Mixed Solutions Of A Class Of Matrix Equation

The problem of solving constrained matrix equations is to find the solution of a matrix equation in a constrained matrix sot. In recent years, the study of it has been a hot topic in the field of numerical algebra, actually, it is widely used in many fields such as vibration theory, network planning, systems engineering, civil planing, economics, iconography. This paper studies the solutions problem, least squares solutions problem and their best approximation problem of the matrix equation A1X1B1+A2X2B2+...+AtXtBt=C by constructing an iterative algorithm, these three problems are called problems I, II and III in this paper, respectively.In this paper. we use iterative methods to study problem I and problem II. We solve problem III by three steps, when SE is the solution of problem I. Firstly, we transform matrix equation into matrix equations which have the same solutions with the matrix equation. Then, we solve the minimum norm solution for matrix equations. Finally, we transform best approximation of problem I into the minimum norm of a specific matrix equation. Through above steps, we obtain the best approximation so-lutions of the problem I:we solve problem III by three steps, when SE is the solution of problem II. Firstly, solving the minimal norm solution of the compatibility matrix equations which is equivalent to the equation what we need to solve. Then, we trans-form best approximation of the compatibility matrix equations into the minimum norm of the specific matrix equations. Thus, we obtain the best approximation solutions of the problem II.The main research results are as follows:1、Given the iterative algorithms of problem I、II and proved their the convergence, then, given numerical examples, and the numerical examples show that the algorithms are feasible and effective?2、Given the equivalent equations of solving matrix equation, thereby establishing a iterative algorithm of problem III by the displacement, and given numerical examples in addition, numerical examples show that the algorithm is feasible and effective.