Minimal polynomial bases for the dual spaces of rational vector spaces with applications to realization theory

Abstract

By considering a convolutional code as the range space of some linear map over a rational field, we define minimal polynomial bases for a rational vector space as a natural extension of the concept of minimal convolutional encoders. Furthermore, by using the notion of dual spaces, it is shown how relatively straightforward it is to construct a minimal polynomial basis for the direct space. An application of this concept to multivariable linear system theory is also noted by constructing left and right standard matrix factorizations of proper rational matrices which generalize to the multivariable case the classical representation of a proper rational function as a ratio of two relatively prime polynomials with a denominator of degree larger or equal to the one of the numerator.