The Hermitian Positive Definite Solutions Of Nonlinear Equation X-A～*X～qA=I(q＞0)

The problem of solving the nonlinear matrix equation, is mainly to determine the solution of the equation by the information of parameters of the equation. From the application point of view, the Hermitian positive solution is more important, about which are we concerned. In the sequel, a solution always means a Hermitian positive definite one. In practice, the equation X — A~*X~qA = I arises in various areas of applications, including control theory, dynamic programming, statistic, the finite difference approximation to an elliptic partial differential equation, and so on. The study of this kind of problem has three basic problems: (1)the theoretic issue on solvability, ie., the necessary and sufficient conditions for the existence of the solution; (2)the numerical solution, ie., the effective numerical ways; (3)the analysis of the perturbation.First, in this paper, we discuss the existence of the solution of the equationby the following theorems.Theorem 1 For any invertible matrix A ,Eq.(1) has a solution if and only if there exist unitary matrices P and Q and diagonal matrices I and > 0 with = I such thatIn this case X = P is a solution of Eq.(1).Theorem 2 If A is invertible, Eq.(1) has a solution X with q > 1,then...