{
  "abstract": "The semidefinite linear complementarity problem associated with the multiplicative map $M_A:\\mathbb{S}^n\\to\\mathbb{S}^n$ defined by $M_A(X)=AXA^T$, is considered, in the case when $A\\in\\mathbb{R}^{n\\times n}$ has rank one. A linear map is said to have the $Q_0$-property if every feasible instance of the associated complementarity problem admits a complementary solution. We prove two sharp, complementary results. When $A=xy^T$ with $x,y\\in\\mathbb{R}^n$ \\emph{linearly independent}, we construct a matrix $Q\\in\\mathbb{S}^n$ for which the associated semidefinite complementarity problem is feasible, but possesses no complementary solution; proving that $M_A$ \\emph{does not have} the $Q_0$-property. Conversely, when $A=uu^T$ for some nonzero $u\\in\\mathbb{R}^n$, we prove that $M_A$ \\emph{has} the $Q_0$-property. These two results provide a complete characterization of the $Q_0$-property within the family of rank-one multiplicative maps.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37197",
  "author_total": 1,
  "authors": [
    {
      "name": "K. C. Sivakumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aztkhfzd3713v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37197",
  "fulltext_url": null,
  "id": "aztkhfzd3713v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37197",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7214996764",
  "pdf_url": "https://arxiv.org/pdf/2609.37197",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aztkhfzd3713v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37197",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37197"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Complementarity Property of Rank-One Multiplicative Maps",
  "title_text": "A Complementarity Property of Rank-One Multiplicative Maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/aztkhfzd3713v",
  "venue": "arXiv",
  "versions": []
}