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Felicia_b
Pe: 2013-07-24T14:05:51+03:00 In: In:

Inegalitate cu determinant

Fie

    \[{A_1},{A_2} \in {\mathbb{M_2}}\left( \mathbb{C} \right)\]

astfel incat

    \[\det \left( {{A_i}} \right) \in {\mathbb{R_ +} }\;,i \in \left\{ {1,2} \right\}\]

.Sa se arate ca oricare ar fi

    \[n \in {\mathbb{N^*}}\]

,are loc inegalitatea:

    \[\det \left( {A_1^n + A_2^n} \right) + \det \left( {A_1^n - A_2^n} \right) \ge {2^{2 - n}}{\left( {\det \left( {{A_1}} \right) + \det \left( {{A_2}} \right)} \right)^n}\]

.

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1 raspuns

  1. TheodorMunteanu

    FOlosim urmatoarea identitate cunoscuta:\det(X+Y)+\det(X-Y)=2(\det X+\det Y)valabila pentru matrici de ordin 2(demonstratia e formata din calcule ordinare)
    \det(A_1^n+A_2^n)+\det(A_1^n-A_2^n)=2(\det A_1^n+\det A_2^n)
    mai folosim inegalitatea \frac{x^n+y^n}{2}\geq \left(\frac{x+y}{2}\right)^ncare se realizeaza din convexitatea lui f(x)=x^n,n\geq 2

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