{
  "abstract": "This paper studies the element-wise convergence behavior of subspace iteration. All results are established for both \\(\\mathbb{F}=\\mathbb{R}\\) and \\(\\mathbb{F}=\\mathbb{C}\\). For a diagonal matrix \\(Λ=\\mathrm{diag}(λ_1,\\ldots,λ_n)\\) with \\(|λ_1|>\\cdots>|λ_n|>0\\), we analyze the modulus of each entry of the iterative matrix sequence generated by subspace iteration, both without and with the Rayleigh--Ritz procedure. Under mild assumptions on the initial matrix \\(X\\in\\mathbb{F}^{n\\times m}\\), we first derive exact asymptotic expressions for \\(|Q_k(i,j)|\\) in the subspace iteration without the Rayleigh--Ritz procedure: entries with \\(i\\neq j\\) decay as \\(|λ_{\\max\\{i,j\\}}/λ_{\\min\\{i,j\\}}|^k\\), and the deviation of \\(|Q_k(j,j)|\\) from \\(1\\) decays as \\(\\max\\{|λ_j/λ_{j-1}|,|λ_{j+1}/λ_j|\\}^{2k}\\), with explicit coefficients determined by the LU factorization of \\(X\\). Similar conclusions could also be established for upper triangular matrices. For the Rayleigh--Ritz variant, we obtain element-wise bounds for the normalized Ritz vectors \\(Z_k\\). Specifically, for \\(|Z_k(i,j)|\\), off-diagonal entries with \\(i\\geq m+1\\) decay as \\((|λ_i|/|λ_j|)^k\\), off-diagonal entries with \\(i\\leq m\\) decay as \\((|λ_{m+1}|^2/(|λ_i||λ_j|))^k\\), and the deviation of \\(|Z_k(j,j)|\\) from \\(1\\) decays as \\((|λ_{m+1}|/|λ_j|)^{2k}\\). These results give an element-wise description of the convergence behavior of subspace iteration. After an orthogonal or unitary change of basis, the results could be further generalized.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21540",
  "author_total": 1,
  "authors": [
    {
      "name": "Mingyang Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5szva6rzyqm0y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21540",
  "fulltext_url": null,
  "id": "5szva6rzyqm0y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21540",
  "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",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7213915019",
  "pdf_url": "https://arxiv.org/pdf/2609.21540",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5szva6rzyqm0y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2609.21540",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21540"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Element-wise Convergence Behavior of Subspace Iteration",
  "title_text": "Element-wise Convergence Behavior of Subspace Iteration",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:26Z",
  "url": "https://research.draconitic.com/paper/5szva6rzyqm0y",
  "venue": "arXiv",
  "versions": []
}