Qi's problems on classifications of third- and fourth-order symmetric tensors by eigenvalues

TL;DR

This paper investigates the classification of complex symmetric tensors of third and fourth order using eigenvalues, establishing canonical forms and spectral invariants, and explores differences between complex and real cases.

Contribution

It provides a complete classification of complex symmetric tensors via eigenvalues and spectral invariants, and extends the approach to related linear PDEs.

Findings

Complete classification of complex symmetric tensors using eigenvalues.

Spectral invariants uniquely determine equivalence classes in the complex domain.

Classification does not extend to the real domain, where invariants are insufficient.

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 $2\times2\times2$ and $2\times2\times2\times2$ complex symmetric tensors, we establish their complete set of equivalence classes via a one-to-one correspondence with the canonical forms of their associated binary cubics and quartics. We then prove that these equivalence classes are uniquely determined by spectral invariants, specifically, the number of eigenpair classes and the multiplicities of zero eigenvalues, over the complex domain. We demonstrate that this classification does not hold in the real domain, where distinct equivalence classes can share identical spectral invariants. Finally, we extend this approach to derive canonical forms and complete classification for complex third- and fourth-order linear partial differential equations in two variables using their bijective relationship to binary forms.