{"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/on-the-condition-number-of-the-shifted-real","title":"On the condition number of the shifted real Ginibre ensemble","arxiv_id":"2105.13719","date":"2021-05-28","proceeding":null,"authors":["Giorgio Cipolloni","László Erdős","Dominik Schröder"],"abstract":"We derive an accurate lower tail estimate on the lowest singular value $\\sigma_1(X-z)$ of a real Gaussian (Ginibre) random matrix $X$ shifted by a complex parameter $z$. Such shift effectively changes the upper tail behaviour of the condition number $\\kappa(X-z)$ from the slower $\\mathbf{P}(\\kappa(X-z)\\ge t)\\lesssim 1/t$ decay typical for real Ginibre matrices to the faster $1/t^2$ decay seen for complex Ginibre matrices as long as $z$ is away from the real axis. This sharpens and resolves a recent conjecture in [arXiv:2005.08930] on the regularizing effect of the real Ginibre ensemble with a genuinely complex shift. As a consequence we obtain an improved upper bound on the eigenvalue condition numbers (known also as the eigenvector overlaps) for real Ginibre matrices. The main technical tool is a rigorous supersymmetric analysis from our earlier work [arXiv:1908.01653].","url_abs":"https://arxiv.org/abs/2105.13719v3","url_pdf":"https://arxiv.org/pdf/2105.13719v3.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":"on-the-condition-number-of-the-shifted-real","repo_url":"https://github.com/wirhabenzeit/conditionNumber","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"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}