On the fundamental matrix of the inverse of a polynomial matrix and applications to ARMA representations

The fundamental matrix of the inverse of a particular Toeplitz structured matrix pencil is used in order to provide an algorithm for the computation of the fundamental matrix of the inverse of a polynomial matrix. The dynamical interpretation of this pencil is explored and its fundamental matrix is used to provide closed formulas for the solutions of AutoRegressive Moving Average (ARMA) representations. Finally, using this pencil the notion of reachability for descriptor state space systems is extended to ARMA representations.