Iterative Solutions For Several Constrained Matrix Equation Problems

The constrained matrix equation problem is to find a solution for the matrix equation in a constrained matrix set. It has been widely used in many fields such as structural design, system identification, automatic control theory, finite elements, vibration theory linear optimal control and so on.This thesis mainly discusses the iterative methods for the following problems:Problem I GiventhatProblem II Given such thatProblem III Given such thatProblem IV Suppose problem I or II or III is compatible, let S_E denote the set of solutions, for given matrix X-0∈R^n×n, find X(?)∈S_E such that where the notation denotes the Frobenius norm.The main achievements are as follows:1. When S are symmetric, anti-symmetric, Centro symmetric or bi-symmetric matrices, the sufficient and necessary solvable conditions for problem I- III and the expressions of these general solutions are given in references. With the property of gradient matrix, we bring forward iterative algorithms for the least square problems with reference to problem I- III, prove the finite termination of them. We also prove that these algorithms are suitable in case that problem I- III is compatible. By choosing certain initial matrices, we get the solutions for problem IV. Finally, we give several numerical examples to show the effectiveness of our algorithms.2. When S are closed convex cones, the algorithms for problem I- III in the references are complicated and unrealizable. By the approximation theory over the closed convex cone and the convex analysis theory, we give a sufficient solvable condition for the least square problems with reference to problem I- III, bring forth an iterative algorithm for these problems, and prove the global convergence and linear convergent speed for the algorithm. For common closed convex cones such as non-negative matrices and positive semi-definite matrices, we provide numerical examples to show the effectiveness of our algorithm.3. When S are symmetrizable matrices, the expressions of general solutions for problem I- IV in the references are so complicated that it is hard to get them. Enlightened by the conjugated gradient method for system of linear equations, we give an iterative algorithm which can automatically judge the compatibility of problem I- III during the process of iteration, prove its convergence and finite limitation, discuss the solution of problem IV in case that problem I- III is compatible, and give several numerical examples to show the effectiveness of our algorithms at last.