半正定Hermite矩阵乘积迹的不等式的推广及应用

The Matrix Similarity of Holder's and Minkowski's Inequalities

本文证明了对半正定Hermite矩阵A_1,A_2,…。A_m成立(3),(4),这里sum from i=1 to m 1/a_1≥1。实现了将离散形式的Hólder不等式和Minkowski不等式推广到矩阵上。

Let A_1,A_2,…, A_m are positive semi-definite Hermitian matrixes. In the paper, we proved the following theorems: Theorem 1 Let a_1,a_2,…a_m are positive numbers, and 1/a_1+1/a_2+…1/a_m≥1,then |tr(A_1A_2…A_m)|≤[tr(A_1^(a_m) )]^(1/a_1)…[tr(A_m^(a_m)]^(1/a_m) Theorem 2 Let p≥1,then [tr(A_1+ A_2+…+A_m)~p]^(1/p)≤[tr(A_1~p)]^(1/p)+…+[tr(A_m^p)]~1/p Of course,our resultes are superior to [1],[4],and stronger than[4].