Orthognal Iterative Methods For Several Matrix Equation Problems

The constrained matrix equation(or equations)problem is solving the matrix equation (or equations)problem under constrained conditions.Whether the constrained conditions or the matrix equation(equations)is(or are)different,we can get different constrained matrix equation(or equations)problem.This thesis mainly researches the following questions.Problem Ⅰ:Given A∈Rm×n,B∈Rm×n,C∈Rm×n,S1(?)Rn×n,S2(?)Rm×m,find X∈S1 Y∈S2 such that AX+YB=CProblem Ⅱ:Suppose the above problems is consistent,and its solution set is SE, givenX∈Rn×n,Y∈Rm×m,find[X,Y]∈SE,such thatProblem Ⅲ:Given A∈Rp×m,B∈Rp×m,C∈Rm×l,D∈Rm×l,S(?)Rm×m,find X∈S such thatProblem Ⅳ:Suppose the above problems is consistent,and its solution set is SE, given X∈Rm×n,findX∈SE such thatThe mainly research work of this paper is as follows:1.When[S1,S2]is different constrained matrix[Rn×n,Rm×m],[S,Rn×n,ScRm×m],and [AS,Rn×n,ScRm×m],firstly,we construct the orthogonal projection iterative algorithm for problem I using the special structure and properties of double-structure matrix space and the theory of the orthogonal projection;secondly,the convergence of the method is proved and the method’s estimation is acquired by using singular value decomposition,the orthogonal invariance of the F-norm of the matrix and properties of solutions of two variable matrix equation;its optimal approximation solution can also be found with the method which only need to be made slight changes;finally,numerical examples are given to verify the effectiveness of the method;and when[S1,S2]is[Rn×n,Rm×m],compare the orthogonal projection iterative algorithm of solving problem Ⅰ with some other iterative algorithm such as gradient iterative algorithm, the orthogonal projection iterative algorithm is highest one.2. When S are real matrices, centrosymmetry matrices and reflexive matrices, firstly, firstly we construct the orthogonal projection iterative algorithm for problem Ⅲ; secondly, the convergence of the method is proved and the method’s estimation is acquired by using properties of solutions of matrix equations; its optimal approximation solution can also be found with the method which only need to be made slight changes; finally, numerical examples are given to verify the effectiveness of the method; and when S are real matrices, compare the orthogonal projection iterative algorithm of solving problem Ⅲ with some other iterative algorithm such as gradient iterative algorithm, the orthogonal projection iterative algorithm is highest one.