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  Volume 3, Issue 5, Article 78
 
A Note on the Trace Inequality for Products of Hermitian Matrix Power
    Authors: Zhong Peng Yang, Xiao Xia Feng,  
    Keywords: Hermitian matrix, Trace, Inequality, Skew Hermitian matrix.  
    Date Received: 06/06/02  
    Date Accepted: 02/07/02  
    Subject Codes:

15A42,15A57

  
    Editors: Drumi Bainov,  
 
    Abstract:

Da-wei Zhang [J.M.A.A., 237 (1999):721-725] obtained the inequality between $ %% tr(AB)^{2^k}$ and $ trA^{2^k}B^{2^k}$ for Hermitian matrices $ A$ and $ B$, where $ k$ is natural number. Here it is proved that these results hold when the power index of the product of Hermitian matrices $ A$ and $ B$ is nonnegative even number. In the meantime, it is pointed out that the relation between $ tr(AB)^{m}$ and $ trA^{m}B^{m}$ is complicated when the power index $ m$ is a nonnegative odd number, therefore the above inequality can't be generalized to all nonnegative integers. As an application, we not only improve the results of Xiaojing Yang [J.M.A.A., 250 (2000), 372-374], Xinmin Yang [J.M.A.A., 263 (2001):327-333] and Fozi M. Dannan [J.Ineq. Pure and Appl. Math., 2(3) Art.34 (2001)], moreover give the complete resolution for the question of the trace inequality about the powers of Hermitian and skew Hermitian matrices that is proposed by Zhengming Jiao.

         
       
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