On the Sylvester matrix equation over quaternions

TL;DR

This paper investigates the quaternion Sylvester matrix equation, focusing on the conditions for solutions and exploring cases with infinitely many or no solutions, especially for triangular and Jordan block matrices.

Contribution

It extends the understanding of quaternion Sylvester equations by analyzing solution existence and uniqueness, particularly for special matrix structures.

Findings

Characterization of solution existence and uniqueness conditions.

Analysis of cases with infinitely many solutions or no solutions.

Special focus on triangular and Jordan block matrices.

Abstract

The Sylvester equation $AX-XB=C$ is considered in the setting of quaternion matrices. Conditions that are necessary and sufficient for the existence of a unique solution are well-known. We study the complementary case where the equation either has infinitely many solutions or does not have solutions at all. Special attention is given to the case where $A$ and $B$ are respectively, lower and upper triangular two-diagonal matrices (in particular, if $A$ and $B$ are Jordan blocks)