Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dominant H-Eigenvectors of Tensor Kronecker Products Do Not Decouple

Ayush Kulkarni, Charles Colley, David F. Gleich

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.

Dominant H-Eigenvectors of Tensor Kronecker Products Do Not Decouple

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.

Paper Structure

This paper contains 5 sections, 2 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Counterexamples
  3. The simple counterexample
  4. A counterexample over the real numbers that is not a counterexample over complex numbers
  5. Discussion and special cases

Key Result

THEOREM 3.1

Let $\ushort{\boldsymbol{{A}}}$ and $\ushort{\boldsymbol{{B}}}$ be diagonal tensors. Then $\ushort{\boldsymbol{{B}}} \otimes \ushort{\boldsymbol{{A}}}$ is also diagonal. Moreover, the $H$-eigenvalues of $\ushort{\boldsymbol{{B}}} \otimes \ushort{\boldsymbol{{A}}}$ are simply the products of the $H$-

Theorems & Definitions (3)

  • THEOREM 3.1
  • THEOREM 3.2
  • Proof 1