The Number of Eigenvalues of a Tensor

TL;DR

This paper investigates the count and properties of eigenvalues and eigenvectors of tensors, establishing finiteness results for symmetric tensors and exploring algebraic relations involving characteristic polynomials.

Contribution

It provides the first comprehensive determination of the number of eigenvalues and eigenvectors for generic tensors and analyzes the algebraic structure of their characteristic polynomials.

Findings

Number of eigenvectors of a generic tensor is explicitly determined.

Symmetric tensors have finitely many normalized eigenvalues.

Relations between characteristic polynomial coefficients, discriminants, and resultants are established.

Abstract

Eigenvectors of tensors, as studied recently in numerical multilinear algebra, correspond to fixed points of self-maps of a projective space. We determine the number of eigenvectors and eigenvalues of a generic tensor, and we show that the number of normalized eigenvalues of a symmetric tensor is always finite. We also examine the characteristic polynomial and how its coefficients are related to discriminants and resultants.