# Dominant H-Eigenvectors of Tensor Kronecker Products Do Not Decouple

## TL;DR

This paper presents a counterexample showing that the dominant H-eigenvector of a tensor Kronecker product does not always decouple into the eigenvectors of the individual tensors, contrasting with matrix and Z-eigenvector cases.

## Contribution

It provides the first known counterexample for H-eigenvectors and clarifies conditions where decoupling does or does not occur.

## Key findings

- Counterexample disproves decoupling for H-eigenvectors
- Decoupling holds for diagonal and nonnegative tensors
- Largest H-eigenvalue can exceed the product of component eigenvalues

## Abstract

We illustrate a counterexample to an open question related to the dominant H-eigenvector of a Kronecker product of tensors. For matrices and Z-eigenvectors of tensors, the dominant eigenvector of a Kronecker product decouples into a product of eigenvectors of the tensors underlying the Kronecker product. This does not occur for H-eigenvectors and indeed, the largest H-eigenvalue can exceed the product of the H-eigenvalues of the component tensors. Beyond this general counterexample, we show this decoupling does hold in the case of diagonal tensors as well as nonnegative tensors.