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.