Eigenvalues and equivalence classes of third-order symmetric tensors

TL;DR

This paper shows that third-order real symmetric tensors cannot be classified solely by their eigenvalues and introduces a new method using Harrison's center theory to classify small tensors via canonical forms.

Contribution

It provides a novel classification approach for third-order symmetric tensors using Harrison's center theory and explicitly computes characteristic polynomials for $2 imes 2 imes 2$ tensors.

Findings

Eigenvalues are insufficient for tensor classification.

Explicit characteristic polynomials for small tensors are derived.

Different tensors can share eigenvalues but belong to different classes.

Abstract

This paper demonstrates that third-order real symmetric tensors cannot be classified up to equivalence by their eigenvalues only, thereby resolving a problem posed by Qi in 2006. By applying Harrison's center theory, we derive equivalence classes of $2 \times 2 \times 2$ symmetric tensors via the one-to-one correspondence with the canonical forms of their associated binary cubics. For such tensors, we compute the explicit characteristic polynomials and discover two previously unknown coefficients using the combination resultant. Pairs of third-order real symmetric tensors of all dimensions with identical eigenvalues but belonging to different equivalence classes are constructed to illustrate the inapplicability of eigenvalues for classification.