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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 398085, 14 pages
http://dx.doi.org/10.1155/2012/398085
Research Article

On the Hermitian -Conjugate Solution of a System of Matrix Equations

1School of Mathematics and Science, Shijiazhuang University of Economics, Shijiazhuang, Hebei 050031, China
2Department of Mathematics, Shanghai University, Shanghai, Shanghai 200444, China
3Department of Mathematics, Ordnance Engineering College, Shijiazhuang, Hebei 050003, China

Received 3 October 2012; Accepted 26 November 2012

Academic Editor: Yang Zhang

Copyright © 2012 Chang-Zhou Dong et al. 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.

Abstract

Let be an by nontrivial real symmetric involution matrix, that is, . An complex matrix is termed -conjugate if , where denotes the conjugate of . We give necessary and sufficient conditions for the existence of the Hermitian -conjugate solution to the system of complex matrix equations and present an expression of the Hermitian -conjugate solution to this system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least squares Hermitian -conjugate solution with the least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally, an algorithm and numerical examples are given.