Iterative methods for solving linear matrix equation and linear matrix system

Abstract

In this paper, an efficient iterative method is presented to solve the linear matrix equation (X) = E with real matrix X. By this iterative method, the solvability of the linear matrix equation can be determined automatically. When the matrix equation is consistent, then, for any initial matrix X ₀, a solution can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm solution can be obtained by choosing a special kind of initial matrix. We also propose an iterative algorithm to obtain the solution or the least norm solution of the consistent matrix system. The given numerical examples demonstrate the efficiency of these two algorithms.

Keywords:

• iterative method
• linear matrix equation
• linear matrix system
• least norm solution
• linear operator

Acknowledgements

The authors are very much indebted to the valuable comments and helpful suggestions of the two referees. The research of this paper was supported by the Science and Technology Commission of Shanghai Municipality through Grant (No. 04JC14031), the China Postdoctoral Science Foundation (No. 20060400634) and the Shanghai Priority Academic Discipline Foundation.