Extension of the Bessmertnyi Realization Theorem for Rational Functions of Several Complex Variables

TL;DR

This paper extends Bessmertnyi's realization theorem for rational functions of several complex variables using an operator-theoretic approach with Schur complements and Kronecker products, simplifying the realization process.

Contribution

It introduces a new operator-theoretic method employing Schur complements and Kronecker products to extend the realization theorem for multivariable rational functions.

Findings

Simplifies the realization process using elementary algebraic operations.

Provides a more natural and potentially extendable construction.

Potential applications in multidimensional systems and electrical network models.

Abstract

We prove a realization theorem for rational functions of several complex variables which extends the main theorem of M. Bessmertnyi, "On realizations of rational matrix functions of several complex variables," in Vol. 134 of Oper. Theory Adv. Appl., pp. 157-185, Birkh\"{a}user Verlag, Basel, 2002. In contrast to Bessmertnyi's approach of solving large systems of linear equations, we use an operator theoretical approach based on the theory of Schur complements. This leads to a simpler and more "natural" construction to solving the realization problem as we need only apply elementary algebraic operations to Schur complements such as sums, products, inverses, and compositions. A novelty of our approach is the use of Kronecker product as opposed to the matrix product in the realization problem. As such our synthetic approach leads to a solution of the realization problem that has potential for further extensions and applications within multidimensional systems theory especially for those linear models associated with electric circuits, networks, and composites.