Fundamental equivalence of discrete-time AR epresentations

We examine the problem of equivalence of discrete time auto-regressive representations (DTARRs) over a finite time interval. Two DTARRs are defined as fundamentally equivalent (FE) over a finite time interval [0,N] if their solution spaces or behaviours are isomorphic. We generalize the concept of strict equivalence (SE) of matrix pencils to the case of general polynomial matrices and in turn we show that FE of DTARRs implies SE of the underlying polynomial matrices.