The Iteration Method For The Solutions Of The Constrained Matrix Equation

The constrained matrix equation problems have been widely applied in biology,electricity, molecular spectroscopy, solid mechanics, structural design, finite elements, parametre identification, vibration theory, automatic control theory, linear optimal control and so on. Solving the constrained matrix equation problems is one of the important study constents in numerical linear algebra.The problems which will be mainly discussed in the M.S thesis are as following:Problemâ… Given A, B∈R^m×n, find X∈S, such that AX = BProblemâ…¡Given A∈R^m×n, B∈R^n×p, C∈R^m×p, find X∈S, such that AXB = CProblemâ…¢Given A₁∈R^m₁×n,B₁∈R^n×p₁,C₁∈R^{m₁×p₁}, A₂∈R^m₂×n,B₂∈R^n×p₂, C₂∈R^{m₂×p₂}, find X∈S, such thatProblemâ…£When Problemâ… orâ…¡orâ…¢consistent, let S_E denote the set of its solutions, given X₀∈R^n×n, find X∈S_E, such thatWhere ||·||denote the Frobenius norm, S is R^n×n or a subset of R^n×n satisfying some constraint conditions.The main achivements are as follows:1. When S are symmetric reflexive matrices, symmetric antireflexive matrices or symmetrizable matrices, the sufficient and necessary conditions under the solvable and the general provided form of the solution of Problemâ… ^â…£have been obtained by means of matrices decompositions in many references. In this thesis, the solvability of the problems can be determined automatically during the iteratiion by iterative methods. Proof that when Problemâ… ^â…¢is solvable, its solution can be obtained within finite iterative steps, when set the special initial matrix, then its least-norm solution can be obtained within finite steps. Furthermore, the least-norm solution of the optimal approximation and numerical experiments are given.2. When S are centrosymmetric matrices or reflexive matrices, in this thesis, iterative solutions and the optimal approximation solutions of Problemâ… on linear manifold are provided by iterative methods successfully and numerical experiments are given.