Proper splittings and reduced solutions of matrix equations

Abstract

In this article we apply proper splittings of matrices to develop an iterative process to approximate solutions of matrix equations of the form TX = W. Moreover, by using the partial order induced by positive semidefinite matrices, we obtain equivalent conditions to the convergence of this process. We also include some speed comparison results of the convergence of this method. In addition, for all matrix T we propose a proper splitting based on the polar decomposition of T.