Commuting Outer Inverse-Based Solutions to the Yang–Baxter-like Matrix Equation

Abstract

This paper investigates new solution sets for the Yang–Baxter-like (YB-like) matrix equation involving constant entries or rational functional entries over complex numbers. Towards this aim, first, we introduce and characterize an essential class of generalized outer inverses (termed as {2,5}-inverses) of a matrix M, which commute with M. Towards this aim, first, we introduce and characterize an essential class of generalized outer inverses (termed as {2,5}-inverses) of a matrix, which commute with it. This class of {2,5}-inverses is defined based on resolving appropriate matrix equations and inner inverses. In general, solutions to such matrix equations represent optimization problems and require the minimization of corresponding matrix norms. We decided to analytically extend the obtained results to the derivation of explicit formulae for solving the YB-like matrix equation. Furthermore, algorithms for computing the solutions are developed corresponding to the suggested methods in some computer algebra systems. The main features of the proposed approach are highlighted and illustrated by numerical experiments.