The Hetmitian Positive Definite Solutions Of Nonlinear Equation X-A～*X～(-p)A=I(p＞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^-pA =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 equationX-A^*X^-pA = I, p>0by the following theorems.Theorem 1 Eq.(1) has a solution for any A ∈ C^n×n.Theorem 2 For any invertible matrix A ∈ C^n×n, there exist unitary matrices P and Q and diagonal matrices Γ > I and Σ > 0 with Γ² - Σ² = I such thatA = P^*Γ^pQΣP.In this case X = P^*Γ²P is a solution of Eq.(1).Theorem 3 If A is normal, in other words, there exists a unitary matrix P such that A = P^*ΛP where Λ = diag(λ₁, λ₂, ...... , λ_n), λ_i,i = 1,2, .... , n are the eigenvalues, then Eq.(1) has the following solutionX = P^*diag(μ₁,μ₂,..... , μ_n)Pwhere μ_i is the unique positive solution of the equationμ_i-|λ_i|²Î¼_i^-p=1for i = 1, 2, ? â–  ? , n.Theorem 4 If A is a unitary matrix and p < Po, where p0 > 0 satisfiesPo + ± — Po>then Eq.(l) has only one solution X — 81, where 5 is the unique positive solution of the following equation6 = 1 + 6^p.What's more, if the equation has a solution, we can know some properties of the solution according the following theorems.Theorem 5 There do not exist two comparable solutions to Eq.(l), that is , it is impossible that for any two solutions X? and X? (X? ï¿¡ X<2>) of Eq.(l) X<1> < X{2) or A'<2> < X<1>.Theorem 6 For the solution X of Eq.(l), we have/ + — A* A 1 and the convergence behaviors of the basic fixed point iteration solutions are investigated by the following theorems.Theorem 7 If3 > (Pk)1/{p+1) orthen1. Eq.(l) has a unique solution X and the solution satisfies [31 < X < al2. The solution can be obtained by the. following matrix sequence:Xn+l = I + A*X^pA, n = 0,1,2, ??? for any XQ e [01, al].3. The estimatesandhold where q = PJï¿¡3l < L Theorem 8where 7 is a unique positive solution in (1, +00) of the equationP7P"1(7-1)2 = ?,then 1. Eq.(l) has a unique solution satisfyingrjI