Hermitian Positive Definite Solutions And Numerical Methods Of Non-linear Matrix Equation X±A~*X~qA=1(0<q<1)

In practice, the nonlinear matrix equations arises in various area of applica-tions, including control theory, transportation theory, dynamic programming, ladder network analysis, statistical programming, stochastic filtering, engineering calcula-tion, and etc. Solving the matrix equations has become an important subject in the research of non-linear analysis and numerical algebra. The problem of solving the nonlinear matrix equation is mainly to determine the solution of the equation by the information of parameters of it. In this paper, we study the Hermitian positive definite solution of the matrix equation X±A*XqA=I(0＜q＜1), where A is an n x n nonsigular complex matrix,I is an n x n identity matrix, A*is the conjugate transpose of the matrix A. In this paper, first we propose a new method to prove the existence of the unique Hermite solution of the matrix equation X-A*XqA=1(0＜q＜1), and more accessible conditions for the unique solution of the matrix equation X+A*XqA=I(0＜q＜1). Then we introduct how to get the numercial solutions of the matrix equation by continuation method of homotopy. At last, several examples are given to verify the solutions. The main results of this paper are as follows:Theorem1K is a cone made up of n x n positive semidefinite matrix, K°is the set made up of n x n positive definite matrix. If X0∈K, A1, A2,…,Am-1are n×n nonsingular matrixes,0＜q1≤q2≤…≤qm-1＜9m, then there exists a unique X∈K°,so that X0+A1*Xq1A1+A2*Xq2A2+…+Am-1*Xqm-1Am-1=Xqm.Corollary1If A is an nonsingular matrix,then the matrix equation X A*XqA=I(0＜q＜1)has a unique solution.Theorem2In the matrix equation X+A*XqA=I,if A=αN,where α is a complex number,N is a unitary matrix,then the matrix equation has a unique solution X=δI.Where δ is the unique solution of the equation x+|α|2xq=1Theorem3Suppose X is the solution of the matrix equation X+A*XqA=I,(i)If the matrix A satisfies A*A≤(?)I,thenλmax(X)≥(?);(ii)If the matrix A satisfiesλmax(A*A)≥(?),then λmin(X)≤(?).Theorem4If the matrix equation X+A*XqA=I has a solution in[(?)I,I] or [0,(?)I],then the solution is unique.Theorem5If A satisfies one of the following conditions: then the matrix equation X+A*XqA=I has a unique solution X.Theorem6There exists component point0=t0＜tq＜1＜…＜tN=1in[0,1] and integer sequence jk,k=1,2,…,N-1,such that is defined,where G(X(t),t)=I+tA*XqA,moreover,when j�'α, converges to X(1).Theorem7If the matrix A satisfies A*A≤(?)I,(?)qA*A+(?)(AA*)-(?)≥I,then there exists component point0=t0＜t1＜…＜tN=1in[0,1]and integer sequence jk,k=1,2,…,N-1,such that is defined,where G(X(t),t)=I-tA*XqA,moreover,when J�'∝, XN,j+a=G(XN,j,1),j=0,1,. converges to X(1).Theorem8If the matrix A satisfies λmax(A*A)≥(?),(?)qλmin(A*A)+(?)≥1,ηi(i=1,2,…,n)are singular-values ofA,η1=≤η2≤…≤ηn,then there exists component point0=t0＜t1＜…＜tN=1in[0,1] and integer sequence jk,k=1,2,…,N-1,such that is defined,where G(X(t),t)=[B(t)(I-X)-1B*(t)]-(?),B(t)=M(t∧+(1-t)ηn)N,δ is the positive solution ofx+ηn2xq=1,moreover,when j�'∝, XN,j+1=G(XN,j,1),j=0,1,. converge to X(1).