{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:26:59Z","timestamp":1787336819000,"version":"build-2736575974"},"reference-count":13,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2005,1]]},"abstract":"<jats:p>Let $\\kappa$ be the condition number of an m-by-n matrix with independent standard Gaussian entries, either real ($\\beta = 1$) or complex ($\\beta = 2$). The major result is the existence of a constant C (depending on m, n, and $\\beta$) such that $P[\\kappa &gt; x] &lt; C \\, x^{-\\beta}$ for all x. As $x \\rightarrow \\infty$, the bound is asymptotically tight. An analytic expression is given for the constant C, and simple estimates are given, one involving a Tracy--Widom largest eigenvalue distribution. All of the results extend beyond real and complex entries to general $\\beta$.<\/jats:p>","DOI":"10.1137\/040614256","type":"journal-article","created":{"date-parts":[[2005,12,5]],"date-time":"2005-12-05T22:29:59Z","timestamp":1133821799000},"page":"547-560","source":"Crossref","is-referenced-by-count":40,"title":["Tails of Condition Number Distributions"],"prefix":"10.1137","volume":"27","author":[{"given":"Alan","family":"Edelman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Brian D.","family":"Sutton","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479803429764"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1988-0929546-7"},{"key":"R3","unstructured":"I. Dumitriu,\n                      Eigenvalue Statistics for Beta\u2010Ensembles\n                      , Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 2003."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1063\/1.1507823"},{"key":"R5","unstructured":"A. Edelman,\n                      Eigenvalues and Condition Numbers of Random Matrices\n                      , Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 1989."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1009210543"},{"key":"R7","unstructured":"P. Koev and A. Edelman,\n                      Efficient evaluationof the hypergeometric function of matrix argument\n                      , Math. Comp., submitted."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316559"},{"key":"R9","unstructured":"P.\u2010O. Persson,\n                      Numerical Methods for Eigenvalue Distributions of Random Matrices\n                      , http:\/\/arxiv.org\/abs\/math\u2010ph\/0501068 (19 March 2005)."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176992819"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1985-15391-1"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"Craig Tracy, Harold Widom, Airy kernel and Painlev\u00e9 II, CRM Proc. Lecture Notes, Vol. 31, Amer. Math. Soc., Providence, RI, 2002, 85\u2013962003k:82038","DOI":"10.1090\/crmp\/031\/07"},{"key":"R13","volume-title":"Collected works. Vol. V: Design of computers, theory of automata and numerical analysis","author":"von Neumann John","year":"1963"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/040614256","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:32:07Z","timestamp":1787333527000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/040614256"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,1]]},"references-count":13,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2005,1]]}},"alternative-id":["10.1137\/040614256"],"URL":"https:\/\/doi.org\/10.1137\/040614256","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,1]]}}}