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Problema matrice simetrica

 
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Arhimede


Joined: 02 Nov 2013
Posts: 6

PostPosted: Fri Mar 07, 2014 10:12 am    Post subject: Problema matrice simetrica Reply with quote

Fie A\epsilon\:  M_{4}(\mathbb{R}) o matrice simetrica astfel incat \det(A+I_{4})=4\cdot \sqrt{\det(A^{2}+I_{4})}.
Arătaţi că: (A-I_{4})^{4}=O_{4}
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enescu
Thales


Joined: 20 May 2008
Posts: 171

PostPosted: Sun Mar 09, 2014 11:59 pm    Post subject: Reply with quote

De unde-i problema?
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Arhimede


Joined: 02 Nov 2013
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PostPosted: Mon Mar 10, 2014 5:19 pm    Post subject: Reply with quote

O fisa de la ora de excelenta Very Happy
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Laurian Filip
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Joined: 25 Nov 2007
Posts: 469
Location: Bucuresti

PostPosted: Mon Mar 10, 2014 11:20 pm    Post subject: Reply with quote

Fie \lambda_1, \lambda_2, \lambda_3, \lambda_4 valorile proprii ale lui A.

Egalitatea de mai sus devine \prod_{i=1}^4(\lambda_i +1)^2=\prod_{i=1}^4(2(\lambda_i^2+1))

Cum A este simetrica stim ca y_i\in\mathbb{R} si deci (\lambda_i - 1)^2 \geq 0 \Longleftrightarrow 2(\lambda_i^2+1)\geq \(\lambda_i+1)^2.
Acest lucru ne indica ca egalitatea de mai sus poate avea loc doar daca
\lambda_1 = \lambda_2 = \lambda_3 = \lambda_4= 1, de unde concluzia este evidenta.
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Arhimede


Joined: 02 Nov 2013
Posts: 6

PostPosted: Tue Mar 11, 2014 11:03 pm    Post subject: Reply with quote

Multumesc!
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