Bessmertny\u{i} realizations of symmetric multivariate rational matrix functions over any field

TL;DR

This paper proves that multivariable rational matrix functions over any field have Bessmertny realizations, with symmetric versions over fields of characteristic not two, using state-space methods from systems theory.

Contribution

It establishes the existence of Bessmertny realizations for multivariable rational functions over any field, including symmetric realizations in certain cases, and develops methods for their construction.

Findings

Every square matrix of multivariable rational functions has a Bessmertny realization.

Symmetric Bessmertny realizations exist over fields with characteristic not two.

Complete characterization of functions with symmetric realizations over fields of characteristic two.

Abstract

In this paper, we prove the following. First, every square matrix whose entries are multivariable rational functions over a field $\mathbb{F}$ has a Bessmertny\u{i} realization, i.e., is the Schur complement of an affine linear square matrix pencil with coefficients in $\mathbb{F}$. Second, if the matrix is also symmetric and the characteristic of the field $\mathbb{F}$ is not two then it has a symmetric Bessmertny\u{i} realization (i.e., the pencil can be chosen to consist of symmetric matrices) and counterexamples are given to prove this statement is false in general for fields of characteristic two. Third, for fields of characteristic two (e.g., binary or Boolean field), we completely characterize those functions that have a symmetric Bessmertny\u{i} realization. Finally, analogous results hold when restricted to the class of homogeneous degree-one rational functions. To solve these realization problems, i.e., finding such structured Bessmertny\u{i} realizations for a given multivariate rational function, we use state-space methods from systems theory to produce realizations for algebraic operations on Schur complements such as sums, products, inverses, and symmetrization, which become the elementary building blocks of our constructions. Further complications arise over fields of characteristic two, so a large part of the paper is devoted to developing additional methods to decide if the symmetric realization problem can be solved and, if so, to construction the symmetric realization for a given symmetric rational matrix function. Our motivations are discussed in the context of multidimensional linear systems theory on generalizing state-space representations for rational functions including the Givone-Roesser and Fornasini-Marchesini realizations.