Proper splittings and reduced solutions of matrix equations

TL;DR

This paper introduces an iterative method based on proper matrix splittings to efficiently approximate solutions to matrix equations, analyzing convergence conditions and proposing a new splitting using polar decomposition.

Contribution

It develops a novel iterative process for matrix equations using proper splittings and provides convergence criteria and speed comparisons, including a new splitting based on polar decomposition.

Findings

Convergence conditions are characterized via positive semidefinite partial order.

The proposed method demonstrates competitive convergence speed.

A new splitting based on polar decomposition is introduced.

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.