Eigenvalues and equivalence classes of third-order symmetric tensors

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.