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Concours National Commun
Epreuve 2 : MP
Session 2003
K K =R C n
p∈N∗Mn,p(K)
Kn p p =nMn,p(K)Mn(K)
nKMn(K)
In
AMn,p(K)tA A A ∈ Mn(K)SpK(A)
AKT r(A)rg(A)
A B Mn(K) ΦA,B
ΦA,B :Mn(K)−→ Mn(K)
X7→ AX +XB
B=AΦA,B ΦA
ere
C∈ Mn(K)SpK(C) = SpK(tC)
ΦA,B
V∈ Mn,1(K)W∈ Mn,1(K)AtB
a b
VtW v1, . . . , vnV
w1, . . . , wnW V tW
VtWΦA,B
λΦA,B Y∈ Mn(K)
k AkY=Y(λIn−B)k
PKP(A)Y=Y P (λIn−B)
PAAK
PA= (−1)nY
µ∈SpK(A)
(X−µ)βµ.

Y PA(λIn−B)=0 PA(λIn−B)
a∈SpK(A) (λ−a)In−B
PAKSpK(ΦA,B ) = SpK(A) + SpK(B)
(Y1, . . . , Yp)Mn,1(K)Z1, . . . , Zp
Mn,1(K)
p
X
i=1
Yt
iZi= 0 Z1, . . . , Zp
A B Mn(K)
(U1, . . . , Un) (W1, . . . , Wn)AtB
(Ut
iWj)16i,j6nΦA,B
A B n
<, >: (M, N )7→ T r(tMN )Mn(R)
C D n T r(DC) = T r(CD)
ΦA,B
(Mn(R), <, >)
eme
K=RS∈ Mn(R)
Mn(R)
S
ΦS
X∈ Mn(R)XΦS(X)
A= a b
b c !
A a > 0ac −b2>0
U∈ M2,1(R)x y tUAU
a, b, c, x y a > 0ac −b2>0A
A Xλ= λ0
0 1 !
λ > 0 ΦA(Xλ)b
λ
P D
S=P DP −1.
λ1, . . . , λnD=diag(λ1, . . . , λn)

X∈ Mn(R)M= ΦS(X)
Y=P−1XP N =P−1MP ni,j N yi,j
Y
N= ΦS(Y)yi,j λk
N
M
N
UMn,1(R)u1, . . . , un
tUY U =X
16i,j6n
ni,j
λi+λj
uiuj
α > 0t7→ tα−1
]0,1]
U(s)Mn,1(R)u1sλ1−1
2, . . . , unsλn−1 2
s∈]0,1] s7→tU(s)NU(s)
]0,1]
tUNU
UtUY U > 0
X
eme
K=CΦA,B
B=−AMn(C)
A= ∆ ∆ diag(µ1, . . . , µn)µi
n= 2 ΦA,−A
ΦA,−A
AMn(C)n
AMn(C)
ΦA,−A
n
A7→ ΦA,−AMn(C)
q∈N∗q6n A = (ai,j )16i,j6n7→ det ((ai,j )16i,j6q)
Mn(C)
Mn(C)n
Mn(C)
r r 6n
A∈ Mn(C)

Or={C∈ Mn(C), rg(C)> r} Mn(C)
(Ap)pMm(C)m>2s > 0
A A s
AMn(C)
ΦA,−An
FIN
c
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