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Journal of Applied Mathematics
Volume 2013, Article ID 593549, 11 pages
Research Article

Solving Optimization Problems on Hermitian Matrix Functions with Applications

Department of Computer Science and Information, Guizhou University, Guiyang 550025, China

Received 17 October 2012; Accepted 20 March 2013

Academic Editor: K. Sivakumar

Copyright © 2013 Xiang Zhang and Shu-Wen Xiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the extremal inertias and ranks of the matrix expressions , where , and are known matrices and and are the solutions to the matrix equations , , and , respectively. As applications, we present necessary and sufficient condition for the previous matrix function to be positive (negative), non-negative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equations , , , and , and give an expression of the general solution to the above-mentioned system when it is solvable.