A new system of sylvester-like matrix equations with arbitrary number of equations and unknowns over the quaternion algebra

Journal article, 2025

In this paper, we derive some practical necessary and sufficient conditions for the existence of a solution to a new system of coupled two-sided Sylvester-like matrix equations with arbitrary number of equations and unknowns over the quaternion algebra. As applications, we give some practical necessary and sufficient conditions for the existence of an η-Hermitian solution to a system of quaternion matrix equations in terms of ranks. We also use graphs to represent linear mappings associated with some Sylvester-type systems. As a special case of the main theorem, we prove a conjecture is correct, which was proposed in [Linear Algebra Appl. 2016;496:549–593]. The main findings of this paper widely extend almost all the known results in the literature.