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matrice "3-nilpotenta" si adjuncta ei

 
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Dragos Fratila
Newton


Joined: 04 Oct 2007
Posts: 335
Location: Paris

PostPosted: Fri Mar 18, 2011 6:31 pm    Post subject: matrice "3-nilpotenta" si adjuncta ei Reply with quote

Fie A o matrice complexa de dimensiune n\times n cu n>3 astfel incat A^3=0.
Demonstrati ca adjuncta A^*=0.

Generalizare: daca A^{n-1}=0 atunci A^*=0

[e evidenta problema folosind numai instrumente de liceu?]
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DrAGos Calinescu
Thales


Joined: 07 Dec 2008
Posts: 142
Location: Pitesti

PostPosted: Fri Mar 18, 2011 7:05 pm    Post subject: Reply with quote

A^3=O_n\ \Longrightarrow\ \text{rang}\ A\le n-1
Daca \text{rang}\ A=n-1\ \Longrightarrow\ \text{rang}\ A^2\ge n-2\ \Longrightarrow\ \text{rang}\ A^3\ge n-3>0 (din Sylvester) contradictie.
Daca \text{rang}\ A\le n-2\ \Longrightarrow\ A*=O_n(toti minorii de ordin n-1 sunt nuli)
Pentru generalizare se procedeaza la fel.
Daca \text{rang}\ A=n-1 obtinem \text{rang}\ A^{n-1}\ge n-(n-1)=1 contradictie\Longrightarrow\ \text{rang}\ A\le n-2 de unde concluzia dorita.
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