Matrix Linear Bilateral Equations Over Different Domains, Methods for the Construction of Solutions, and Description of Their Structure

We present a survey of the methods aimed at solving matrix linear bilateral equations (and, in particular, Sylvester-type equations) over different domains and at describing the structure of their solutions. Our main attention is focused on the extension and generalization of the results obtained by the authors earlier. On the basis of the standard form of polynomial matrices with respect to semiscalar equivalence, we develop a method aimed at solving Sylvester-type matrix polynomial equations. The structure of these solutions is investigated. We select solutions of bounded degrees and present the conditions for the uniqueness of these solutions. A method for constructing the solutions of Sylvester matrix equations over adequate rings is developed and criteria for the uniqueness of solutions of a certain type are obtained. We also establish the conditions for the existence of solution to the Sylvester-type matrix equation in rings of triangular and block-triangular matrices over the commutative domain of principal ideals.