Concours Blanc - Lycée Jean Perrin

TSI2
no1
n>2
Mn(R)Mn(C)n
InOnMn(R)Mn(C)
A= (ai,j )∈ Mn(R)Mn(C) det(A)ATr(A)A
Tr(A) = Pn
i=1 ai,i
A∈ Mn(R)Mn(C)A χA(x) = det(xIn−A)
2
A2A∈ M2(R)
χA(x) = x2−Tr(A)x+ det(A)
AM2(C)
Tr(A)2−4 det(A)6= 0 ∃λ0∈CA=λ0I2
AM2(R)
Tr(A)2−4 det(A)>0∃λ0∈RA=λ0I2
(uk)k∈N(vk)k∈N
u0= 1
v0= 2 ∀k∈N,uk+1 = 4uk−2vk vk+1 =uk+vk
k∈NXk=uk
vk
AM2(R)k Xk+1 =AXk
kNXkA, X0k
A P M2(R)
P−1AP =2 0
0 3 =D
kNAkk
ukvkk
3
(a, b, c)∈R3M(a, b, c)M3(R)
M(a, b, c) = 
a+c b c
b a + 2c b
c b a +c
F F ={M(a, b, c) ; (a, b, c)∈R3}
I=M(1,0,0) J=M(0,1,0) K=M(0,0,1)

>4
M(a, b, c)a, b, c I, J K
FM2(R)
M(a, b, c)∈F M(a, b, c)
J K
J
K
J K
Vect 
1
0
1
,
0
1
0
= Vect 
1
√2
1
,
1
−√2
1
PM3(R)P−1JP =D1P−1KP =D2
D1D2
M(a, b, c)F D
>4
U
U=
0100
0010
0001
1000
U, U2, U3U4
U
UC
PM4(C)
U=P
1 0 0 0
0i0 0
0 0 −1 0
0 0 0 −i
P−1
a, b, c d
V(a, b, c, d) = 
a b c d
d a b c
c d a b
b c d a
V(a, b, c, d)a, b, c, d I4, U, U2U3
V(a, b, c, d)
a, b, c d
V(a, b, c, d)
det V(a, b, c, d)= (a+b+c+d)(a+ib −c−id)(a−b+c−d)(a−ib −c+id)
n>4
n4n>4
B= (e1, . . . , en)Cn
uCnu(e2) = e1, u(e3) = e2,··· , u(en) = en−1u(e1) = en
∀k∈ {2, . . . , n}, u(ek) = ek−1u(e1) = en

TSI2
U u BCnU
ω n xωCn
xω=
n
X
k=1
ωk−1ek.
u(xω)ω xω
u u
un
AMn(C)
χA(x) = xn−an−1xn−1−an−2xn−2− ··· − a0A
A
An−an−1An−1−an−2An−2− ··· − a0In=On
TMn(C)PMn(C)
A=PTP−1
λ1, . . . , λnT
E1, . . . , EnCn
E1=
1
0
0
, . . . , En=
0
0
1
T χT(x) = (x−λ1)(x−λ2)···(x−λn)
T A
(i, j) 1 n
(T−λiIn)(T−λjIn)=(T−λjIn)(T−λiIn)
k1n−1
(T−λk+1In)Ek+1 ∈Vect{E1, . . . , Ek}
k1n Mk= (T−λ1In)(T−λ2In)···(T−λkIn)
Mk=
k
Y
j=1
(T−λjIn)
k1n MkEk= 0
k
n
Y
j=1
(T−λjIn) = On
n
Y
j=1
(A−λjIn) = On
χT=χA
AMn(R)
χA(x) = xn−an−1xn−1−an−2xn−2− ··· − a0

X0∈ Mn,1(R)
X=
a0
a1
an−1
AnX0=an−1An−1X0+an−2An−2X0+··· +a0X0
X˜ AX =B˜ A
Mn(R)B
(An−1X0, An−2X0, . . . , X0)
A n
|λ1|>|λ2|>··· >|λn|
F(yk)k∈N
y0, y1, . . . , yn−1
yk+n=an−1yk+n−1+an−2yk+n−2+··· +a0ykk>0
j1n(λk
j)k∈NF
Fdim F=n
(λk
1)k∈N,...,(λk
n)k∈N
(yk)k∈NF
n(α1, . . . , αn)k
yk=
n
X
j=1
αjλk
j
y0, y1, . . . , yn−1α1
(yk)k∈Nk+∞
yk
lim
k→+∞
yk+1
yk
=λ1