In this paper, the Hermitian positive definite solutions of the matrix equation \(X^s +A^* X^{ - t}A = Q\), where A is an \(n \times n\) nonsingular complex matrix, Q is an \(n \times n\) Hermitian positive definite matrix and \(s, t> 0\), are discussed. Some conditions for the existence of Hermitian positive definite solutions of this equation are derived. In addition, two iterative methods to obtaining the maximum or minimum Hermitian positive definite solutions of this equation are proposed. In addition, a necessary and sufficient condition for the existence of these solutions is presented. Theoretical results are illustrated by some numerical examples.

在本文中，矩阵方程\（X ^ s + A ^ * X ^ {-t} A = Q \）的Hermitian正定解，其中A是一个\（n × n \）非奇异复矩阵，Q是一个\（n × n）埃尔米特正定矩阵，并讨论\（s，t> 0 \）。推导了该方程的埃尔米特正定解存在的一些条件。此外，提出了两种迭代方法来获得该方程的最大或最小厄米正定解。此外，为这些解决方案的存在提供了充要条件。一些数值例子说明了理论结果。