TMMATE 2009, Problema 1

Post by **Laurian Filip** » Thu Apr 23, 2009 7:26 pm

Aratati ca pentru orice \( A\in M_2 (\mathbb{R}) \) are loc inegalitatea \( \tr(A^4) \leq (\tr(A^2))^2 \).

Andrei Eckstein

Post by **Beniamin Bogosel** » Thu Apr 23, 2009 9:09 pm

\( A^4=\tr{(A^2)}A^2-\det(A^2)I\Rightarrow \tr(A^4)=(\tr(A^2))^2-2(\det(A))^2\leq (\tr(A^2))^2 \)

Marius Mainea · Post by **Marius Mainea** » Thu Apr 23, 2009 9:13 pm

Daca \( \lambda_1 \) si \( \lambda_2 \) sunt valorile proprii ale matricei A (cazul real sau complex), atunci

\( \lambda_1^4+\lambda_2^4\le (\lambda_1^2+\lambda_2^2)^2 \)

Theodor Munteanu · Post by **Theodor Munteanu** » Thu Apr 23, 2009 11:25 pm

De ce si cazul pentru valorile proprii complexe?

Post by **Laurian Filip** » Fri Apr 24, 2009 5:54 am

Pentru ca trebuie sa te folosesti de faptul ca sunt conjugate, altfel relatia nu ar fi adevarata. Ia de exemplu \( \lambda_1 =1+i \) si \( \lambda_2=1+2i \).