{
  "abstract": "For Hermitian matrices $A$ and $B$, we prove that $(A\\join B)\\oplus(A\\meet B)$ is unitarily equivalent to $A\\oplus B$, where the join and meet are taken in Olson's spectral order. Taking traces answers a question of Bourin and Lee: for positive semidefinite matrices, $\\Tr(A\\join B)=\\Tr(A+B)$ if and only if $A\\meet B=0$. Iterating the direct-sum identity gives a trace formula for a finite family of positive matrices. The trace of the spectral supremum equals the trace of the sum precisely when the ranges form an algebraic direct sum. For each fixed $1<p<\\infty$, the spectral supremum and the sum have equal Schatten $p$-norms if and only if the ranges are pairwise orthogonal. Finally we establish an elegant formula for the Frobenius inner product and the spectral order.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39659",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohamed Amine Aouichaoui"
    },
    {
      "name": "Eun-Young Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bcjgsmft13px2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39659",
  "fulltext_url": null,
  "id": "bcjgsmft13px2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39659",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7215050182",
  "pdf_url": "https://arxiv.org/pdf/2609.39659",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bcjgsmft13px2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39659",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39659"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Grassmann formula for the spectral order of matrices",
  "title_text": "A Grassmann formula for the spectral order of matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/bcjgsmft13px2",
  "venue": "arXiv",
  "versions": []
}