Solvability And Effective Algorithms Based On Semi-tensor Product And Special Structural Constraint Matrix Equations

The solution of matrix equation is widely used in image restoration,linear programming,parameter identification and other fields,and the constrained matrix equation is one of the most active topics in the field of numerical algebra.In this paper,the solution of special matrix equation is studied,and an effective solution method is put forward.Its main job has two parts.Firstly,a class of matrix equations is solved based on the semi-tensor product.The semi-tensor product,which was initially proposed by Professor Cheng Daizhan has been extensively applied in Boolean control networks,graph coloring,game theory,cryptographic algorithms and so on.In this article,motivated by the existing work by Yao,we further investigate the solvability of the matrix equation AXB=C with respect to semi-tensor product.The case of matrix-vector equation,in which the required unknown X be a vector,is studied first.Compatible condition for matrix dimensions,necessary and sufficient conditions and concrete solving methods are established.Based on this,the solvability of the matrix the equal case,in which the unknown X be a matrix,under semi-tensor products are then studied.For each part,several elementary examples are presented to illustrate the efficiency of the results.Secondly,the solutions of two kinds of constraint matrix equations are given.Constraint matrix equation problems have important applications in the fields of parameter identification,automated control,biology and so on.In this paper,we consider the low-rank symmetric positive semidefinite numerical solutions for the matrix equations AX=B and AXAT=C respectively.Firstly,we characterize the feasible set by X=YYT,Y?Rn×l and then transform low-rank symmetric positive semidefinite solutions into unconstrained optimization problems and use the nonmonotone conjugate gradient method to establish iterative methods for computing the symmetric positive semidefinite solutions.Finally,numerical examples show that the methods are feasible and effective.