Kronecker differences

TL;DR

This paper introduces Kronecker differences as a novel algebraic operation that acts as an inverse to Kronecker sums, offering new insights into matrix and tensor decompositions.

Contribution

It defines and explores Kronecker differences, establishing their relationship with Kronecker quotients and providing a new nonlinear perspective on tensor decomposition.

Findings

Kronecker differences can be characterized by specific matrix families.

A connection between Kronecker differences and Kronecker quotients is established.

The approach offers a new nonlinear view on tensor decomposition.

Abstract

Over the real numbers, the Kronecker sum is the unique operation on matrices which exponentiates to the Kronecker product. Kronecker quotients provide an algebraic view of decompositions of matrices in terms of Kronecker products. This article explores families of operations, Kronecker differences, which are a kind of "inverse" for Kronecker sums. The correspondence between Kronecker differences and Kronecker quotients is explored. Furthermore, we show that a certain class of Kronecker differences may be characterized by families of matrices with these families again being expressed as Kronecker products. This approach provides a different "nonlinear" view towards tensor decomposition.