Uniqueness And Numerical Method For Hermitian Positive Definite Solutions Of Non-linear Matrix Equation X-A^*X^-2A=I

Nonlinear matrix equation is one of the important substance of the nu-merical algebra. This kind of equation arises in various areas of application, including dynamic programming, control theory, ladder networks, stochastic filtering statistics, and so on. The Hermitian positive definite solutions of nonlinear matrix equation X-A*X-2A=I is studied, where A is an n x n nonsingular complex matrix, I is an n×n identity matrix, A*denotes the conjugate transpose of the matrix A. In this paper, Some properties of solutions and some new sufficient conditions for the existence of a unique Hermitian solution of the mauix equation are given, and the numerical solution of seeking Hermitian positive definite solution by continuation method of homotopy are obtained. The main results in this paper are listed here:Theorem1when A=aN, where a is a complex number,N is a unitary matrix. then Eq.(1.15) has only one solution X=δI, whereδis the unique positive solution of the following equationTheorem2If X, Y<21are solutions of Eq.(1.15), then X=Y Theorem3Let α is the maximal positive solution of the following equation β is the minimal positive solution of the following equationand we have β≤α,(0.7)(α-l)β2=λmax(A*A),(β-1)α2=λmin(A*A).(0.8)Now we assumeThen Eq.(1.15) has a unique solution X with βI≤X≤αI<2ITheorem5If Eq.(1.15) has a positive definite solution X∈[2I,∞], thusX is unique.Theorem7If A satisfies then we have X-tA*X-2A=I exists a unique hermitian positive solution X <2I, where t∈[0,1].Theorem8If A satisfies (0.9), thus there exists component point0=t0<t1<…<tN=1in t∈[0,1] and integer sequence jk, k=1,…, N—1such that is defined, where G(X(t),t)=I+tA*X-2A, moreover, when jâ†'∞, XN,j+1=G(XN,,j,1) j=0,1,…Converge to X(1).Theorem9If A satisfies then we know X-B*(t)X-2B(t)=I exists a unique Hermite positive solu-tion X and X>21, where0≤t≤1, A=MAN, ηi is singular value of A, B(t)=M(tΛ+(1-t)maxηiI)N.Theorem10If A satisfies (0.10), thus there exists component point0=t0<t1<…<tN=I and integer sequence jk, k=1,…, N-1such that xn,j+1=G(xN,j1), j=0,1, converge to X(1).