{
  "abstract": "A real symmetric matrix is called $d$-locally positive semidefinite if all of its $d \\times d$ principal submatrices are positive semidefinite. We investigate the spectral geometry of $d$-locally positive semidefinite matrices. The set of vectors of eigenvalues of $d$-locally positive semidefinite matrices of size $n \\times n$ is fully understood and known to be convex when $d \\in \\{1,n-1,n\\}$ \\cite{blekherman2022hyperbolic}. In the smallest remaining case $n=4, d=2$, non-convexity of the set of vectors of eigenvalues was proved in \\cite{kozhasov2023eigenvalues}, but even in this case the full description was unknown. We provide a basic semialgebraic description of the set of vectors of eigenvalues for $n=4, d=2$ by establishing a Fischer-type inequality for $2$-locally positive semidefinite matrices of size $4 \\times 4$ and prove non-convexity for $n \\geq 4$ and $d \\in \\{2, n-2\\}$. Non-convexity is established via solving certain non-smooth and non-convex min-max point configuration problems in the complex plane, which could be interesting in themselves. Similar problems were considered in \\cite{nesterenko2024submatrices,sengupta2026submatrices} in the context of matrix decomposition and approximation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.15444",
  "author_total": 5,
  "authors": [
    {
      "name": "Jose Acevedo"
    },
    {
      "name": "Grigoriy Blekherman"
    },
    {
      "name": "Sebastian Debus"
    },
    {
      "name": "Seokbin Lee"
    },
    {
      "name": "Cordian Riener"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/763qc63z28z4b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.15444",
  "fulltext_url": null,
  "id": "763qc63z28z4b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.15444",
  "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",
      "math.AG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7203660346",
  "pdf_url": "https://arxiv.org/pdf/2608.15444",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/763qc63z28z4b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.15444",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.15444"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Eigenvalues of locally positive semidefinite matrices: Non-convexity and Geometry",
  "title_text": "Eigenvalues of locally positive semidefinite matrices: Non-convexity and Geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/763qc63z28z4b",
  "venue": "arXiv",
  "versions": []
}