Nondegeneracy of eigenvectors and singular vector tuples of tensors

TL;DR

This paper investigates the nondegeneracy properties of eigenvectors and singular vector tuples of tensors, establishing conditions under which these vectors are nondegenerate for generic and orthogonally decomposable tensors.

Contribution

It proves that all eigenvectors and singular vector tuples of generic tensors and orthogonally decomposable tensors are nondegenerate, advancing understanding of tensor eigenstructure.

Findings

Eigenvectors of generic tensors are nondegenerate.

Singular vector tuples of orthogonally decomposable tensors are nondegenerate.

Nondegeneracy holds for all nonzero Z-eigenvectors of such tensors.

Abstract

In this article, nondegeneracy of singular vector tuples, Z-eigenvectors and eigenvectors of tensors is studied. They have found many applications in diverse areas. The main results are: (i) each (Z-)eigenvector/singular vector tuple of a generic tensor is nondegenerate, and (ii) each nonzero Z-eigenvector/singular vector tuple of an orthogonally decomposable tensor is nondegenerate.