{
  "abstract": "We study the set $\\mathrm{Gq}(P,Q)$ of real $n\\times p$ matrices $X$ satisfying $X^\\top Q X = P$, where the nonsingular matrices $P$ and $Q$ are both symmetric or both skew-symmetric and the set is nonempty. Special choices of $P$ and $Q$ yield the Stiefel, generalized Stiefel, indefinite Stiefel, and symplectic Stiefel manifolds, as well as the quadratic matrix Lie groups. We prove that $\\mathrm{Gq}(P,Q)$ is a closed embedded submanifold of $\\mathbb{R}^{n \\times p}$ and that it is compact exactly when $P$ and $Q$ are both symmetric and definite of the same sign. We derive closed-form expressions for its tangent space and, under a canonical-like Riemannian metric, for the normal space, the orthogonal projection, and the Riemannian gradient and Hessian. We extend the Cayley, quasi-geodesic, and polar retractions to $\\mathrm{Gq}(P,Q)$, with explicit domains, vector transports, and closed-form second-order corrections. These objects are the ingredients of Riemannian first- and second-order methods. In trace minimization tests with symmetric positive-definite, skew-symmetric, and symmetric indefinite pairs $(P,Q)$, the Riemannian conjugate gradient and trust-region methods reach the same objective value with different retractions and keep the iterates on $\\mathrm{Gq}(P,Q)$ up to rounding error.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28159",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhen Peng"
    },
    {
      "name": "Bin Gao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0rvdmnsry8z3r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28159",
  "fulltext_url": null,
  "id": "0rvdmnsry8z3r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28159",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7223127644",
  "pdf_url": "https://arxiv.org/pdf/2610.28159",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0rvdmnsry8z3r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28159",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28159"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Generalized quadratic matrix manifolds: Riemannian geometry and computations",
  "title_text": "Generalized quadratic matrix manifolds: Riemannian geometry and computations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/0rvdmnsry8z3r",
  "venue": "arXiv",
  "versions": []
}