Journal of the
Korean Mathematical Society JKMS

In this paper, the Hermitian positive definite solutions of the matrix equation \({X^s} + {A^*}{X^{ - t}}A = Q\), where \(Q\) is an \(n\times n\) Hermitian positive definite matrix, \(A\) is an \(n\times n\) nonsingular complex matrix and \(s, t \in [1,\infty)\) 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.

Keywords: iterative algoritheorem, nonlinear matrix equation, positive definite solution, fixed point theorem

MSC numbers: 65F30, 15A24, 15B48, 47H10