J. Korean Math. Soc. 54 (2017), No. 6, pp. 1667–1682 https://doi.org/10.4134/JKMS.j160429 pISSN: 0304-9914 / eISSN: 2234-3008 HERMITIAN POSITIVE DEFINITE SOLUTIONS OF THE MATRIX EQUATION X s + A * X -t A = Q Mohsen Masoudi, Mahmoud Mohseni Moghadam, and Abbas Salemi Abstract. In this paper, the Hermitian positive definite solutions of the matrix equation X s + A * X -t A = Q, where Q is an n × n Hermitian positive definite matrix, A is an n × n nonsingular complex matrix and s, t ∈ [1, ∞) are discussed. We find a matrix interval which contains all the Hermitian positive definite solutions of this equation. Also, a necessary and sufficient condition for the existence of these solutions is presented. Iterative methods for obtaining the maximal and minimal Hermitian positive definite solutions are proposed. The theoretical results are illustrated by numerical examples. 1. Introduction and preliminaries We consider Hermitian positive definite solutions of the nonlinear matrix equation X s + A * X -t A = Q, (1.1) where, A is an n × n nonsingular complex matrix, Q is an n × n Hermitian positive definite matrix and s, t ∈ [1, ∞). This form of the nonlinear matrix equation and same configuration to them, can be appeared in control theory [11, 13], ladder networks [2, 3], dynamic programming [19], quantum mechanics [17], stochastic filtering and statistics [5]. The existence of Hermitian positive definite solutions of the matrix equation (1.1), has been investigated in some special cases. The case s = t = 1 has been systematically investigated by several authors [2, 3, 10, 11]. The cases s =1,t ∈ N in [16], s =1,t ∈ (0, ∞) in [18, 20], s =1,t ≥ 1 in [9], s, t ∈ N in [6, 7, 8, 21, 22] and s> 0,t> 0 in [24] have been studied. In this paper, we consider the Hermitian positive definite solutions of the matrix equation (1.1), where s ≥ 1 and t ≥ 1. Also, we find a matrix interval Received June 18, 2016; Revised September 29, 2016; Accepted December 26, 2016. 2010 Mathematics Subject Classification. 65F30, 15A24, 15B48, 47H10. Key words and phrases. iterative algoritheorem, nonlinear matrix equation, positive def- inite solution, fixed point theorem. This research has been partially supported by the SBUK Center of Excellence in Linear Algebra and Optimization, Kerman, Iran. c 2017 Korean Mathematical Society 1667