{"id":"https://openalex.org/W2963816141","doi":"https://doi.org/10.1080/10586458.2018.1459962","title":"How Many Eigenvalues of a Product of Truncated Orthogonal Matrices are Real?","display_name":"How Many Eigenvalues of a Product of Truncated Orthogonal Matrices are Real?","publication_year":2018,"publication_date":"2018-05-09","ids":{"openalex":"https://openalex.org/W2963816141","doi":"https://doi.org/10.1080/10586458.2018.1459962","mag":"2963816141"},"language":"en","primary_location":{"id":"doi:10.1080/10586458.2018.1459962","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10586458.2018.1459962","pdf_url":null,"source":{"id":"https://openalex.org/S119636414","display_name":"Experimental Mathematics","issn_l":"1058-6458","issn":["1058-6458","1944-950X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Experimental Mathematics","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/A5019351501","display_name":"Peter J. Forrester","orcid":"https://orcid.org/0000-0002-9328-1387"},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"The University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]},{"id":"https://openalex.org/I4210123005","display_name":"ARC Centre of Excellence for Mathematical and Statistical Frontiers","ror":"https://ror.org/02vcqg248","country_code":"AU","type":"facility","lineage":["https://openalex.org/I1337719021","https://openalex.org/I165779595","https://openalex.org/I2801453606","https://openalex.org/I4210123005","https://openalex.org/I4210132349"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"P. J. Forrester","raw_affiliation_strings":["ARC Centre of Excellence for Mathematical and Statistical Frontiers, School of Mathematics and Statistics, The University of Melbourne, Victoria, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"ARC Centre of Excellence for Mathematical and Statistical Frontiers, School of Mathematics and Statistics, The University of Melbourne, Victoria, Australia","institution_ids":["https://openalex.org/I165779595","https://openalex.org/I4210123005"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5022135648","display_name":"J. R. Ipsen","orcid":"https://orcid.org/0000-0001-6577-6684"},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"The University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]},{"id":"https://openalex.org/I4210123005","display_name":"ARC Centre of Excellence for Mathematical and Statistical Frontiers","ror":"https://ror.org/02vcqg248","country_code":"AU","type":"facility","lineage":["https://openalex.org/I1337719021","https://openalex.org/I165779595","https://openalex.org/I2801453606","https://openalex.org/I4210123005","https://openalex.org/I4210132349"]}],"countries":["AU"],"is_corresponding":true,"raw_author_name":"J. R. Ipsen","raw_affiliation_strings":["ARC Centre of Excellence for Mathematical and Statistical Frontiers, School of Mathematics and Statistics, The University of Melbourne, Victoria, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"ARC Centre of Excellence for Mathematical and Statistical Frontiers, School of Mathematics and Statistics, The University of Melbourne, Victoria, Australia","institution_ids":["https://openalex.org/I165779595","https://openalex.org/I4210123005"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5086915639","display_name":"Santosh Kumar","orcid":"https://orcid.org/0000-0002-8405-7308"},"institutions":[{"id":"https://openalex.org/I26604189","display_name":"Shiv Nadar University","ror":"https://ror.org/05aqahr97","country_code":"IN","type":"education","lineage":["https://openalex.org/I26604189"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"S. Kumar","raw_affiliation_strings":["Department of Physics, Shiv Nadar University, Gautam Buddha Nagar, Uttar Pradesh, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics, Shiv Nadar University, Gautam Buddha Nagar, Uttar Pradesh, India","institution_ids":["https://openalex.org/I26604189"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5022135648"],"corresponding_institution_ids":["https://openalex.org/I165779595","https://openalex.org/I4210123005"],"apc_list":null,"apc_paid":null,"fwci":0.6324,"has_fulltext":false,"cited_by_count":16,"citation_normalized_percentile":{"value":0.72493521,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"29","issue":"3","first_page":"276","last_page":"290"},"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.9950000047683716,"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"}},{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9927999973297119,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8894249200820923},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.7201515436172485},{"id":"https://openalex.org/keywords/pfaffian","display_name":"Pfaffian","score":0.7155168056488037},{"id":"https://openalex.org/keywords/random-matrix","display_name":"Random matrix","score":0.6615356206893921},{"id":"https://openalex.org/keywords/orthogonal-matrix","display_name":"Orthogonal matrix","score":0.6210645437240601},{"id":"https://openalex.org/keywords/truncation","display_name":"Truncation (statistics)","score":0.5639007687568665},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5328208804130554},{"id":"https://openalex.org/keywords/complex-conjugate","display_name":"Complex conjugate","score":0.48953208327293396},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4546821415424347},{"id":"https://openalex.org/keywords/unit-circle","display_name":"Unit circle","score":0.4279887080192566},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.4168539047241211},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3836510181427002},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3109176754951477},{"id":"https://openalex.org/keywords/orthogonal-basis","display_name":"Orthogonal basis","score":0.20034539699554443},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10517182946205139}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8894249200820923},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.7201515436172485},{"id":"https://openalex.org/C2775956649","wikidata":"https://www.wikidata.org/wiki/Q1189744","display_name":"Pfaffian","level":2,"score":0.7155168056488037},{"id":"https://openalex.org/C64812099","wikidata":"https://www.wikidata.org/wiki/Q176604","display_name":"Random matrix","level":3,"score":0.6615356206893921},{"id":"https://openalex.org/C44292817","wikidata":"https://www.wikidata.org/wiki/Q333871","display_name":"Orthogonal matrix","level":3,"score":0.6210645437240601},{"id":"https://openalex.org/C106195933","wikidata":"https://www.wikidata.org/wiki/Q7847935","display_name":"Truncation (statistics)","level":2,"score":0.5639007687568665},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5328208804130554},{"id":"https://openalex.org/C45542137","wikidata":"https://www.wikidata.org/wiki/Q381040","display_name":"Complex conjugate","level":2,"score":0.48953208327293396},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4546821415424347},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.4279887080192566},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.4168539047241211},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3836510181427002},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3109176754951477},{"id":"https://openalex.org/C187064257","wikidata":"https://www.wikidata.org/wiki/Q3306808","display_name":"Orthogonal basis","level":2,"score":0.20034539699554443},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10517182946205139},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/10586458.2018.1459962","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10586458.2018.1459962","pdf_url":null,"source":{"id":"https://openalex.org/S119636414","display_name":"Experimental Mathematics","issn_l":"1058-6458","issn":["1058-6458","1944-950X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Experimental Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":46,"referenced_works":["https://openalex.org/W1546545543","https://openalex.org/W1579981923","https://openalex.org/W1582957647","https://openalex.org/W1596739703","https://openalex.org/W1597841999","https://openalex.org/W1764542523","https://openalex.org/W1802110487","https://openalex.org/W1999320979","https://openalex.org/W2000027618","https://openalex.org/W2001095588","https://openalex.org/W2005031553","https://openalex.org/W2037676563","https://openalex.org/W2058074009","https://openalex.org/W2064654110","https://openalex.org/W2068705582","https://openalex.org/W2072338329","https://openalex.org/W2080059175","https://openalex.org/W2082523211","https://openalex.org/W2093144809","https://openalex.org/W2094406305","https://openalex.org/W2101336297","https://openalex.org/W2119842426","https://openalex.org/W2275896185","https://openalex.org/W2293064825","https://openalex.org/W2346110991","https://openalex.org/W2421399543","https://openalex.org/W2473782587","https://openalex.org/W2480353178","https://openalex.org/W2962817727","https://openalex.org/W2962931600","https://openalex.org/W2963041888","https://openalex.org/W2963046116","https://openalex.org/W2964033018","https://openalex.org/W3098102273","https://openalex.org/W3098110976","https://openalex.org/W3098759207","https://openalex.org/W3100668510","https://openalex.org/W3100836770","https://openalex.org/W3101425201","https://openalex.org/W3102442796","https://openalex.org/W3102610798","https://openalex.org/W3103362295","https://openalex.org/W3103550897","https://openalex.org/W3106109711","https://openalex.org/W3106139507","https://openalex.org/W4248145103"],"related_works":["https://openalex.org/W4316128600","https://openalex.org/W3099762357","https://openalex.org/W2075947579","https://openalex.org/W2741223735","https://openalex.org/W3102442796","https://openalex.org/W4381743238","https://openalex.org/W4234710384","https://openalex.org/W977443235","https://openalex.org/W4298305744","https://openalex.org/W1970289613"],"abstract_inverted_index":{"A":[0],"truncation":[1,105],"of":[2,39,47,61,75,91,96,110,143,151],"a":[3,12,36,78],"Haar":[4,117],"distributed":[5,118],"orthogonal":[6,43,119],"random":[7,44],"matrix":[8,13,156],"gives":[9],"rise":[10],"to":[11,56,82,159],"whose":[14],"eigenvalues":[15,74,153],"are":[16,24],"either":[17],"real":[18,62,97,152],"or":[19],"complex":[20],"conjugate":[21],"pairs,":[22],"and":[23,112,147],"supported":[25],"within":[26],"the":[27,59,70,73,89,104,115,140,144,148,155],"closed":[28],"unit":[29],"disk.":[30],"This":[31],"is":[32,55],"also":[33],"true":[34],"for":[35,51,58,88,139],"product":[37],"Pm":[38,76],"m":[40],"independent":[41],"truncated":[42],"matrices.":[45],"One":[46],"most":[48],"basic":[49],"questions":[50],"such":[52],"asymmetric":[53],"matrices":[54],"ask":[57],"number":[60,95,109,150],"eigenvalues.":[63,98],"In":[64],"this":[65],"paper,":[66],"we":[67,134],"will":[68,100,124,135],"exploit":[69],"fact":[71],"that":[72,102],"form":[77,142],"Pfaffian":[79],"point":[80],"process":[81],"obtain":[83],"an":[84,107],"explicit":[85],"determinant":[86],"expression":[87],"probability":[90],"finding":[92],"any":[93],"given":[94],"We":[99],"see":[101],"if":[103],"removes":[106],"even":[108],"rows":[111],"columns":[113],"from":[114],"original":[116],"matrix,":[120],"then":[121],"these":[122],"probabilities":[123],"be":[125],"rational":[126],"numbers.":[127],"Finally,":[128],"based":[129],"on":[130],"exact":[131],"finite":[132],"formulas,":[133],"provide":[136],"conjectural":[137],"expressions":[138],"asymptotic":[141],"spectral":[145],"density":[146],"average":[149],"as":[154],"dimension":[157],"tends":[158],"infinity.":[160]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1}],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-10-10T00:00:00"}
