The Hermitian Positive Definite Solutions Of Matrix Epuation X+A^*X^-qA=Q When 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,a bout which are we concerned.In the sequel,a solution always means a hermitian positive definite one.In practice,the equation X+A^*X^-qA=Q arises in various areas of applications,including control theory,ladder networks,dynamic program-ming,statistics,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.In this paper,we are concerned with the Hermitian positive definite solutions of the nonlinear matrix equationX+A^*X^-qA=Q,q＞0(1) where X is a unknown matrix,Q is the Hermitian positive definite matrix,A is an n×n complex matrix and q is a positive real number.First,in this paper,we discuss the existence of the solution of the equation(1) by the following theorems.Theorem 1 Eq.(1)has a solution if and only if there exist unitary matrices P,T,F,and diagonal matrixΓ＞PQP^*,and Hermitian matrix∑＞0,withΓ+∑²= PQP^*,such thatA=P^*Γ^q/2F∑P In this case X=P^*ΓP is a solution for Eq.(1).Theorem 2 Suppose X is a solution of Eq.(1)and the matrix A is invertible, then 1)0＜X＜Q, 2)([21]引理2.1)If A^*X^-qA≤Q for all(?)X∈[0,Q],then Eq.(1)has a solution in [0,Q]. 3)At the same time,this paper gives the domain of the solutions,when the equa-tion have solutions.Theorem 3 Suppose that,and X is a solution of Eq.(1) 1)If matrix A is invertible,then and 2)If matrix A is singular,then andTheorem 4 Suppose that A is invertible and, 1)If q≥1,then Eq.(1)has a maximal solution(?)and a minimal solution(?)in [α₁I,β₁I].(?)=lim_{n→∞}(?)_n,(?)=lim_{n→∞}(?)_n,where Moreover, For any solution X of Eq.(1),X≥(?),that is(?)is the smallest solution of all the solutions for Eq.(1). 2)If 0＜q≤1,the Eq.(1)has a maximal solution(?)and a minimal solution(?)in [β₂I,α₂I].(?)=lim_{n→∞}(?)_n,(?)=lim_{n→∞}(?)_n,where Moreover,For any solution X of Eq.(1),X≤(?),that is(?)is maximal solution for Eq.(1).Second,this paper offers two different iterative methods to approximate the solutions of the Eq.(1)with q＞1.Theorem 5 Suppose there ix a matrix A and numbersα,βfor which the conditions 1), 2), are satisfied.Then there is a solution of Eq.(1).The solution can be obtained by the following matrix sequence: whereη∈[β,α].Theorem 6 Suppose there ix a matrix A and numbersα,βfor which the conditions 1), 2), are satisfied.Then there is a solution of Eq.(1).The solution can be obtained by the following matrix sequence: whereη∈[β,α].But it is difficult to fined numbersα,βin the two theorems above.So we give the following Theorems.Theorem 7 If,then Eq.(1)has a solution X_L satis-fyingβ₂I≤X_L≤α₂I.Moreover,if,then 1)The solution X_L is unique. 2)X_L can be obtained by the iteration and the estimates hold,where. 3)There does not exist any solution X for Eq.(1)such that X＞X_L.Especially,if 0＜q≤1,X_L must be the maximal solution.Theorem 8 If A is invertible and,then Eq.1)has a solution X_l satisfyingα₁I≤X_l≤β₁I.Moreover,if q＞1 and ,then the solution X_l is unique and must be the minimal solu-tion.X_l can bi obtained by the iteration where X₀∈[α₁I,β₁I].And the estimates hold where. Moreover,three qualities of the approximate solution have been givon in this paper:Theorem 9 If A is a normal matrix and AQ=QA,then AX_n=X_nA,n= 0,1,2,…,where X_n can be obtained by the iterationTheorem 10 If A is a normal matrix and AQ=QA,then QX_n=X_nQ,n= 0,1,2,…,where X_n can be obtained by the iterationTheorem 11 If A is a normal matrix and AQ=QA,then X_n+1X_n=X_nX_n+1, where X_n can be obtained by the iterationLast,in this paper,the perturbation analysis for the unique solution of the matrix equation are studied according to the following theorems.Theorem 12 Let A,(?),Q,(?)∈C,Q and(?)be Hermitian definite.If and then X_L and(?)_L are the maximal solutions of Eq.(1)and respectively,and satisfy where,â‘ if,then;â‘¡ifTheorem 13 Let(?)approximate the maximal solution X_L,the residual If then where.