Centrale Maths 1 MP 2017

Thme de l'preuve Sur la partie symtrique d'une matrice
Principaux outils utiliss thorme spectral, calcul matriciel par blocs, systmes diffrentiels, exponentielle de matrice
Mots clefs partie symtrique, matrice dfinie positive, matrice positivement stable, matrice singulire

Corrig

 : 👈 gratuite pour tous les corrigs si tu cres un compte
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

nonc complet

(tlcharger le PDF)

Rapport du jury

(tlcharger le PDF)

nonc obtenu par reconnaissance optique des caractres


JS
9 ?2m`2b

*H+mHi`B+2b miQ`Bb2b

kyRd

Ji?KiB[m2b R
am` H T`iB2 bvKi`B[m2 /mM2 Ki`B+2

LQiiBQMb
aB  2i  bQMi /2b 2MiB2`b Mim`2Hb MQM MmHb- QM MQi2   H2bT+2 p2+iQ`B2H /2b 
Ki`B+2b `2HH2b   HB;M2b 2i
 +QHQMM2b 2i   H2bT+2 p2+iQ`B2H /2b Ki`B+2b +``2b   X PM /}MBi /2 7QM MHQ;m2 
  2i
  X
G i`MbTQb2 /mM2 Ki`B+2  /2   2bi MQi2  X PM `TT2HH2 [mmM2 Ki`B+2  /2   2bi 
/Bi2
bvKi`B[m2 bB    2i [m2HH2 2bi /Bi2 MiBbvKi`B[m2 bB   X
G2 bQmb@2bT+2 p2+iQ`B2H /2   +QMbiBim /2b Ki`B+2b bvKi`B[m2b 2bi MQi   X G2 
bQmb@2bT+2 p2+iQ`B2H
/2   +QMbiBim /2b Ki`B+2b MiBbvKi`B[m2b 2bi MQi   X
G2 ;`QmT2 /2b Ki`B+2b Q`i?Q;QMH2b   HB;M2b 2i  +QHQMM2b 2bi MQi 0  X
PM MQi2  H Ki`B+2 B/2MiBi /Mb   X
SQm` iQmi2 Ki`B+2 +``2     - QM MQi2      2i       X BMbB-  2bi mM2 Ki`B+2
bvKi`B[m2-  2bi mM2 Ki`B+2 MiBbvKi`B[m2 2i     X PM /Bi [m2  2bi H T`iB2 
bvKi`B[m2 /2  2i
[m2  2bi b T`iB2 MiBbvKi`B[m2X
SQm`     - QM MQi2 TQ  H2 bT2+i`2 `2H /2 - +2bi@@/B`2 H2Mb2K#H2 /2b pH2m`b 
T`QT`2b `2HH2b /2 X
lM2 Ki`B+2 bvKi`B[m2 `2HH2 2bi /Bi2 TQbBiBp2 bB b2b pH2m`b T`QT`2b bQMi 
TQbBiBp2b 2i 2HH2 2bi /Bi2 /}MB2 TQbBiBp2
bB b2b pH2m`b T`QT`2b bQMi bi`B+i2K2Mi TQbBiBp2bX
PM MQi2   H2Mb2K#H2 /2b Ki`B+2b bvKi`B[m2b TQbBiBp2b /2   2i   H2Mb2K#H2 /2b 
Ki`B+2b
bvKi`B[m2b /}MB2b TQbBiBp2b /2   X
P#D2+iB7
GQ#D2+iB7 /m T`Q#HK2 2bi /im/B2` +2`iBM2b T`QT`Bib /2b Ki`B+2b `2HH2b 
+``2b /QMi H T`iB2 bvKi`B[m2
2bi /}MB2 TQbBiBp2X
G T`2KB`2 T`iB2 TTQ`i2 [m2H[m2b `bmHiib T`HBKBMB`2bX
G /2mtBK2 T`iB2- Q QM im/B2 H2b Ki`B+2b @bBM;mHB`2b- 2i H i`QBbBK2 T`iB2- [mB 
i`Bi2 /2b Ki`B+2b
TQbBiBp2K2Mi bi#H2b- bQMi H`;2K2Mi BM/T2M/Mi2bX

A _bmHiib T`HBKBMB`2b
AX 
.BbiM+2 /2   
PM KmMBi   /m T`Q/mBi b+HB`2 +MQMB[m2 /QMM T`    US    Q US /bB;M2 H i`+2X PM 
MQi2
 H MQ`K2 2m+HB/B2MM2 bbQ+B2X
AXXRV JQMi`2` [m2   2i   bQMi /2mt bQmb@2bT+2b p2+iQ`B2Hb bmTTHK2MiB`2b 
Q`i?Q;QMmt /Mb
  2i T`+Bb2` H2m`b /BK2MbBQMbX
AXXkV aQBi     X JQMi`2` [m2 TQm` iQmi2 Ki`B+2     -         X S`+Bb2`  [m2HH2
+QM/BiBQM bm`     - +2ii2 BM;HBi 2bi mM2 ;HBiX
AX" 
oH2m`b T`QT`2b /2 
PM +QMbB/`2     X
AX"XRV aB     2i      - H Ki`B+2     TT`iB2Mi    2i QM +QMpB2Mi /2 HB/2MiB}2`
m MQK#`2 `2H ;H  bQM mMB[m2 +Q2{+B2MiX
p2+ +2ii2 +QMp2MiBQM- KQMi`2` [m2     bB 2i b2mH2K2Mi bB     -       2i [m2
    bB 2i b2mH2K2Mi bB      \^-      X
AX"XkV SQm` iQmi2 pH2m` T`QT`2 `2HH2  /2 - KQMi`2` [m2 NJO TQ     NBY TQ  X
1M //mB`2 [m2 bB     HQ`b  2bi BMp2`bB#H2X
AX"XjV

PM bmTTQb2 [m2   

X

V JQMi`2` [mBH 2tBbi2 mM2 mMB[m2 Ki`B+2  /2 

 i2HH2 [m2    X

#V JQMi`2` [mBH 2tBbi2 mM2 Ki`B+2  /2   i2HH2 [m2 EFU   EFU  EFU 
+V 1M //mB`2 [m2 EFU   EFU  X

kyRd@yj@jy R9,kR,Rk

S;2 Rf9

X

AX"X9V

PM bmTTQb2  BMp2`bB#H2 2i- +QM7Q`KK2Mi mt MQiiBQMb /m T`Q#HK2- 

i`B[m2 /2 HBMp2`b2 /2 X JQMi`2` [m2 EFU   EFU 
PM TQm`` +QMbB/`2`     X

AX* 
AX*XRV

 EFU  X

/bB;M2 H T`iB2 bvK@

S`iB2 bvKi`B[m2 /2b Ki`B+2b Q`i?Q;QMH2b

aQBi   0  X JQMi`2` [m2 H2b pH2m`b T`QT`2b /2  bQMi /Mb < >X

AX*XkV .QMM2` mM 2t2KTH2 /2 Ki`B+2 bvKi`B[m2  /Mb   i2HH2 [m2 TQ   < > 2i TQm` 
H[m2HH2 BH
M2tBbi2 Tb /2 Ki`B+2   0  p`B}Mi   X
AX*XjV aQBi     X

V PM bmTTQb2 [m2 TQ   < > 2i [m2 TQm` iQmi2 pH2m` T`QT`2  /2  /Mb > <- H2bT+2 T`QT`2 /2 bbQ+B   2bi /2 /BK2MbBQM TB`2X JQMi`2` [mBH 2tBbi2   0  i2HH2 [m2   X #V _+BT`Q[m2K2Mi- KQMi`2` [m2 bBH 2tBbi2   0  i2HH2 [m2   - HQ`b TQ   < > 2i 
TQm` iQmi2
pH2m` T`QT`2  /2  /Mb > <- H2bT+2 T`QT`2 /2  bbQ+B   2bi /2 /BK2MbBQM TB`2X AA Ji`B+2b @bBM;mHB`2b .Mb H bmBi2 /2 +2ii2 T`iB2- QM MQi2     [mQM KmMBi /m T`Q/mBi b+HB`2  ]  /}MB T` ] Q- +QKK2 m AX"XR- QM B/2MiB}2 H Ki`B+2     bQM mMB[m2 +Q2{+B2MiX aB     - QM MQi2   H2Mb2K#H2 /2b Ki`B+2b /2   /2 `M; ;H  X lM2 Ki`B+2 /2   2bi /Bi2 bBM;mHB`2 bB 2HH2 M2bi Tb BMp2`bB#H2X aB  2bi mM bQmb@2bT+2 p2+iQ`B2H MQM `/mBi  \^ /2  2i bB     - QM /Bi [m2  2bi @bBM;mHB`2 bBH 2tBbi2    MQM MmH i2H [m2   -     X .Mb H2 +b +QMi`B`2- QM /Bi [m2  2bi @`;mHB`2X AAX *b Q  2bi mM ?vT2`THM AAXXRV JQMi`2` [mmM2 Ki`B+2 /2   2bi bBM;mHB`2 bB 2i b2mH2K2Mi bB 2HH2 2bi @bBM;mHB`2X .Mb +2ii2 bQmb@T`iB2 AAX- QM bmTTQb2 /bQ`KBb   X aQBi    mM ?vT2`THM /2  2i bQBi    mM p2+i2m` mMBiB`2 MQ`KH  X AAXXkV JQMi`2` [m2  2bi @bBM;mHB`2 bB 2i b2mH2K2Mi bBH 2tBbi2 mM p2+i2m` MQM MmH  /2  2i mM `2H  i2Hb [m2   X AAXXjV 1M //mB`2 [m2  2bi @bBM;mHB`2 bB 2i b2mH2K2Mi bB H Ki`B+2 2bi bBM;mHB`2X .Mb H2b [m2biBQMb bmBpMi2b-  2bi mM2 Ki`B+2 BMp2`bB#H2 /2   X p2+     -     -     AAXX9V JQMi`2` [mBH 2tBbi2 mM2 Ki`B+2 i2HH2 [m2 , X AAXX8V 1M //mB`2 [m2 EFU       EFU  X AAXXeV JQMi`2` [m2 bB EFU     - HQ`b BH 2tBbi2 mM ?vT2`THM  /2  i2H [m2  2bi @bBM;mHB`2X AAXXdV 1M //mB`2 [m2 bB EFU   - HQ`b BH 2tBbi2 mM ?vT2`THM  /2  i2H [m2  2bi @bBM;mHB`2X AAXX3V PM bmTTQb2 [m2 X JQMi`2` [m2  2bi @`;mHB`2 TQm` iQmi ?vT2`THM  /2  X AAX"  1t2KTH2 PM i`Bi2` H2t2KTH2 AAX"XRV JQMi`2` [m2   2bi BMp2`bB#H2 TQm` iQmi `2H X AAX"XkV *H+mH2` 2i KQMi`2` [m2 2bi bBM;mHB`2 TQm`   - AAX"XjV .i2`KBM2` mM ?vT2`THM  i2H [m2   bQBi @bBM;mHB`2X kyRd@yj@jy R9,kR,Rk S;2 kf9 - X AAX*  *b Q  2bi /2 /BK2MbBQM PM bmTTQb2 B+B   X aQBi  mM bQmb@2bT+2 p2+iQ`B2H /2  /2 /BK2MbBQM   X PM +QMbB/`2   mM2 #b2 /2   2i QM TQb2 AAX*XRV JQMi`2` [m2  2bi @bBM;mHB`2 bB 2i b2mH2K2Mi bBH 2tBbi2 mM HK2Mi MQM MmH  /2  2i /2mt `2Hb   i2Hb [m2       X AAX*XkV 1M //mB`2 [m2  2bi @bBM;mHB`2 bB 2i b2mH2K2Mi bB H Ki`B+2 2bi bBM;mHB`2X .Mb H2b [m2biBQMb bmBpMi2b-  2bi mM2 Ki`B+2 BMp2`bB#H2 /2   X AAX*XjV JQMi`2` [mBH 2tBbi2 mM2 Ki`B+2 p2+     -     - 2i     i2HH2 [m2 AAX*X9V 1M //mB`2 [m2 EFU   EFU     EFU  X AAX*X8V JQMi`2` [mBH 2tBbi2     i2HH2 [m2 EFU       bB 2i b2mH2K2Mi bBH 2tBbi2 i2HH2 [m2 EFU       X AAX*XeV JQMi`2` [m2 bB        HQ`b EFU AAX*XdV 1M //mB`2 [m2 bB AAX*X3V 1M +QM+Hm`2 [m2 bB /2  X - HQ`b EFU      X - HQ`b  2bi @`;mHB`2 TQm` iQmi bQmb@2bT+2 p2+iQ`B2H  /2 /BK2MbBQM AAX.  1t2KTH2 PM `2T`2M/ H2t2KTH2 /2 H bQmb@T`iB2 AAX" p2+   X AAX.XRV *QKK2Mi +?QBbB`        /2 7QM [m2 EFU        \ AAX.XkV .i2`KBM2` mM bQmb@2bT+2 p2+iQ`B2H  /2  i2H [m2 EJN    2i i2H [m2 bQBi @bBM;mHB`2X AAX1  *b ;M`H aQBi  mM bQmb@2bT+2 p2+iQ`B2H /2  /2 /BK2MbBQM   - Q       X AAX1XRV JQMi`2` [m2  2bi @bBM;mHB`2 bB EFU        TQm` mM2 Ki`B+2      [m2 HQM /}MB`X PM bmTTQb2 /bQ`KBb [m2     X AAX1XkV JQMi`2` [m2 bB     2bi MQM MmH HQ`b          X AAX1XjV 1M //mB`2 [m2 H2b pH2m`b T`QT`2b `2HH2b /2      bQMi bi`B+i2K2Mi TQbBiBp2bX AAX1X9V 1M //mB`2 [m2 EFU       X AAX1X8V 1M //mB`2 [m2  2bi @`;mHB`2 TQm` iQmi bQmb@2bT+2 p2+iQ`B2H   \^ /2  X AAA Ji`B+2b TQbBiBp2K2Mi bi#H2b PM /Bi [mmM2 Ki`B+2  /2   2bi TQbBiBp2K2Mi bi#H2 bB iQmi2b b2b pH2m`b T`QT`2b +QKTH2t2b QMi mM2 T`iB2 `2HH2 bi`B+i2K2Mi TQbBiBp2X AAAX  1t2KTH2b AAAXXRV aQBi     X JQMi`2` [m2  2bi TQbBiBp2K2Mi bi#H2 bB 2i b2mH2K2Mi bB US 2i EFU   X AAAXXkV V G bQKK2 /2 /2mt Ki`B+2b TQbBiBp2K2Mi bi#H2b /2   2bi@2HH2 M+2bbB`2K2Mi TQbBiBp2K2Mi bi#H2 \ #V aQBi -  /Mb   /2mt Ki`B+2b TQbBiBp2K2Mi bi#H2b [mB +QKKmi2MiX JQMi`2` [m2 2bi TQbBiBp2K2Mi bi#H2X kyRd@yj@jy R9,kR,Rk S;2 jf9 AAAXXjV aQBi     i2HH2 [m2  bQBi /}MB2 TQbBiBp2X V aQBi J mM2 Ki`B+2 +QHQMM2 /2   - Q  2i  TT`iB2MM2Mi    X PM TQb2     J 2i QM B/2MiB}2 H Ki`B+2      m MQK#`2 +QKTH2t2 ;H  bQM mMB[m2 +Q2{+B2MiX JQMi`2` [m2- bB   - HQ`b 3F    - Q 3F  /bB;M2 H T`iB2 `2HH2 /2   X #V JQMi`2` [m2  2bi TQbBiBp2K2Mi bi#H2X AAAXX9V .QMM2` mM 2t2KTH2 /2 Ki`B+2  TQbBiBp2K2Mi bi#H2 i2HH2 [m2  M2bi Tb /}MB2 TQbBiBp2X AAAX"  .Mb +2ii2 bQmb@T`iB2 AAAX"- QM i#HBi mM `bmHii bm` H2tTQM2MiB2HH2 /2 Ki`B+2 [mB b2` miBH2 T` H bmBi2X PM `TT2HH2 [m2- TQm` iQmi2 Ki`B+2     - H2tTQM2MiB2HH2 /2  2bi /}MB2 T` FYQ G 7QM+iBQM   FYQ  2bi /}MB2 2i /2 +Hbb2  bm`  2i b 7QM+iBQM /`Bp2 2bi /QMM2 T` FYQ   FYQ /2 THmb- FYQ  FYQ    TQm` iQmi `2H X AAAX"XRV aQBi    i2H [m2 3F   X aQBi  mM2 7QM+iBQM  pH2m`b +QKTH2t2b /2 +Hbb2 bm`  X PM bmTTQb2 [m2 H 7QM+iBQM     2bi #Q`M2 bm`  X JQMi`2` [m2  2bi #Q`M2 bm`  X PM TQm`` +QMbB/`2` H[miBQM /Bz`2MiB2HH2    X AAAX"XkV aQBi     mM2 Ki`B+2 i`BM;mHB`2 bmT`B2m`2  +Q2{+B2Mib +QKTH2t2bX PM bmTTQb2 [m2 H2b +Q2{+B2Mib /B;QMmt /2  bQMi /2b MQK#`2b +QKTH2t2b /2 T`iB2 `2HH2 bi`B+i2K2Mi TQbBiBp2X aQBi  w  /2b 7QM+iBQMb  pH2m`b +QKTH2t2b- /}MB2b 2i /2 +Hbb2  bm`  2i bQBi- TQm` iQmi PM bmTTQb2 [m2- TQm` iQmi    - X JQMi`2` [m2 H2b 7QM+iBQMb  - Q     - bQMi #Q`M2b bm`  X AAAX"XjV aQBi     mM2 Ki`B+2 TQbBiBp2K2Mi bi#H2 /2 pH2m`b T`QT`2b +QKTH2t2b  w 2i bQBi  mM `2H i2H [m2     NJO 3F  X JQMi`2` [m2 H 7QM+iBQM   F FYQ  2bi #Q`M2 bm`  X PM TQm`` TTHB[m2` H [m2biBQM AAAX"Xk  mM2 Ki`B+2 i`BM;mHB`2  b2K#H#H2     X AAAX*  lM2 +`+i`BbiBQM /2b Ki`B+2b TQbBiBp2K2Mi bi#H2b aQBi     mM2 Ki`B+2 TQbBiBp2K2Mi bi#H2X PM +QMbB/`2 H2M/QKQ`T?BbK2  /2   i2H [m2 AAAX*XRV JQMi`2` [m2  2bi TQbBiBp2K2Mi bi#H2- +2bi@@/B`2 [m2 b Ki`B+2 /Mb mM2 #b2 [m2H+QM[m2 /2 2bi TQbBiBp2K2Mi bi#H2X AAAX*XkV V JQMi`2` [mBH 2tBbi2 mM2 mMB[m2 Ki`B+2     i2HH2 [m2 #V JQMi`2` [m2  2bi bvKi`B[m2 2i [m2 EFU   X AAAX*XjV SQm` iQmi `2H - QM TQb2    FYQ V JQMi`2` [m2- TQm` iQmi `2H - #V JQMi`2` [m2- TQm` iQmi `2H - +V ZmQ#iB2Mi@QM 2M 7BbMi i2M/`2  p2`b [m2biBQM AAAX*Xk 2bi /}MB2 TQbBiBp2X FYQ  2i       EX 2i [m2- bB   - X X /Mb H;HBi T`+/2Mi2 \ 1M //mB`2 [m2 H Ki`B+2  /2 H r r r 6AL r r r kyRd@yj@jy R9,kR,Rk X S;2 9f9