0
0 0 −1
Xn

A
A B
An
f
fa b
c d=−a c b−d
fM2(R)
f
f
fa b
c d=d2b 2c a 
A=
030
1−2 4
111
fR3
A
A
P P −1AP
u=
1
−1
0
v=
1
1
1
w=
3
−1
−1
(u, v, w)R3
f
P−1P−1AP
A2A I3A−1
A=
201
111
−2 0 −1
A
α, β ∈RαA +βI3
n>2
An=
2−1 (0)
−1
−1
(0) −1 2
∈ Mn(R)
Dn= det(An)
An
0A