Dominant H-Eigenvectors of Tensor Kronecker Products Do Not Decouple

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.