{"id":"https://openalex.org/W2901185990","doi":"https://doi.org/10.1137/18m1232462","title":"Schur's Lemma for Coupled Reducibility and Coupled Normality","display_name":"Schur's Lemma for Coupled Reducibility and Coupled Normality","publication_year":2019,"publication_date":"2019-01-01","ids":{"openalex":"https://openalex.org/W2901185990","doi":"https://doi.org/10.1137/18m1232462","mag":"2901185990"},"language":"en","primary_location":{"id":"doi:10.1137/18m1232462","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1232462","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":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1811.08467","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057160666","display_name":"Dana Lahat","orcid":"https://orcid.org/0000-0001-6179-0219"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Dana Lahat","raw_affiliation_strings":["Signal et Communications"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Signal et Communications","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5055562421","display_name":"Christian Jutten","orcid":"https://orcid.org/0000-0002-4477-4847"},"institutions":[{"id":"https://openalex.org/I106785703","display_name":"Institut polytechnique de Grenoble","ror":"https://ror.org/05sbt2524","country_code":"FR","type":"education","lineage":["https://openalex.org/I106785703","https://openalex.org/I899635006"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Christian Jutten","raw_affiliation_strings":["Institut polytechnique de Grenoble - Grenoble Institute of Technology"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut polytechnique de Grenoble - Grenoble Institute of Technology","institution_ids":["https://openalex.org/I106785703"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5008910180","display_name":"Helene Shapiro","orcid":null},"institutions":[{"id":"https://openalex.org/I118020396","display_name":"Swarthmore College","ror":"https://ror.org/012dg8a96","country_code":"US","type":"education","lineage":["https://openalex.org/I118020396"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Helene Shapiro","raw_affiliation_strings":["Department of Mathematics and Statistics, Swarthmore College"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Swarthmore College","institution_ids":["https://openalex.org/I118020396"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.06141634,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"40","issue":"3","first_page":"998","last_page":"1021"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11673","display_name":"Advanced Topics in Algebra","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11673","display_name":"Advanced Topics in Algebra","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9962000250816345,"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/combinatorics","display_name":"Combinatorics","score":0.7756302356719971},{"id":"https://openalex.org/keywords/schurs-lemma","display_name":"Schur's lemma","score":0.7714778184890747},{"id":"https://openalex.org/keywords/irreducibility","display_name":"Irreducibility","score":0.7093977332115173},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.6903638243675232},{"id":"https://openalex.org/keywords/lemma","display_name":"Lemma (botany)","score":0.5357064008712769},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.526751697063446},{"id":"https://openalex.org/keywords/vector-space","display_name":"Vector space","score":0.43829160928726196},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.11451882123947144}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7756302356719971},{"id":"https://openalex.org/C158385407","wikidata":"https://www.wikidata.org/wiki/Q1816952","display_name":"Schur's lemma","level":5,"score":0.7714778184890747},{"id":"https://openalex.org/C2776823524","wikidata":"https://www.wikidata.org/wiki/Q6073561","display_name":"Irreducibility","level":2,"score":0.7093977332115173},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.6903638243675232},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.5357064008712769},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.526751697063446},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.43829160928726196},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.11451882123947144},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.0},{"id":"https://openalex.org/C46757340","wikidata":"https://www.wikidata.org/wiki/Q43238","display_name":"Poaceae","level":2,"score":0.0},{"id":"https://openalex.org/C78540521","wikidata":"https://www.wikidata.org/wiki/Q1498246","display_name":"Gegenbauer polynomials","level":4,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1137/18m1232462","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1232462","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:arXiv.org:1811.08467","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1811.08467","pdf_url":"https://arxiv.org/pdf/1811.08467","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"mag:2901185990","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1811.08467.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:works.swarthmore.edu:fac-math-stat-1256","is_oa":false,"landing_page_url":"https://works.swarthmore.edu/fac-math-stat/257","pdf_url":null,"source":{"id":"https://openalex.org/S4377196506","display_name":"Works - Scholarship, Research, & Creative Expression (Swarthmore College)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I118020396","host_organization_name":"Swarthmore College","host_organization_lineage":["https://openalex.org/I118020396"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics &amp; Statistics Faculty Works","raw_type":"text"},{"id":"pmh:oai:HAL:hal-02305142v1","is_oa":false,"landing_page_url":"https://hal.science/hal-02305142","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications, 2019, 40 (3), pp.998-1021. &#x27E8;10.1137/18M1232462&#x27E9;","raw_type":"Journal articles"},{"id":"doi:10.48550/arxiv.1811.08467","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1811.08467","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1811.08467","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1811.08467","pdf_url":"https://arxiv.org/pdf/1811.08467","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5175147346","display_name":null,"funder_award_id":"FACTORY-CoG-6681839","funder_id":"https://openalex.org/F4320338335","funder_display_name":"H2020 European Research Council"},{"id":"https://openalex.org/G6175499886","display_name":"Systemes et Algorithmes Pervasifs au confluent des mondes physique et num\u00e9rique","funder_award_id":"ANR-11-LABX-0025","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"},{"id":"https://openalex.org/G8832998507","display_name":null,"funder_award_id":"ANR11-LABX-0025","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"}],"funders":[{"id":"https://openalex.org/F4320320883","display_name":"Agence Nationale de la Recherche","ror":"https://ror.org/00rbzpz17"},{"id":"https://openalex.org/F4320338335","display_name":"H2020 European Research Council","ror":"https://ror.org/0472cxd90"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":9,"referenced_works":["https://openalex.org/W1539811621","https://openalex.org/W1568175246","https://openalex.org/W2000045479","https://openalex.org/W2085991972","https://openalex.org/W2099741732","https://openalex.org/W2242568341","https://openalex.org/W2401909077","https://openalex.org/W2621428331","https://openalex.org/W2900782715"],"related_works":["https://openalex.org/W2885964212","https://openalex.org/W2608961613","https://openalex.org/W3119106883","https://openalex.org/W3212366520","https://openalex.org/W3183290475","https://openalex.org/W3167361675","https://openalex.org/W2024119920","https://openalex.org/W2890705220","https://openalex.org/W2067076293","https://openalex.org/W2158230504","https://openalex.org/W2803656990","https://openalex.org/W1966078622","https://openalex.org/W2153399488","https://openalex.org/W2952906647","https://openalex.org/W2186074037","https://openalex.org/W3120686177","https://openalex.org/W2083241480","https://openalex.org/W1588214017","https://openalex.org/W2918409046","https://openalex.org/W1971880286"],"abstract_inverted_index":{"Let":[0,99],"$\\mathcal":[1,11,37,46,58,64,77,100,156,182],"A":[2],"=":[3,102,142],"\\{A_{ij}":[4],"\\}_{i,":[5,104],"j":[6,105],"\\in":[7,33,73,106,126],"\\mathcal":[8,34,61,74,80,93,107,127,158,184],"I}$,":[9],"where":[10,24,117],"I$":[12],"is":[13,26,48,119],"an":[14,40],"index":[15],"set,":[16],"be":[17,39,110,131],"a":[18,111,132,167],"doubly":[19,112],"indexed":[20,113],"family":[21,114],"of":[22,115,134,152,169,174],"matrices,":[23,116],"$A_{ij}$":[25],"$n_i":[27,136],"\\times":[28,121,137],"n_j$.":[29],"For":[30,123],"each":[31,124],"$i":[32,72,125],"I$,":[35,75,128],"let":[36,129],"V_i$":[38,81],"$n_i$-dimensional":[41],"vector":[42],"space.":[43],"We":[44,149,164],"say":[45],"A$":[47],"reducible":[49],"in":[50],"the":[51],"coupled":[52,161,187,190],"sense":[53],"if":[54],"there":[55],"exist":[56],"subspaces,":[57],"U_i":[59,65,78],"\\subseteq":[60,92],"V_i$,":[62],"with":[63],"\\neq":[66,79],"\\{0\\}$":[67],"for":[68,82,95,145,155,172,181],"at":[69,83],"least":[70,84],"one":[71,85],"and":[76,177,189],"$i$,":[86],"such":[87],"that":[88],"$A_{ij}":[89,140],"(\\mathcal":[90],"U_j)":[91],"U_i$":[94],"all":[96,146],"$i,":[97,147],"j$.":[98,148],"B":[101],"\\{B_{ij}":[103],"I}$":[108],"also":[109,165],"$B_{ij}$":[118],"$m_i":[120],"m_j$.":[122],"$X_i$":[130],"matrix":[133],"size":[135],"m_i$.":[138],"Suppose":[139],"X_j":[141],"X_i":[143],"B_{ij}$":[144],"prove":[150,178],"versions":[151,180],"Schur's":[153,170],"lemma":[154,171],"A,":[157,183],"B$":[159,185],"satisfying":[160,186],"irreducibility":[162,191],"conditions.":[163,192],"consider":[166],"refinement":[168],"sets":[173],"normal":[175],"matrices":[176],"corresponding":[179],"normality":[188]},"counts_by_year":[{"year":2018,"cited_by_count":1}],"updated_date":"2026-06-22T08:00:12.763002","created_date":"2025-10-10T00:00:00"}
