The complete equivalence canonical form of four matrices over an   arbitrary division ring
  In this paper, we give the complete structures of the equivalence canonical form of four matrices over an arbitrary division ring. As applications, we derive some practical necessary and sufficient conditions for the solvability to some systems of generalized Sylvester matrix equations using the ranks of their coefficient matrices. The results of this paper are new and available over the real number field, the complex number field, and the quaternion algebra. 
Zhuo-Heng He, Qing-Wen Wang, Yang Zhang
math.RA