A Note on the Trace Inequality for Products of Hermitian Matrix Power

Volume 3, Issue 5, 2002

Article 78

A NOTE ON THE TRACE INEQUALITY FOR PRODUCTS OF HERMITIAN MATRIX POWER

DEPARTMENT OF MATHEMATICS
PUTIAN COLLEGE
PUTIAN, FUJIAN 351100
PEOPLE'S REPUBLIC OF CHINA
E-Mail: yangzhongpeng@sina.com

DEPARTMENT OF MATHEMATICS
XI'AN JIAOTONG UNIVERSITY
XI'AN, SHANXI, 710049
PEOPLE'S REPUBLIC OF CHINA
E-Mail: fengxiaoxia0110@163.com

Received 6 June, 2002; Accepted 2 July, 2002.
Communicated by: D. 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 and , where is natural number. Here it is proved that these results hold when the power index of the product of Hermitian matrices and 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 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.

Key words:
Hermitian matrix, Trace, Inequality, Skew Hermitian matrix.

2000 Mathematics Subject Classification:
15A42, 15A57.

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