Properties of finite-dimensional sets of solutions of 2D difference equations

Finite-dimensional autonomous behaviors [2,6] are the sets of solutions of certain two-dimensional (2D) difference equations, endowed with the property of constituting finite-dimensional vector spaces. For this class of behaviors, there are two possible representations: kernel descriptions (corresponding to right factor prime polynomial matrix operators) or 2D state-space descriptions (associated with a pair of nonsingular commuting system matrices). In this paper, we both explore the internal properties of these state-space representations, and analyze the algebraic connections between the properties of the kernel descriptions and the properties of the corresponding state-space realizations.