Rationality, minimality and uniqueness of representation of matrix formal power series

In some multivariate time-series models a matrix power series is involved. These models can be identified as rational models if these series correspond to a matrix rational function. Moreover, it is necessary to answer some questions about minimality and uniqueness of representation. The main results of this paper fall within the sphere of matrix Padé approximation. On the basis of formal power series, matrix rational functions of arbitrary dimensions are characterized. Furthermore, we study certain minimality types, that are, global minimum degrees and row minimum degrees. In addition, given that the rational representation of the function for the same pair of degrees need not be unique, we have obtained conditions to study the uniqueness of said representation and, also, to find a “canonical” unique representation. Moreover, we consider an application to special series which is associated with time-series models; such series leads to new theoretical results relating to matrix Padé approximation. Finally, we comment on some illustrative examples.