A note on the consistency of a system of $\star$-Sylvester equations

TL;DR

This paper establishes necessary and sufficient conditions for the consistency of a system of $ abla$-Sylvester equations over fields with characteristic not 2, considering transpose or conjugate transpose operations.

Contribution

It provides a complete characterization of when such systems are consistent, extending previous results to multiple equations and different transpose operations.

Findings

Derived necessary and sufficient conditions for system consistency

Unified treatment for transpose and conjugate transpose cases

Applicable over fields with characteristic not 2

Abstract

Let ${\mathbb F}$ be a field with characteristic not $2$, and $A_i\in{\mathbb F}^{m\times n},B_i\in{\mathbb F}^{n\times m},C_i\in{\mathbb F}^{m\times m}$, for $i=1,...,k$. In this short note, we obtain necessary and sufficient conditions for the consistency of the system of $\star$-Sylvester equations $A_iX-X^\star B_i=C_i$, for $i=1,...,k$, where $\star$ denotes either the transpose or, when ${\mathbb F}={\mathbb C}$, the conjugate transpose.