| 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 Hcrmitian positive solution is more important,about which arc we concerncd.In the sequel,a solution always means a Hcrmitian positive definite one. In practice,the cquation X+A*XqA=I arises in various arcas 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 equation by the following theorems.Theorem 1 Eq.(1)has a solution for any A∈Cn×n.Theorem 2 For any invertible matrix A∈Cn×n,Eq.(1)has a solution if and only if there exist unitary matrices P and Q and diagonal matrices 0＜F＜I and∑＞0 withΓ+∑2=I such that In this case X=P*ΓP is a solution of Eq.(1).Theorem 3 If A is normal,there exists a unitary P such thatA=P*∧P,∧= diag(λ1,λ2,…,λn),λi(i=1,2…,n)are the eigenvalues,then Eq.(1).has the following solution X=P*diag(μ1,μ2,…,μn)P,whereμiis the unique positive solution of the equationμi+λi2μiq=1 for i=1,2,…,n. Theorem 4 If A is invertible,Eq.(1)has a solution X with 0＜q＜1,thenTheorem 5 If A is a invertible and normal matrix,Eq.(1)has a solution X with 0＜q＜1 if and only ifTheorem 6 If A is singular,X is a solution of Eq.(1),thenλmax(X)=1What's more,if the equation has a solution,we can know some properties of the solution according to the following theorems.Theorem 7 There do not exist two compareable solutions to Eq.(1).that is,it is impossible that for any two solution X1and X2(X1≠X2)of Eq.(1).X1≥X2or X1≤X2.Theorem 8 If A is a unitary matrix and 0＜q≤q0,where q0 satisfies(q0)q0/(1+q0)+ 1=q0,then Eq.(1)has only one solution X=(1/δ)I.whereδis the unique positive solution of the following equationδ=1+δ1-q.Second,this paper offers approximate the solutions of the Eq.(1)with q＞1 and the convergence behaviors of the basie fixed point iteration solutions are investigated by the following theorems.Theorem 9 If ,then1.Eq.(1)has a unique solution X and the solution satisfiesαI≤X≤βI.2.The solution can be obtained by the following matrix sequence; for any X0∈[αI,βI].3.The estimates and hold where p=q‖A*A‖2βq-1＜1.Theorem 10 then1.Eq.(1)has a unique solution satisfying whereμ,εsatisfics and2.The solution can be obtained by the following matrix sequence; for any X0∈[μI,εI].3.The estimates and hold where p=(μ1-q)/(q(?))＜1,Theorem 11 For A*A＜I,Eq.(1)with 0＜q＜1 exists a unique positive definite solution,and if X is the solution of Eq.(1),it is in[αI,βI].Theorem 12 For Eq.(1)with 0＜q＜1,there exists a unique positive definite solution X and for any X0＞0,the iteration converges to X,that is,(?)Xn=X.Theorem 13 Suppose that X and(?)are the solutions of the matrix equations and respectively.LetΔA=(?)-A.If‖ΔA‖2≤(‖A‖22+ζ)1/2-‖A‖2whereζ=((2α+q-qα-1)2/4α2), then whereTheorem 14 Let(?)＞0 be an approximation to the solution X of the equation X+A*XqA=I(0＜q＜1).If the residual R(?)≡I-A*(?)qA-(?)satisfies that withθ1≡1+‖(?)-1‖2‖R(?)‖2-(1-q)‖(?)q/2A(?)-1/2‖22. then with...
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