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Matrice complexe A, B si pentru care A comuta cu AB-BA

 
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bae
Bernoulli


Joined: 02 Oct 2007
Posts: 223

PostPosted: Mon Feb 18, 2008 4:04 am    Post subject: Matrice complexe A, B si pentru care A comuta cu AB-BA Reply with quote

Fie A,B\in M_n(\mathbb{C}) si C=AB-BA. Daca AC=CA, atunci sa se arate ca pentru orice \lambda\in\mathbb{C} avem \det(\lambda C+B)=\det(B). Sa se deduca de aici ca matricea C nu este inversabila.

GM 1/1999
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Bogdan Cebere
Thales


Joined: 04 Nov 2007
Posts: 145

PostPosted: Mon Apr 14, 2008 1:38 pm    Post subject: Reply with quote

Ar fi mai bine daca C ar comuta cu B. Dar vad ca nici in Gazeta nu a fost publicata o solutie.

Last edited by Bogdan Cebere on Mon Apr 14, 2008 4:21 pm; edited 1 time in total
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opincariumihai
Thales


Joined: 09 May 2009
Posts: 142
Location: BRAD

PostPosted: Wed Jun 02, 2010 2:07 pm    Post subject: Reply with quote

Rezultatul problemei ( este semnata de N. Boboc ) functioneaza si vad ca are legatura cu o problema postata pe forum ( voi redacta in curand )...
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andy crisan
Pitagora


Joined: 28 Dec 2008
Posts: 87
Location: Pitesti/Londra

PostPosted: Wed Jun 02, 2010 7:21 pm    Post subject: Reply with quote

Se arata, mai intai, ca AB-BA este nilpotenta, concluzia reiesind din aceasta proprietate.
Sa aratam acum ca AB-BA este nilpotenta.
(AB-BA)^{m+1}=C^{m+1}=C^m(AB-BA)=AC^mB-C^mBA(caci Asi C comuta). Trecem la urma in aceasta ultima relatie si obtinem:tr(C^m)=tr(AC^mB-C^mBA)=0 \forall m\in\mathbb{N}^*. Aplicand sumele lui Newton obtinem ca polinomul lui C este polinomul nul, in concluzie C este nilpotenta.
Pentru a arata identitatea de determintanti a se vedea http://mateforum.ro/viewtopic.php?t=4825 in ultimul post.
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opincariumihai
Thales


Joined: 09 May 2009
Posts: 142
Location: BRAD

PostPosted: Wed Jun 02, 2010 7:45 pm    Post subject: Reply with quote

La postul de care vorbesti ai demonstrat ca
Daca A,X\in\mathcal{M}(\mathbb{C}) cu X nilpotenta si cu AX=XA atunci \det(A+X)=\det(A).

La problema noastra B nu comuta cu C
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andy crisan
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Joined: 28 Dec 2008
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PostPosted: Sat Jun 05, 2010 12:08 pm    Post subject: Reply with quote

Da nu am citit cu atentie. Aveti dreptate. Am sa caut si alta solutie. Desi eu tind sa cred ca B si C ar trebui sa comute caci nu vad o "jonglare" cu matricele din paranteze.
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