The product of the eigenvalues of a symmetric tensor

TL;DR

This paper investigates the relationship between E-eigenvalues of symmetric tensors and their E-characteristic polynomial, deriving a formula for their product that generalizes the determinant-eigenvalues connection for matrices.

Contribution

It provides a new closed-form expression for the product of E-eigenvalues of symmetric tensors, extending classical matrix eigenvalue results to higher-order tensors.

Findings

The leading coefficient of the E-characteristic polynomial relates to the $ ilde{Q}$-discriminant.

A closed formula for the product of E-eigenvalues is established.

The results generalize the determinant-eigenvalues relationship from matrices to symmetric tensors.

Abstract

We study E-eigenvalues of a symmetric tensor $f$ of degree $d$ on a finite-dimensional Euclidean vector space $V$, and their relation with the E-characteristic polynomial of $f$. We show that the leading coefficient of the E-characteristic polynomial of $f$, when it has maximum degree, is the $(d-2)$-th power (respectively the $((d-2)/2)$-th power) when $d$ is odd (respectively when $d$ is even) of the $\widetilde{Q}$-discriminant, where $\widetilde{Q}$ is the $d$-th Veronese embedding of the isotropic quadric $Q\subseteq\mathbb{P}(V)$. This fact, together with a known formula for the constant term of the E-characteristic polynomial of $f$, leads to a closed formula for the product of the E-eigenvalues of $f$, which generalizes the fact that the determinant of a symmetric matrix is equal to the product of its eigenvalues.