# Realizations of non-commutative rational functions around a matrix centre, I: synthesis, minimal realizations and evaluation on stably finite algebras

## TL;DR

This paper extends classical non-commutative rational function realization theory to arbitrary matrix centers, establishing existence, uniqueness, and evaluation methods, with implications for matrix and symmetric cases.

## Contribution

It introduces a generalized framework for minimal realizations of nc rational functions centered at any matrix point, including evaluation over stably finite algebras.

## Key findings

- Proves existence and uniqueness of minimal realizations at arbitrary matrix centers.
- Provides a method to evaluate nc rational functions on all matrix sizes and stably finite algebras.
- Offers a new proof of the equivalence theorem for rational expressions over matrices and stably finite algebras.

## Abstract

In this paper we generalize classical results regarding minimal realizations of non-commutative (nc) rational functions using nc Fornasini-Marchesini realizations which are centred at an arbitrary matrix point. We prove the existence and uniqueness of a minimal realization for every nc rational function, centred at an arbitrary matrix point in its domain of regularity. Moreover, we show that using this realization we can evaluate the function on all of its domain (of matrices of all sizes) and also with respect to any stably finite algebra. As a corollary we obtain a new proof of the theorem by Cohn and Amitsur, that equivalence of two rational expressions over matrices implies the expressions are equivalent over all stably finite algebras. Applications to the matrix valued and the symmetric cases are presented as well.