On positive definite solution of a nonlinear matrix equation

In this paper, an efficient iterative method is presented to solve a new nonlinear matrix equation * r X A X A I   with real matrices and r 1  . Some properties of the positive definite solutions for the nonlinear matrix equation are derived. Moreover, necessary and sufficient conditions for the existence of the positive definite solutions are derived. The error estimation of the iterative method is also given. Finally, two numerical examples are given to demonstrate the efficiency of the algorithm. Key-Words: : Nonlinear matrix equation, positive definite solution, existence, iterative method, fixed-point theorem.