On Classifications of Third- and Fourth-Order Symmetric Tensors by Eigenvalues

Abstract

This paper addresses two fundamental problems posed by Qi [18] regarding the sufficiency of eigenvalues for the classification of symmetric tensors in the two-dimensional setting. For

2\times 2\times 2

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and

2\times 2\times 2\times 2

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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 (PDEs) in two variables using their bijective relationship to binary forms.

Keywords

Tensor eigenvalues / Symmetric tensors / Canonical forms / Classifications / Partial differential equations (PDEs) / 15A69 / 15A21 / 15A18 / 13P15 / 35C05

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Funding

Natural Science Foundation of Xiamen(3502Z202371014)

Science Fund for Distinguished Young Scholars of Fujian Province(2024J02018)

National Natural Science Foundation of China(12371037)

RIGHTS & PERMISSIONS

Shanghai University

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