Algebraic conditions for the solvability to some systems of matrix equations

Abstract

Although solvability conditions for a system of two linear equations are well-known even in the case of rings and, for three linear equations, in the case of matrices, in the case of four linear equations there are no results. In this paper, we consider systems of four linear matrix equations , and present some necessary and sufficient conditions for their solvability as well as an expression for the general solution. There are two advantages to our results: the presented solvability conditions in many cases can be presented in a purely algebraic form and the method used in the proof allows for a generalization of the obtained results to some more general structures such as algebras of bounded linear operators or rings, under some additional assumptions concerning regularity only. We present several applications of our results.