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Kronecker differences

Keegan Doig Anderson, Yorick Hardy, Bertin Zinsou

TL;DR

This work develops Kronecker differences as nonlinear inverses to Kronecker sums, connecting them to Kronecker quotients and exploring when such differences admit uniform, linear representations. It establishes a canonical linear form δ(A,B)=tr12(αT(A⊗Im−Bm⊗Im)) under a solvability condition and shows how uniform Kronecker differences can be generated by (υn, γm,n), yielding a multiplicative/prime-like structure for the ηn tensors. The paper also demonstrates duality between Kronecker quotients and differences and proves a uniform-difference framework with trace-operators as the organizing principle. These results offer a nonlinear, tensor-decomposition perspective on Kronecker-type operations and open questions about broader nonlinear generalizations and exponential relationships.

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.

Kronecker differences

TL;DR

This work develops Kronecker differences as nonlinear inverses to Kronecker sums, connecting them to Kronecker quotients and exploring when such differences admit uniform, linear representations. It establishes a canonical linear form δ(A,B)=tr12(αT(A⊗Im−Bm⊗Im)) under a solvability condition and shows how uniform Kronecker differences can be generated by (υn, γm,n), yielding a multiplicative/prime-like structure for the ηn tensors. The paper also demonstrates duality between Kronecker quotients and differences and proves a uniform-difference framework with trace-operators as the organizing principle. These results offer a nonlinear, tensor-decomposition perspective on Kronecker-type operations and open questions about broader nonlinear generalizations and exponential relationships.

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.
Paper Structure (9 sections, 34 theorems, 168 equations)

This paper contains 9 sections, 34 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Kronecker products and Kronecker quotients
  3. Kronecker sums and Kronecker differences
  4. Properties of Kronecker sums and Kronecker differences
  5. A canonical form
  6. Uniform Kronecker differences
  7. Conclusion
  8. Trace, transpose and orthogonality structures in tensor products
  9. Additional lemmata

Key Result

Lemma 1

Let $\mathbb{F}$ be a field. Then there exists a uniform Kronecker quotient over $\mathbb{F}$.

Theorems & Definitions (85)

  • Definition 1: Kronecker quotient kronquotientleopardi05a
  • Definition 2
  • Remark 1
  • Lemma 1
  • proof
  • Definition 3
  • Definition 4
  • Remark 2
  • Remark 3: Duality 1
  • Definition 5
  • ...and 75 more