The complete equivalence canonical form of four matrices over an arbitrary division ring

TL;DR

This paper establishes the complete equivalence canonical form for four matrices over any division ring and applies it to determine solvability conditions for generalized Sylvester matrix equations.

Contribution

It introduces a new canonical form for four matrices over arbitrary division rings and derives solvability criteria for related matrix equations.

Findings

Complete canonical form for four matrices over division rings

Necessary and sufficient conditions for solvability of generalized Sylvester equations

Applicable over real numbers, complex numbers, and quaternions

Abstract

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.