{
  "abstract": "The index of a matrix is a fundamental invariant in the analysis of singular matrices and their generalized inverses. While sharp results are available for block triangular matrices, the corresponding theory for anti-triangular block matrices is less developed. In this paper, we study matrices of the form 𝑀 = [ 𝐴 𝐵 𝐶 0 ] , under algebraic constraints on the blocks. Building on additive decompositions involving von Neumann inverses, we relate the index of M to invariance properties of the index and minimal polynomial of expressions of the form 𝐴 2 ⁢ 𝐴 − + 𝐼 − 𝐴 ⁢ 𝐴 − , where 𝐴 − is any von Neumann inverse of A . This connection provides an effective mechanism to control the index of M through suitable factorizations and associated block products. As a consequence, we derive explicit lower and upper bounds for the index of M in terms of the index of A and of BC , and characterize situations in which these bounds are attained. Under additional annihilation or orthogonality conditions on the blocks, we obtain closed-form representations for the Drazin inverse of M . Applications to adjacency matrices of directed graphs illustrate the sharpness of the bounds and the applicability of the results to structured matrices arising in graph-theoretic settings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2604.08137",
  "author_total": 4,
  "authors": [
    {
      "name": "Faustino Maciala"
    },
    {
      "name": "Xavier Mary"
    },
    {
      "name": "C. Mendes Araújo"
    },
    {
      "name": "Pedro Patrício"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2ck7brdfn0njj.bib",
  "cited_by": 0,
  "doi": "10.1016/j.amc.2026.130346",
  "fulltext_url": null,
  "id": "2ck7brdfn0njj",
  "issns": [
    "0096-3003"
  ],
  "landing_url": "https://hal.science/hal-05828531v1",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7153338162",
  "pdf_url": "https://www.sciencedirect.com/science/article/pii/S0096300326003978/pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": "Elsevier",
  "ris": "https://research.draconitic.com/paper/2ck7brdfn0njj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:00Z",
      "native_id": "hal-05828531",
      "source": "hal",
      "url": "https://hal.science/hal-05828531v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "On the index of an anti-triangular block matrix",
  "title_text": "On the index of an anti-triangular block matrix",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:00Z",
  "url": "https://research.draconitic.com/paper/2ck7brdfn0njj",
  "venue": "Applied Mathematics and Computation",
  "versions": []
}