Constructing Multidimensional Difference Equations From a State-Space Representation Using the Generalized Cayley–Hamilton Theorem

We show that applying a generalization of the Cayley-Hamilton Theorem to a state-space representation of a single-output, multidimensional (mD), linear, shift-invariant, causal, autonomous system, with distinct eigentuples results in an equivalent difference equation representation. The proposed method is also applicable to parameterize a set of mD shift-invariant difference equations in terms of a given set of eigenvalues. Lastly, a closed-form expression in terms of the eigenvalues of the state-transition matrices is derived.