{
  "abstract": "We study Hamiltonian and symplectic tensor structures in the T-product algebra. We define T-Hamiltonian and T-symplectic tensors and characterize them through their Fourier-domain slices. For T-Hamiltonian tensors we establish the standard block form and the spectral symmetry of T-eigenvalues, while for T-symplectic tensors we derive the inverse and exponential-map properties. Our main result is a constructive T-Williamson normal form for tensors whose Fourier-domain slices are real symmetric positive-definite matrices. We also show that, under the Hermitian symplectic convention adopted here, this decomposition does not extend directly to arbitrary Hermitian positive-definite Fourier-domain slices, and we derive a real-valued recovery criterion under Fourier conjugate symmetry. Numerical experiments verify the construction, exhibit runtime trends consistent with the slice-wise complexity $O(pn^3)$, and illustrate the framework on a Fourier-domain encoding of covariance-matrix families arising in continuous-variable quantum dynamics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.20829",
  "author_total": 2,
  "authors": [
    {
      "name": "Susana Lopez-Moreno"
    },
    {
      "name": "Taehyeong Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5r5c54b1v5qvz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.20829",
  "fulltext_url": null,
  "id": "5r5c54b1v5qvz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.20829",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2605.20829",
  "pmcid": null,
  "pmid": null,
  "published": "2026-05-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5r5c54b1v5qvz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2605.20829",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.20829"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Hamiltonian and Symplectic Tensors in the T-product Algebra",
  "title_text": "Hamiltonian and Symplectic Tensors in the T-product Algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/5r5c54b1v5qvz",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1007/s40314-026-03934-8",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}