The Structure of Symmetric Solutions of the Matrix Equation AX=B over a Principal Ideal Domain

V M Prokip

doi:10.1155/2017/2867354

Abstract

We investigate the structure of symmetric solutions of the matrix equation AX=B, where A and B are m-by-n matrices over a principal ideal domain R and X is unknown n-by-n matrix over R. We prove that matrix equation AX=B over R has a symmetric solution if and only if equation AX=B has a solution over R and the matrix ABT is symmetric. If symmetric solution exists we propose the method for its construction.

Highlights

Let R be a principal ideal domain with an identity element and let Rm×n denote the set of m × n matrices over R
We investigate the structure of symmetric solutions of the matrix equation AX = B, where A and B are m-by-n matrices over a principal ideal domain R and X is unknown n-by-n matrix over R
We prove that matrix equation AX = B over R has a symmetric solution if and only if equation AX = B has a solution over R and the matrix ABT is symmetric

Summary

Introduction

Let R be a principal ideal domain with an identity element and let Rm×n denote the set of m × n matrices over R. Methods for constructed symmetric solutions of linear matrix equation AXB = C are proposed in [9, 13, 15, 16]. More details on this problem and many references to original literature can be found in [6, 17,18,19,20,21].

Preparatory Results

The Main Result

The Representation of Symmetric Solutions