{"id":"https://openalex.org/W2130076109","doi":"https://doi.org/10.1137/040616413","title":"Condition Numbers of Gaussian Random Matrices","display_name":"Condition Numbers of Gaussian Random Matrices","publication_year":2005,"publication_date":"2005-01-01","ids":{"openalex":"https://openalex.org/W2130076109","doi":"https://doi.org/10.1137/040616413","mag":"2130076109"},"language":"en","primary_location":{"id":"doi:10.1137/040616413","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040616413","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5061737717","display_name":"Zizhong Chen","orcid":"https://orcid.org/0000-0003-2578-4940"},"institutions":[{"id":"https://openalex.org/I75027704","display_name":"University of Tennessee at Knoxville","ror":"https://ror.org/020f3ap87","country_code":"US","type":"education","lineage":["https://openalex.org/I75027704"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Zizhong Chen","raw_affiliation_strings":["Department of Computer Science, The University of Tennessee, 203 Claxton Complex, 1122 Volunteer Boulevard, Knoxville, TN, 37996-3450"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science, The University of Tennessee, 203 Claxton Complex, 1122 Volunteer Boulevard, Knoxville, TN, 37996-3450","institution_ids":["https://openalex.org/I75027704"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5075517045","display_name":"Jack Dongarra","orcid":"https://orcid.org/0000-0003-3247-1782"},"institutions":[{"id":"https://openalex.org/I75027704","display_name":"University of Tennessee at Knoxville","ror":"https://ror.org/020f3ap87","country_code":"US","type":"education","lineage":["https://openalex.org/I75027704"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jack J. Dongarra","raw_affiliation_strings":["Department of Computer Science, The University of Tennessee, 203 Claxton Complex, 1122 Volunteer Boulevard, Knoxville, TN, 37996-3450"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science, The University of Tennessee, 203 Claxton Complex, 1122 Volunteer Boulevard, Knoxville, TN, 37996-3450","institution_ids":["https://openalex.org/I75027704"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5061737717"],"corresponding_institution_ids":["https://openalex.org/I75027704"],"apc_list":null,"apc_paid":null,"fwci":4.0232,"has_fulltext":false,"cited_by_count":224,"citation_normalized_percentile":{"value":0.93861473,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":100},"biblio":{"volume":"27","issue":"3","first_page":"603","last_page":"620"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9934999942779541,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11680","display_name":"Advanced Algebra and Geometry","score":0.9912999868392944,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7759684324264526},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7224675416946411},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.5530354976654053},{"id":"https://openalex.org/keywords/independent-and-identically-distributed-random-variables","display_name":"Independent and identically distributed random variables","score":0.5400048494338989},{"id":"https://openalex.org/keywords/random-matrix","display_name":"Random matrix","score":0.4136776924133301},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.15236902236938477},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.14945045113563538},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.07975298166275024},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.05891701579093933}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7759684324264526},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7224675416946411},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.5530354976654053},{"id":"https://openalex.org/C141513077","wikidata":"https://www.wikidata.org/wiki/Q378542","display_name":"Independent and identically distributed random variables","level":3,"score":0.5400048494338989},{"id":"https://openalex.org/C64812099","wikidata":"https://www.wikidata.org/wiki/Q176604","display_name":"Random matrix","level":3,"score":0.4136776924133301},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.15236902236938477},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.14945045113563538},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.07975298166275024},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.05891701579093933}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1137/040616413","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040616413","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.244.4547","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.244.4547","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0810.0800v1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.67.2210","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.67.2210","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://icl.cs.utk.edu/news_pub/submissions/condition_numbers_of_gaussian_random_matrices.pdf","raw_type":"text"},{"id":"pmh:oai:pure.atira.dk:openaire_cris_publications/1360ba12-a96f-420b-b7c2-93b21490149c","is_oa":false,"landing_page_url":"https://research.manchester.ac.uk/en/publications/1360ba12-a96f-420b-b7c2-93b21490149c","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Chen, Z & Dongarra, J J 2005, 'Condition numbers of Gaussian random matrices', SIAM Journal on Matrix Analysis and Applications, vol. 27, no. 3, pp. 603-620. https://doi.org/10.1137/040616413","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:pure.atira.dk:publications/1360ba12-a96f-420b-b7c2-93b21490149c","is_oa":false,"landing_page_url":"https://www.research.manchester.ac.uk/portal/en/publications/condition-numbers-of-gaussian-random-matrices(1360ba12-a96f-420b-b7c2-93b21490149c).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Chen, Z & Dongarra, J J 2005, 'Condition numbers of Gaussian random matrices', SIAM Journal on Matrix Analysis and Applications, vol. 27, no. 3, pp. 603-620. https://doi.org/10.1137/040616413","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1645119126","display_name":null,"funder_award_id":"AC05-00OR22725","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G1719536385","display_name":null,"funder_award_id":"DE-AC05-00OR22725","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G5726405315","display_name":null,"funder_award_id":"DE-AC05","funder_id":"https://openalex.org/F4320306250","funder_display_name":"Battelle"},{"id":"https://openalex.org/G6129992089","display_name":null,"funder_award_id":"DE-AC05-","funder_id":"https://openalex.org/F4320316892","funder_display_name":"UT-Battelle"},{"id":"https://openalex.org/G6864165199","display_name":null,"funder_award_id":"DE-AC05-00OR22725","funder_id":"https://openalex.org/F4320306250","funder_display_name":"Battelle"},{"id":"https://openalex.org/G7995982022","display_name":null,"funder_award_id":"DE-AC05","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G8813984943","display_name":null,"funder_award_id":"DE-AC05-00OR22725","funder_id":"https://openalex.org/F4320316892","funder_display_name":"UT-Battelle"},{"id":"https://openalex.org/G8906985441","display_name":null,"funder_award_id":"00OR22725","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G8943143067","display_name":null,"funder_award_id":"AC05-00OR22725","funder_id":"https://openalex.org/F4320316892","funder_display_name":"UT-Battelle"}],"funders":[{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"},{"id":"https://openalex.org/F4320306250","display_name":"Battelle","ror":"https://ror.org/01h5tnr73"},{"id":"https://openalex.org/F4320316892","display_name":"UT-Battelle","ror":"https://ror.org/04nza6677"},{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":8,"referenced_works":["https://openalex.org/W1970821516","https://openalex.org/W1974637548","https://openalex.org/W1974755392","https://openalex.org/W2018233872","https://openalex.org/W2018239771","https://openalex.org/W2052645364","https://openalex.org/W2091849156","https://openalex.org/W2128854702"],"related_works":["https://openalex.org/W2347733609","https://openalex.org/W2066502241","https://openalex.org/W2605475288","https://openalex.org/W2292976891","https://openalex.org/W4245457074","https://openalex.org/W2128683635","https://openalex.org/W4240383795","https://openalex.org/W2008809748","https://openalex.org/W2140857733","https://openalex.org/W2951620698"],"abstract_inverted_index":{"Let":[0],"$G_{m":[1,34],"\\times":[2,7,26,35,53,65,112],"n}$":[3],"be":[4,28],"an":[5],"$m":[6,42,104],"n$":[8],"real":[9],"random":[10,21,127],"matrix":[11],"whose":[12],"elements":[13],"are":[14,91],"independent":[15,95],"and":[16,23,48,85,99,107],"identically":[17],"distributed":[18],"standard":[19],"normal":[20],"variables,":[22],"let":[24],"$\\kappa_2(G_{m":[25,52],"n})$":[27,54],"the":[29],"2-norm":[30],"condition":[31],"number":[32],"of":[33,96,122],"n}$.":[36],"We":[37],"prove":[38],"that,":[39],"for":[40,102,124],"any":[41,103],"\\geq":[43,46,50,105,109],"2$,":[44,47,110],"$n":[45,108],"$x":[49],"|n-m|+1$,":[51],"satisfies":[55],"${\\scriptsize":[56],"\\frac{1}{\\sqrt{2\\pi}}}":[57,73],"(":[58,74],"{":[59,75],"c":[60,82],"}/{x}":[61,77],")^{|n-m|+1}":[62],"<":[63,71,114],"P{\\scriptsize(\\frac{\\kappa_2(G_{m":[64],"n})}":[66],"{{n}/{(|n-m|+1)}}":[67],">":[68],"x":[69],")}":[70],"{\\scriptsize":[72],"C":[76,88],")^{|n-m|+1},$":[78],"where":[79],"$0.245":[80],"\\leq":[81,83,89],"2.000$":[84],"$5.013$":[86],"$\\leq":[87],"6.414$":[90],"universal":[92],"positive":[93],"constants":[94],"m,":[97],"n,":[98],"x.":[100],"Moreover,":[101],"2$":[106],"$E(\\log\\kappa_2(G_{m":[111],"n}))":[113],"\\log{\\scriptsize":[115],"\\frac{n}{|n-m|+1}}":[116],"+":[117],"2.258.$":[118],"A":[119],"similar":[120],"pair":[121],"results":[123],"complex":[125],"Gaussian":[126],"matrices":[128],"is":[129],"also":[130],"established.":[131]},"counts_by_year":[{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":9},{"year":2021,"cited_by_count":17},{"year":2020,"cited_by_count":20},{"year":2019,"cited_by_count":14},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":9},{"year":2016,"cited_by_count":18},{"year":2015,"cited_by_count":17},{"year":2014,"cited_by_count":15},{"year":2013,"cited_by_count":19},{"year":2012,"cited_by_count":16}],"updated_date":"2026-07-13T07:31:44.756512","created_date":"2025-10-10T00:00:00"}
