{"paper":{"title":"Condition Numbers of Gaussian Random Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"cs.NA","authors_text":"Jack Dongarra, Zizhong Chen","submitted_at":"2008-10-05T04:18:54Z","abstract_excerpt":"Let $G_{m \\times n}$ be an $m \\times n$ real random matrix whose elements are independent and identically distributed standard normal random variables, and let $\\kappa_2(G_{m \\times n})$ be the 2-norm condition number of $G_{m \\times n}$. We prove that, for any $m \\geq 2$, $n \\geq 2$ and $x \\geq |n-m|+1$, $\\kappa_2(G_{m \\times n})$ satisfies $\n  \\frac{1}{\\sqrt{2\\pi}} ({c}/{x})^{|n-m|+1} < P(\\frac{\\kappa_2(G_{m \\times n})} {{n}/{(|n-m|+1)}}> x) <\n  \\frac{1}{\\sqrt{2\\pi}} ({C}/{x})^{|n-m|+1}, $ where $0.245 \\leq c \\leq 2.000$ and $ 5.013 \\leq C \\leq 6.414$ are universal positive constants indepen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0810.0800","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0810.0800/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}