A system of $k$ Sylvester-type quaternion matrix equations with $3k+1$ variables

TL;DR

This paper establishes rank-based solvability conditions for a system of Sylvester-type quaternion matrix equations with multiple variables, providing necessary and sufficient criteria for solutions and applications to related systems.

Contribution

It introduces new rank conditions for the solvability of a complex quaternion matrix system with $3k+1$ variables, extending previous results in quaternion matrix equations.

Findings

Derived rank equalities as solvability criteria

Provided necessary and sufficient conditions for general solutions

Applied results to specific quaternion matrix systems

Abstract

In this paper, we provide some solvability conditions in terms of ranks for the existence of a general solution to a system of $k$ Sylvester-type quaternion matrix equations with $3k+1$ variables $A_{i}X_{i}+Y_{i}B_{i}+C_{i}Z_{i}D_{i}+F_{i}Z_{i+1}G_{i}=E_{i},~i=\overline{1,k}$. As applications of this system, we present rank equalities as the necessary and sufficient conditions for the existence of a general solution to some systems of quaternion matrix equations $A_{i}X_{i}+(A_{i}X_{i})_{\phi}+C_{i}Z_{i}(C_{i})_{\phi}+F_{i}Z_{i+1}(F_{i})_{\phi}=E_{i},~i=\overline{1,k}$.