# Realization of tensor-product and of tensor-Factorization of rational functions

## TL;DR

This paper explores the realization of tensor-products and tensor-factorizations of rational functions, providing explicit formulas and methods to handle their state space representations, especially under specific limiting conditions.

## Contribution

It introduces explicit formulas for tensor-factorization of rational functions and extends classical realization methods to tensor-product cases with dimension inflation.

## Key findings

- Explicit realization formulas for tensor-products of rational functions.
- A method for tensor-factorization under certain limit conditions.
- Extension of classical realization theory to tensor-product scenarios.

## Abstract

We here first study the state space realization of a tensor-product of a pair of rational functions. At the expense of "inflating" the dimensions, we recover the classical expressions for realization of a regular product of rational functions. Then, under an additional assumption that the limit at infinity of a given rational function exists and is equal to identity, an explicit formula for a tensor-factorization of this function, is introduced.