{
  "abstract": "In this paper, the exact joint density of the top $k$ sample eigenvectors is derived for any $p\\times p$ population covariance matrix $\\varSigma$, sample size $n > p -1$, and $1 \\le k \\le p$. Using symmetric functions such as zonal polynomials and a dual summation identity by I. G. Macdonald, the result is expressed as a series in terms of determinants of differential operators. A matrix Kummer transformation converts the resulting local alternating expansion into a globally absolutely convergent series over the entire positive definite matrix space. For $k=2$, the ordered eigenvalue integrals reduce to the Gauss hypergeometric function ${}_2F_1$ with a simple closed form when $n=p+1$. Previously, explicit formulas were only available for a scalar matrix $\\varSigma = σ^2 I_p$ or a general matrix $\\varSigma$ with $p=2$ or $k=1$, obtained by T. W. Anderson and T. Sugiyama, respectively. The frame law also induces an exact Grassmann density that provides a finite-sample benchmark for principal subspace inference.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07028",
  "author_total": 2,
  "authors": [
    {
      "name": "Koki Shimizu"
    },
    {
      "name": "Haoming Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fvss5rqv8xp27.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07028",
  "fulltext_url": null,
  "id": "fvss5rqv8xp27",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07028",
  "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.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7212020603",
  "pdf_url": "https://arxiv.org/pdf/2609.07028",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fvss5rqv8xp27.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07028",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07028"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "The joint density of top k sample eigenvectors and principal subspace inference",
  "title_text": "The joint density of top k sample eigenvectors and principal subspace inference",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:07Z",
  "url": "https://research.draconitic.com/paper/fvss5rqv8xp27",
  "venue": "arXiv",
  "versions": []
}