The Least Square Solution Of A Class Of Constrained Matrix Equation And Its Best Approximation Problem

The problem of solving constrained matrix equation and the corresponding leastsquares problem is one of the important topics in the field of numerical algebra for many years.It has been widely used in the fields of system identification,structural design,automatic control,structural dynamics and vibration theory.In this master's thesis,we study the general solution,tridiagonal solution and corresponding least-squares solution of matrix equation AX+XTB=C by direct method and iteration method.The specific problems are described as follows:problem 1.Given matrix A?Rn×n,B?Rn×n,C?Rn×n,find X?Rn×n(R3n×n),such that AX+XTB=C.problem 2.Given matrix A ?Rn×n,B?Rn×n,C?Rn×n,find X?Rn×n(R3n×n),such that min?AX+XTB-C?.problem 3.Given matrix X*?Rn×n,find X?SE(SL),such that where Rn×n,R3n×n represents real matrix of order n and tridiagonal matrix of order n respectively;SE and SL are the solution sets of problem 1 and problem 2,respectively.The main contents of this paper are as follows:1.We study the general real matrix solution,the least square solution and the corresponding best approximation solution of matrix equation AX+XTB=C.first,we use the direct method to solve the matrix equation AX+XTB=C,and use the matrix column straightening operator and the matrix Kronecker product to transform the matrix equation AX+XTB=C into the linear equation AX=B,The necessary and sufficient conditions for the existence of solutions and the expressions of the solutions are obtained.Then,the general solution of matrix equation AX+XTB=C,the least square solution and the corresponding best approximation solution are studied by using the iterative method.The iterative method can terminate in a finite step without considering the rounding error.2.We study the tridiagonal solution,tridiagonal least square solution and the corresponding best approximation solution of matrix equation AX+XTB=C.first,we use direct method to solve,and obtain the necessary and sufficient conditions for tridiagonal solution of matrix equation AX+XTB=C and the expression of the general solution.Second,we use iterative method to study the tridiagonal matrix solution and its best approximation solution,Without considering the rounding error,the constructed iterative method can calculate a tridiagonal least square solution on the solution set for any initial tridiagonal matrix in finite steps,and for problem 3,it can be equivalent converted to the problem of finding the least norm least square solution of a new matrix equation.3.The numerical examples of direct method and iterative method are given to prove the convergence of iterative algorithm.The theory is proved by numerical examples.