{
  "abstract": "While studying for a linear algebra final, the first named author prepared some test questions for herself to see how well she understood the material, and asked the second named author: \\emph{If $A$ and $A^T$ have the same eigenvalues and eigenvectors, is $A$ a symmetric matrix?} We show how this excellent question is a great springboard to related questions, in particular when do equal eigenvalues and eigenvectors imply the matrices are, if not equal, at least closely related (such as similar or the transpose/complex conjugate transpose of each other)? The answer depends on how we interpret the question, and provides a great opportunity to talk about creating good questions. In particular, we characterize matrices $A$ for which the transpose or conjugate transpose shares the same eigenvectors (regardless of eigenvalues) and, for each eigenvalue, the same eigenpair (equivalently, the same eigenspace). Thus, a square matrix $A$ is Hermitian if and only if $A^*$ has the same eigenpairs as $A$; moreover, if $A$ is a real matrix with real eigenvalues and $A^T$ has the same eigenvectors as $A$, then $A$ is symmetric.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10578",
  "author_total": 3,
  "authors": [
    {
      "name": "Kayla Miller"
    },
    {
      "name": "Steven J. Miller"
    },
    {
      "name": "Fuzhen Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ea6a8pxb19xcy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10578",
  "fulltext_url": null,
  "id": "ea6a8pxb19xcy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10578",
  "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.GM"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7212302747",
  "pdf_url": "https://arxiv.org/pdf/2609.10578",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ea6a8pxb19xcy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.10578",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10578"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Consequences of Matrices Sharing Eigenvalues and Eigenvectors",
  "title_text": "Consequences of Matrices Sharing Eigenvalues and Eigenvectors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/ea6a8pxb19xcy",
  "venue": "arXiv",
  "versions": []
}