mateforum.ro Forum Index mateforum.ro

 
 FAQFAQ   SearchSearch   MemberlistMemberlist   UsergroupsUsergroups   RegisterRegister 
 ProfileProfile   Log in to check your private messagesLog in to check your private messages   Log inLog in 

O identitate simpla cu det si tr

 
Post new topic   Reply to topic    mateforum.ro Forum Index -> Clasa a XI-a -> Algebra
View previous topic :: View next topic  
Author Message
Mateescu Constantin
Newton


Joined: 21 Apr 2009
Posts: 398
Location: Londra/Pitesti

PostPosted: Fri Sep 03, 2010 2:04 pm    Post subject: O identitate simpla cu det si tr Reply with quote

Aratati ca daca A\in\mathcal{M}_2(\mathbb{C}) atunci exista relatia : \fbox{\ \det\ \left\(A^3+A^2+A+I_2\right\)=\left\[1+\tr A+\det A\right\]\ \cdot\ \left\[\left\(1-\det A\right\)^2+\tr^2 A\right\]\ } .
Back to top
View user's profile Send private message
Marius Mainea
Gauss


Joined: 26 May 2008
Posts: 1099
Location: Gaesti (Dambovita)

PostPosted: Fri Sep 03, 2010 7:49 pm    Post subject: Reply with quote

(\lambda_1^3+\lambda_1^2+\lambda_1+1)(\lambda_2^3+\lambda_2^2+\lambda_2+1)=(1+\lambda_1+\lambda_2+\lambda_1\lambda_2)[(1-\lambda_1\lambda_2)^2+(\lambda_1+\lambda_2)^2]
Back to top
View user's profile Send private message Send e-mail
Mateescu Constantin
Newton


Joined: 21 Apr 2009
Posts: 398
Location: Londra/Pitesti

PostPosted: Fri Sep 03, 2010 11:14 pm    Post subject: Reply with quote

Fie P(X)=\det (A-XI_2)=X^2-a\cdot X+b polinomul caracteristic al matricei A , unde a=\tr A , b=\det A .

Atunci : \left\|\ \begin{array}{cccc}
P(\mbox{i}) & = & -1-\mbox{i}\cdot a+b \\\\\\\\\\  
P(-1) & = & 1+a+b \\\\\\\\\\  
P(-\mbox{i}) & = & -1+\mbox{i}\cdot a+b\end{array}\ \right|\ \bigodot\ \Longrightarrow\ P(\mbox{i})\cdot P(-1)\cdot P(-\mbox{i})=(1+a+b)\cdot\left\[(1-b)^2+a^2\right\]\ \Longleftrightarrow

\det\left[(A+I_2)\cdot (A-\mbox{i}\cdot I_2)\cdot (A+\mbox{i}\cdot I_2)\right\]=(1+a+b)\cdot\left\[(1-b)^2+a^2\right\]\ \Longleftrightarrow\ \det\ \left\(A^3+A^2+A+I_2\right)=(1+a+b)\cdot\left\[(1-b)^2+a^2\right\]

Observatie. In aceeasi maniera se poate arata ca pentru o matrice A\in\mathcal{M}_3(\mathbb{C}) are loc relatia :

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \fbox{\ \det\ (A^3+A^2+A+I_3)=-\left\(1+\tr(A)+\tr(A^{\ast})+\det(A)\right\)\ \cdot\ \left\[\left\(\tr(A)-\det(A)\right\)^2+\tr^2(A^{\ast})\right\]\ }
Back to top
View user's profile Send private message
Display posts from previous:   
Post new topic   Reply to topic    mateforum.ro Forum Index -> Clasa a XI-a -> Algebra All times are GMT + 2 Hours
Page 1 of 1

 
Jump to:  
You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot vote in polls in this forum



Powered by phpBB © 2001, 2005 phpBB Group