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

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.