Inegalitate intre determinati de matrice de ordin 2
Moderators: Bogdan Posa, Laurian Filip, Beniamin Bogosel, Radu Titiu, Marius Dragoi
-
Cezar Lupu
- Site Admin
- Posts: 612
- Joined: Wed Sep 26, 2007 2:04 pm
- Location: Bucuresti sau Constanta
- Contact:
Inegalitate intre determinati de matrice de ordin 2
Fie \( A, B, C \) matrice de ordinul \( 2 \) cu elemente numere reale. Notam \( X=AB+BC+CA \), \( Y=BA+CB+AC \) si \( Z=A^2+B^2+C^2 \). Sa se demonstreze ca \( \det(2Z-X-Y)\geq 3\det(X-Y) \).
-
opincariumihai
- Thales
- Posts: 134
- Joined: Sat May 09, 2009 7:45 pm
- Location: BRAD
Fie \( M=A+eB+\overline{e}C \) unde \( e=-1/2+i\sqrt{3}/2 \)
Atunci \( M\overline{M}+\overline{M}M=2Z-X-Y \) si \( M\overline{M}-\overline{M}M=-\sqrt{3}i(X-Y) \) . Tinand cont ca \( \det(M\overline{M}+\overline{M}M)+\det(M\overline{M}-\overline{M}M)=2(\det(M\overline{M})+\det(\overline{M}M))\geq0 \) se obtine inegalitatea din enunt.
Atunci \( M\overline{M}+\overline{M}M=2Z-X-Y \) si \( M\overline{M}-\overline{M}M=-\sqrt{3}i(X-Y) \) . Tinand cont ca \( \det(M\overline{M}+\overline{M}M)+\det(M\overline{M}-\overline{M}M)=2(\det(M\overline{M})+\det(\overline{M}M))\geq0 \) se obtine inegalitatea din enunt.