Reproductive and non-reproductive solutions of the matrix equation AXB=C

TL;DR

This paper explores the general solutions of the matrix equation AXB=C, distinguishing between reproductive and non-reproductive solutions, and introduces new conditions for the equation's consistency.

Contribution

It presents a novel representation of the general solution incorporating both reproductive and non-reproductive solutions based on a given particular solution.

Findings

New form of the general solution for AXB=C

Criteria for the consistency of the matrix equation

Analysis of solutions using reproductivity concept

Abstract

In this article we consider a consistent matrix equation $AXB = C$ when a particular solution $X_{0}$ is given and we represent a new form of the general solution which contains both reproductive and non-reproductive solutions (it depends on the form of particular solution $X_{0}$). We also analyse the solutions of some matrix systems using the concept of reproductivity and we give a new form of the condition for the consistency of the matrix equation $AXB=C$.