Journal of Applied Mathematics & Bioinformatics, vol.1, no.2, 2011, 109-129 ISSN: 1792-7625 (print), 1792-8850 (online) International Scientific Press, 2011 On the Hermitian solutions of the matrix equation X s + A * X -s A = Q Maria Adam 1 , Nicholas Assimakis 2 and Georgia Fotopoulou 3 Abstract In this paper, necessary and sufficient conditions for the existence of the Hermitian solutions of the nonlinear matrix equation X s +A * X -s A = Q are presented, when A is a nonsingular matrix and s an integer. The formulas for the computation of these solutions are presented. An al- gebraic method for the computation of the solutions is proposed; the method is based on the algebraic solution of the corresponding discrete time Riccati equation. The exact number of the Hermitian solutions is also derived. The formula for the computation of the maximal solution of the matrix equation X s - A * X -s A = Q is given as an application of the formulas derived for solving X s + A * X -s A = Q. The results are verified through simulation experiments. Mathematics Subject Classification : 15A24, 15A60, 15A18, 93B25 Keywords: matrix equations, numerical radius, eigenvalue, Riccati equation 1 Department of Computer Science and Biomedical Informatics, University of Central Greece, Lamia 35100, Greece, e-mail: madam@ucg.gr 2 Department of Electronics, Technological Educational Institute of Lamia, Lamia 35100, Greece, e-mail: assimakis@teilam.gr 3 Department of Computer Science and Biomedical Informatics, University of Central Greece, Lamia 35100, Greece, e-mail: gfwtopoulou@gmail.com Article Info: Revised : September 5, 2011. Published online : November 30, 2011