2018/02/28 by Luca Sodomaco
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Characteristic polynomial #Combinatorics #Degree (music) #Discriminant #Eigenvalues and eigenvectors #Elementary symmetric polynomial #Euclidean space #Geometry #Homogeneous polynomial #Mathematical analysis #Mathematics #Matrix Theory and Algorithms #Matrix polynomial #Orthogonal polynomials #Polynomial #Power sum symmetric polynomial #Product (mathematics) #Pure mathematics #Quadric #Symmetric polynomial #Symmetric tensor #Tensor (intrinsic definition) #Tensor decomposition and applications #Tensor product #math.AG #msc:14M20 #msc:15A18 #msc:15A69 #msc:15A72 #msc:65H17
paper · pdf · doi:10.1016/j.laa.2018.05.033
18 pages, 1 figure
arxiv created 2018/03/06 · openalex publication_date 2018/06/01 · arxiv updated 2018/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 \widetildeQ-discriminant, where \widetildeQ is the d-th Veronese embedding of the isotropic quadric Q⊆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.