{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/uniqueness-of-size-2-positive-semidefinite","title":"Uniqueness of size-2 positive semidefinite matrix factorizations","arxiv_id":"2410.18891","date":"2024-10-24","proceeding":null,"authors":["Kristen Dawson","Serkan Hoşten","Kaie Kubjas","Lilja Metsälampi"],"abstract":"We characterize when a size-2 positive semidefinite (psd) factorization of a positive matrix of rank 3 and psd rank 2 is unique. The characterization is obtained using tools from rigidity theory. In the first step, we define s-infinitesimally rigid psd factorizations and characterize 1- and 2-infinitesimally rigid size-2 psd factorizations. In the second step, we connect 1- and 2-infinitesimal rigidity of size-2 psd factorizations to uniqueness via global rigidity. We also prove necessary conditions on a positive matrix of rank 3 and psd rank 2 to be on the topological boundary of all nonnegative matrices with the same rank conditions.","url_abs":"https://arxiv.org/abs/2410.18891v1","url_pdf":"https://arxiv.org/pdf/2410.18891v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"uniqueness-of-size-2-positive-semidefinite","repo_url":"https://github.com/kaiekubjas/uniqueness-of-size-2-psd-factorizations","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}