{
  "abstract": "In this paper, we give a constructive proof of the well-known fact that every ETF of n+1 vectors in Rn equals a signed orthogonal transform of the standard MB frame, establishing an explicit relationship between equiangular tight frames of n+1 vectors in Rn and the standard MB frame via QR factorization. This relationship is utilized to develop a QR-based method for replacing the missing vectors in an equiangular tight frame of n+1 vectors for Rn to complete an equiangular tight frame. A simple recursion and explicit expressions for the entries of the standard MB frame in Rn are also presented. An example in R5 is provided to demonstrate the approach. The preliminary numerical test on randomly generated problems indicates that the method is accurate and stable.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics, Kennesaw State University, Marietta, GA 30060, USA"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Jun Ji",
      "orcid": "0000-0003-3012-2757"
    },
    {
      "affs": [
        0
      ],
      "name": "Long Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/248kmqnfgpchm.bib",
  "cited_by": 0,
  "doi": "10.3390/math14193570",
  "fulltext_url": null,
  "id": "248kmqnfgpchm",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/19/3570",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7216226293",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/248kmqnfgpchm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:29Z",
      "native_id": "10.3390/math14193570",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/19/3570"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2604|Mathematical Analysis and Transform Methods",
      "subfield": "general"
    }
  ],
  "title": "Completion of Equiangular Tight Frames via QR Decomposition",
  "title_text": "Completion of Equiangular Tight Frames via QR Decomposition",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:29Z",
  "url": "https://research.draconitic.com/paper/248kmqnfgpchm",
  "venue": "Mathematics",
  "versions": []
}