The Iterative Method For The Constrained Least-squares Solutions To A Class Of Matrix Equation

The problem of solving the linear matrix equation is one of the importantstudy fields of the numerical linear algebra. It has been widely applied in biology,electricity, molecular spectroscopy, vibration theory, finite elements, structuraldesign, solid mechanics, parametre identification, automatic control theory, linearoptimal control and so on.This master thesis is mainly concerned with the problem how to get theconstraint least-squares solutions of the matrix equation AX = B and its optimalapproximation by applying iteration systematically. The problems are as follows:Problem I Given A∈R^m×n,B∈R^m×n, find X∈S (?) R^n×n, such that||AX - B|| = min.Problem II Let SE denotes the solution set of Problem I, given X0∈Rn×n,find X∈SE, such thatwhere . denotes the Frobenius norm, S is a subset of Rn×n. This master thesishas mainly studied centrosymmetric matrix set, centroskew symmetric matrix set,re?exive matrix set, antire?exive matrix set, bisymmetric matrix set, symmetricand antipersymmetric matrix set, symmetric orthogonal symmetric matrix set,symmetric orthogonal antisymmetric matrix set.The main results are as follows:1. For Problem I, many references have given a series important results bymeans of matrices decompositions. In this thesis, iterative method associatedwith the normal equation is used to find the problems centrosymmetric least-squares solutions, centroskew symmetric least-squares solutions, re?exive least-squares solutions, antire?exive least-squares solutions, bisymmetric least-squaressolutions, symmetric and antipersymmetric least-squares solutions, symmetric or-thogonal symmetric least-squares solutions, symmetric orthogonal antisymmetricleast-squares solutions and their optimal approximation to the linear matrix equa-tion AX = B, and solve them successfully.2. For Problem II, we can convert it to another problem of finding the least-squares solutions with the least norm of a new consistent matrix equation. Onthe base of the solutions of Problem I we can apply the iterative method to get...