Some Researches On Iterative Methods For Solving The Constrained Matrix Equation AX=B In The Field Of Complex Numbers.

The constrained matrix equation problem is to find solutions of a matrix equation or a system of matrix equations in a set of matrices which satisfies some constraint conditions. When the matrix equations are different, or the constrained conditions are different, we can obtain a different constrained matrix equation problem. The constrained matrix equation problem has been one of the most abundant topics in the field of numerical algebra. It has a very wide range of applications in the structural design, parameter analysis, biological, automatic control theory , finite element and so on, but also made a lot of scientific research.The master's thesis mainly researches the following questions:Problem 1. Given A, B∈C^m×n,find X∈S(?)C^n×n, such that AX = B.Problem 2. Given X₀∈S, find (X|?)∈S_E, such thatWhere (?) is Frobenius Norm, S_E is the solution set of problem 1;S includes the Generalized (anti-Hamilton) -Hamilton matrices, the Hermite-Generalized (anti-Hamilton)- Hamilton matrix, anti-Hermite- Generalized (anti-Hamilton) Hamilton matrix.Firstly, we use the theory of the orthogonal projection and the nature of the characteristics of double-structure matrix to construct the iterative method. Secondly, by use of the orthogonal invariance of the F-norm of the matrix, singular value decomposition and the orthogonal projection principle, the convergence of the method is proved and the method's estimation is acquired. If the equations are consistent, the method will converge to the least-norm solution, and if the equations are not consistent, the method will converge to the least-squares least-norm solutions of the equations, the related optimal approximation solution can also be obtained with the method which only need to be made slight changes. Finally, numerical examples are given to verify the effectiveness of the method .