Sylvester-type quaternion matrix equations with arbitrary equations and arbitrary unknowns

Abstract

In this paper, we prove a conjecture which was presented in a recent paper [Linear Algebra Appl. 2016; 496: 549--593]. We derive some practical necessary and sufficient conditions for the existence of a solution to a system of coupled two-sided Sylvester-type quaternion matrix equations with arbitrary equations and arbitrary unknowns $A_i{X}_i{B}_i+C_i{X}_{i+1}{D}_i=E_i,i=\overline{1,k}$. As an application, we give some practical necessary and sufficient conditions for the existence of an $\eta$-Hermitian solution to the system of quaternion matrix equations $A_i{X}_i{A}_i^{\eta \ast }+C_i{X}_{i+1}{C}_i^{\eta \ast }=E_i$ in terms of ranks, $i=\overline{1,k}$.