On solutions and algebraic duality of generalised linear discrete time systems

We study a class of generalised linear discrete time systems whose coefficients are square constant matrices. The main objective of this research is to provide a link between the solutions of a class of singular linear discrete time systems and their proper (and transposed) dual systems. First, we study the prime system and by using the invariants of its matrix pencil we give necessary and sufficient conditions for existence and uniqueness of solutions, and obtain formulas for the solutions. Next, we prove that by using the same matrix pencil we can study the existence and uniqueness of solutions of the proper and transposed dual system. Moreover their solutions, when they exist, can be explicitly represented without resorting to further processes of computations for each one separately. Finally, we provide a numerical example to justify the theory.