On a Non-homogeneous Singular Linear Discrete Time System with a Singular Matrix Pencil

In this article, we study the initial value problem of a non-homogeneous singular linear discrete time system whose coefficients are either non-square constant matrices or square with an identically zero matrix pencil. By taking into consideration that the relevant pencil is singular, we provide necessary and sufficient conditions for existence and uniqueness of solutions. More analytically we study the conditions under which the system has unique, infinite and no solutions. Furthermore, we provide a formula for the case of the unique solution. Finally we provide some numerical examples based on a singular discrete time real dynamical system to justify our theory.