Theory And Computation Of Some Constrained Matrix Equation Problems

The constrained matrix equation problem is to find solutions of a matrix equation (or a system of matrix equations) in a constrained matrix set. The research of it has been an important topic in the field of numerical algebra in recent years. In this thesis, theory and computation of some special constrained matrix equation problems are studied.1. Iterative algorithms for solving two classes of constrained matrix equation problems and associated optimal approximation problemsAn Iterative algorithms for the (least squares) bisymmetric solutions of the matrix equations A1XB1=C1,A2XB2=C2 is proposed. In the sight of operator, more than ten kinds of common constraints on the structure of matrices (such as symmetric constraint, centrosymmetric constraint, reflexive constraint and so on) are reduced to a kind of special operator constraint. Then an iterative method is constructed to find the (least squares) solutions of the general system of linear matrix equations with this operator constraint. By the proposed iterative algorithms, the constrained solutions above can be obtained in finite iteration steps in the absence of round-off errors. Moreover, the associated optimal approximation problems can also be solved.2.Hermitian positive definite solutions of the nonlinear matrix equation Xs+ A*X-tA=QThe solvability and numerical algorithms for the nonlinear matrix equation Xs+ A*X-tA=Q are investigated deeply, where s and t are positive integers. Two necessary and sufficient conditions for the existence of a Hermitian positive definite solution are derived by using matrix decomposition principle. Necessary conditions for the existence of the Hermitian positive definite solutions of the matrix equation Xs+A*X-tA=Q with the case AQ1/2=Q1/2A are studied. Based on the convergence analysis of a fixed-point iteration, some monotonically convergent conditions of the iteration are given. Besides, the matrix equation with two cases:s≥1,0