Null-pole subspaces of nonregular rational matrix functions

Abstract

Given a (not necessarily regular) rational matrix function W and a subset σ of the extended complex plane, we associate with it a free module l σ( W) over the ring of scalar rational functions which are analytic on σ, called the null-pole subspace of W over σ. In the scalar case, the information encoded in this module is equivalent to the knowledge of all the poles and zeros, including their multiplicities, at all points of σ; in the general matrix case, the null-pole subspace encodes the more complicated zero-pole structure of a rational matrix function, which is the key tool for the understanding of many factorization problems. In this paper we show how various other modules which have been introduced in the literature in connection with pole-zero structure (e.g. pole module, zero module) can be read off from the null-pole subspace, and how, conversely, the null-pole subspace can be recovered from its various pieces (pole module, null module, a null-pole coupling operator and left annihilator). We give an analytic, coordinate-dependent description of these modules which has connection with realization theory and which is useful for computations. We also solve the converse problem of finding a rational matrix function which has a given admissible module as its null-pole subspace.