Some Researches On Orthogonal Projection Iterative Methods For Solving Some Constrained Matrix Equations

The constrained matrix equation problem is to find solutions of a matrix equation or a system of matrix equations in a set of matrices which satisfies some constraint conditons. When the constrained condtions are different, or the matrix equations are different, we can get a different constrained matrix equation problem. The constrianed matix equation problem has been one of the topics of very active research in the field of numercial algebra in recent years. Actually , it has been widely usede in structural design, system identification, principal component analysis, exploration , remote sensing, biology, electricity, molecular spectroscopy, automatics control theory, vibration theory, finite elements, circuit theory, linear optimal control, and so on. The orthogonal projection iteration for the solutions of some contrained matrix equations and the related optimal approximation problem are considered in this Ph.D theis.In 2004, Yaxin Peng put foward a kind of iteration method for the constrianed solutions of the matrix equations AX = B and AXB = C in her Ph.D. thesis, and proved theoritically the convegence of the method in finite steps, but failed to give an estimation of the convergence rate of the method, and so faid to give a synthetic evaluation for the method. When the conditons for solving the problem are bad, it is difficult to improve the convegence of the methoed by some effective means. Based on these consideraion, we studied the iteration mehtod for the solutions to the two equations, and the main works and results are as follows.1. The iteration method for the general solutions to the matrix equations AX = B and AXB = C. In chapter 2, we put foward an iteration method for the general solutions to the matrix equations AX = B and AXB = C, proved the convergence of the methods. The estimations of the convergence rate are given. If the equations are consistent, the method will coverge to the least-norm solutions of the equations, and if the equaitons are not consistent, the method will coverge to the least-norm least-squares solutions of the equations. The numerical results indicate that the convergence rates will be impoved obviously if we use the appropriate preconditioned method. The related optimal approximation solution can also be obtained with the method which only need to be made slight changes.2. The iteration method for the symmetric and anti-symmetric solutions to the matrix equations AX = B, AXB = C and the inverse eigenvalue problem AX = XA. In chapter 3, we put foward an iteration method for the the symmetric and anti-symmetric solutions to the matrix equations AX = B, AXB = C and the inverse eigenvalue problem AX = X∧, proved the convergence of the methods. The estimations of the convergence rate are given. If the equations are consistent, the method will coverge to the least-norm solutions of the equations. The numerical results indicate that the convergence rates will be impoved obviously if we use the appropriate preconditioned method. The related optimal approximation solution can also be obtained with the method which only need to be made slight changes.3. The iteration method for the centro-symmetic and centro-skew symmetric solutions to the matrix equation AX = B. In chapter 4, we put foward an iteration method for the centro-symmetic and centro-skew symmetric solutions to the matrix equation AX = B, proved the convergence of the methods. The estimations of the convergence rate are given. If the equations are consistent, the method will coverge to the least-norm solutions of the equations. The numerical results indicate that the convergence rates will be impoved obviously if we use the appropriate preconditioned method. The related optimal approximation solution can also be obtained with the method which only need to be made slight changes.4. The iteration method for the reflexive and anti-reflexive solutions to the matrix equation AX = B. In chapter 5, we put foward an iteration method for the reflexive and anti-reflexive solutions to the matrix equation AX = B, proved the convergence of the methods. The estimations of the convergence rate are given. If the equations are consistent, the method will coverge to the least-norm solutions of the equations. The numerical results indicate that the convergence rates will be impoved obviously if we use the appropriate preconditioned method. The related optimal approximation solution can also be obtained with the method which only need to be made slight changes.5. The iteration method for bi-symmetric, symmetric-anti-symmetric, and bi-anti-symmetric solutions to the matrix equation AX = B. In chapter 6, we put foward an iteration method for bi-symmetric, symmetric-anti-symmetric, and bi-anti-symmetric solutions to the matrix equation AX = B, proved the convergence of the methods. The estimations of the convergence rate are given. If the equations are consistent, the method will coverge to the least-norm solutions of the equations. The numerical results indicate that the convergence rates will be impoved obviously if we use the appropriate preconditioned method. The related optimal approximation solution can also be obtained with the method which only need to be made slight changes. This dissertation is supported by Natural Science Foundation of China(10571047) and Doctorate Foundation of the Ministry of Education of China(20060532014).This dissertation is typeset by software L~AT_EX2_ε.