{"paper":{"title":"A geometric note on subspace updates and orthogonal matrix decompositions under rank-one modifications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Ralf Zimmermann","submitted_at":"2017-11-22T11:33:07Z","abstract_excerpt":"In this work, we consider rank-one adaptations $X_{new} = X+ab^T$ of a given matrix $X\\in \\mathbb{R}^{n\\times p}$ with known matrix factorization $X = UW$, where $U\\in\\mathbb{R}^{n\\times p}$ is column-orthogonal, i.e. $U^TU=I$. Arguably the most important methods that produce such factorizations are the singular value decomposition (SVD), where $X=UW=U\\Sigma V^T$, and the QR-decomposition, where $X = UW = QR$. An elementary approach to produce a column-orthogonal matrix $U_{new}$, whose columns span the same subspace as the columns of the rank-one modified $X_{new} = X +ab^T$ is via applying a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.08235","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}