Eigenvalues and equivalence classes of third-order symmetric tensors

[2512.10263] Eigenvalues and equivalence classes of third-order symmetric tensors Skip to main content Search Submit Donate Log in Mathematics > Rings and Algebras arXiv:2512.10263 (math) [Submitted on 11 Dec 2025] Title: Eigenvalues and equivalence classes of third-order symmetric tensors Authors: Lishan Fang , Hua-Lin Huang , Shengyuan Ruan , Yu Ye View PDF HTML (experimental) 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. Subjects: Rings and Algebras (math.RA) MSC classes: 15A69, 15A21, 15A18, 13P15 Cite as: arXiv:2512.10263 [math.RA] (or arXiv:2512.10263v1 [math.RA] for this version) https://doi.org/10.48550/arXiv.2512.10263 Focus to learn more arXiv-issued DOI via DataCite Journal reference: J. Symbolic Comput. 139 (2027) 102613 Related DOI : https://doi.org/10.1016/j.jsc.2026.102613 Focus to learn more DOI(s) linking to related resources Submission history From: Lishan Fang [ view email ] [v1] Thu, 11 Dec 2025 03:54:37 UTC (19 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.RA < prev | next > new | recent | 2025-12 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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