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 $Qsubseteqmathbb{P}(V)$. 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.

The product of the eigenvalues of a symmetric tensor / Sodomaco L.. - In: LINEAR ALGEBRA AND ITS APPLICATIONS. - ISSN 0024-3795. - ELETTRONICO. - 554:(2018), pp. 224-248. [10.1016/j.laa.2018.05.033]

The product of the eigenvalues of a symmetric tensor

Sodomaco L.
2018

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 $Qsubseteqmathbb{P}(V)$. 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.
2018
554
224
248
Sodomaco L.
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1179696
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