On classifications of third- and fourth-order symmetric tensors by eigenvalues

Abstract

This paper addresses two fundamental problems posed by Qi regarding the sufficiency of eigenvalues for the classification of symmetric tensors in the two-dimensional setting. For $2\times 2\times 2$ and $2\times 2\times 2\times 2$ complex symmetric tensors, we classify their equivalence classes using canonical forms of their associated binary cubics and quartics. We then determine the number of eigenpair classes and the multiplicities of zero eigenvalues for the canonical forms over the complex domain. We demonstrate that these spectral counts do not distinguish all real equivalence classes. Finally, we extend this approach to derive canonical forms and classification for real third- and fourth-order linear partial differential equations in two variables using their bijective relationship to binary forms.

arXiv commentAn earlier version was published in Communications on Applied Mathematics and Computation (doi: 10.1007/s42967-026-00591-w). This revision corrects errors in the published version