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Inegalitate intre determinati de matrice de ordin 2
Posted: Wed Nov 07, 2007 1:13 am
by Cezar Lupu
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 \( \det(2Z-X-Y)\geq 3\det(X-Y) \).
Posted: Sun May 31, 2009 11:24 am
by opincariumihai
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.