| Nonlinear matrix equation is one of the important study fields of the matrix theory and numerical algebra. The kind equation is widely used in many fields such as ladder network, control theory, dynamic programming, random selection and statistics and so on. We pay most attention to the Hermitian positive definite solution of the equation in all the applications, so we only discuss the Hermitian positive definite solution of the equation. In this paper, the Hermitian positive definite solution of the nonlinear matrix equation X-A*X-qA=I is discussed. Where A is an n×n complex matrix,I is an n x n identity matrix, A*denotes the conjugate transpose of the matrix A, q>1. In this paper, some properties of solutions and some weaker sufficient conditions for the existence of a unique Hermitian solution of the equation are given, And the numerical solution of seeking Hermitian positive definite solution by continuation method of homotopy is derived. Based on the theory of condition developed by Rice, the condition number of the Hermitian positive definite solution X to the matrix equation is defined, and an explicit expression is derived. Finally the numerical examples are given to verify the correctness of the conclusions in this paper. The main results in this paper are listed as follow:Lemma1Eq.(1) has a solution X for any A∈Cnxn, and X∈(I,I+A*A]. Lemma2For the real numbers x,y,(1)if0<x,y<(?),then(2)if x,y>(?),thenTheorem1For any matrix A∈Cn×n,there exist unitary matrices P and Q and diagonal matrices Γ>I and∑≥O with Γ一∑2=I,such that A=P*Γq/2Q∑P. In this case,X=P*ΓP is a solution of Eq.(1).Lemma3For any matrix A∈Cn×n satisfy the inequality then X-A*X-q A=I has a solution in[I(q-1/q)qA*A,q/q-1I].Theorem2For any matrix A∈Cn×n,if X,Y<q/q-1,are solutions of Eq.(1),thenX=Y.Theorem3Let α is the maximal positive solution of the following equation β is the minimal positive solution of the following equation and we have β≤α,(α-1)βq.:λmax(A*A),(β-1)αq=λmin(A*A). If then Eq.(1)has a unique solutionx X with βI≤X≤αI<q/q-1I.Theorem4λmax(A*A)<βq/q-1holds if and only ifTheorem5If X,Y are the solution of Eq.(1),and X,Y>q/q-1I,then X=Y. Theorem6If matrix A satisfies then Eq.(1)has the unique hermitian positive solution in [q/q-1I,[(q-1)AA*]1/q].Theorem7If A satisfiesthen for any t∈[0,1]Eq.(1)has the unique hermitian positive solution X with X<q/q-1I.Lemma4Let matrix function F(X)=I+A*X-q.A,if X=F(X)has the unique positive solution X*in[I,q-q-1I],then there exist the neighborhood‖X-X*‖<δ of X.,such that (?)X0∈N(X*,δ),Xn=F(Xn-1)iterative convergence to X*.Theorem8If A satisfies thus there exists component point0=t0<t1<…<tn=1in t∈[0,1]and intteger sequence jk,k=1,…,n-1,such that is defined,where G(X(t),t)=I+tA*X-qA,moreover,when jâ†'∞,the matrix sequence Xn,j+1=G(Xn,j,1),J=0,1,… converge to X(1).Lemma5When A=aU,where a∈C,U is a unitary matrix,then Eq.(1) has the unique X=ωI,where ω is the unique positive solution of the follwing equationTheorem9We assume A=MAN,M,N are unitary matrices,A is diago-nal matrix,η is the largest singular value of A.Let B(t)=M[t∧2/q+(1-t)η2/qI]q/2N,t∈[0,1]. If A satisfies then for (?)t∈[0,1],we have the equation X-B*(t)X-qB(t)=I exists an unique Hermite positive solution X,and X>q/q-1I.Lemma6Let matrix function F1(X)=[A(X-I)-1A*]1/q,if X=F1(X)has the unique positive solution X*in[q/q-1I,∞],then there exist the neighborhood‖X-X*‖<δ of X*,such that (?)X0∈N(X*,δ),Xn=F1(Xn-1)iterative convergence to X*. Theorem10If A satisfies thus there exists component point0=to<t1<…<tn=1and integer sequence jk,k=1,…,n-1,such that is defined,where G(X(t),t)=(?),ζ=1+η2/ζq,when jâ†'∞, the matrix sequence Xn,j+1=G1(Xn,j,1),J=0,1,· converge to X(1). |
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