{"id":"https://openalex.org/W2025514573","doi":"https://doi.org/10.1137/090765018","title":"The Canonical Generalized Polar Decomposition","display_name":"The Canonical Generalized Polar Decomposition","publication_year":2010,"publication_date":"2010-01-01","ids":{"openalex":"https://openalex.org/W2025514573","doi":"https://doi.org/10.1137/090765018","mag":"2025514573"},"language":"en","primary_location":{"id":"doi:10.1137/090765018","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090765018","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/A5069339254","display_name":"Nicholas J. Higham","orcid":"https://orcid.org/0000-0001-5956-4976"},"institutions":[{"id":"https://openalex.org/I4210131439","display_name":"Applied Mathematics (United States)","ror":"https://ror.org/03seew607","country_code":"US","type":"company","lineage":["https://openalex.org/I4210131439"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Nicholas J. Higham","raw_affiliation_strings":["Applied Mathematics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Applied Mathematics","institution_ids":["https://openalex.org/I4210131439"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103132248","display_name":"Christian Mehl","orcid":"https://orcid.org/0000-0003-2146-4974"},"institutions":[{"id":"https://openalex.org/I28407311","display_name":"University of Manchester","ror":"https://ror.org/027m9bs27","country_code":"GB","type":"education","lineage":["https://openalex.org/I28407311"]},{"id":"https://openalex.org/I4577782","display_name":"Technische Universit\u00e4t Berlin","ror":"https://ror.org/03v4gjf40","country_code":"DE","type":"education","lineage":["https://openalex.org/I4577782"]}],"countries":["DE","GB"],"is_corresponding":false,"raw_author_name":"Christian Mehl","raw_affiliation_strings":["School of Mathematics The University of Manchester Manchester, M13 9PL, UK","Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester",", Technische Universit\u00e4t Berlin, 10623 Berlin, Germany","School of Mathematics, The University of Manchester, Manchester, M13 9PL, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics The University of Manchester Manchester, M13 9PL, UK","institution_ids":["https://openalex.org/I28407311"]},{"raw_affiliation_string":"Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester","institution_ids":["https://openalex.org/I28407311"]},{"raw_affiliation_string":", Technische Universit\u00e4t Berlin, 10623 Berlin, Germany","institution_ids":["https://openalex.org/I4577782"]},{"raw_affiliation_string":"School of Mathematics, The University of Manchester, Manchester, M13 9PL, UK","institution_ids":["https://openalex.org/I28407311"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5059791596","display_name":"Fran\u00e7oise Tisseur","orcid":"https://orcid.org/0000-0002-1011-2570"},"institutions":[{"id":"https://openalex.org/I4210131439","display_name":"Applied Mathematics (United States)","ror":"https://ror.org/03seew607","country_code":"US","type":"company","lineage":["https://openalex.org/I4210131439"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Fran\u00e7oise Tisseur","raw_affiliation_strings":["Applied Mathematics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Applied Mathematics","institution_ids":["https://openalex.org/I4210131439"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.7483,"has_fulltext":false,"cited_by_count":37,"citation_normalized_percentile":{"value":0.89857645,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"31","issue":"4","first_page":"2163","last_page":"2180"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994000196456909,"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"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994000196456909,"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/T11673","display_name":"Advanced Topics in Algebra","score":0.9948999881744385,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9925000071525574,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.813228964805603},{"id":"https://openalex.org/keywords/polar-decomposition","display_name":"Polar decomposition","score":0.8125585317611694},{"id":"https://openalex.org/keywords/invertible-matrix","display_name":"Invertible matrix","score":0.6374192237854004},{"id":"https://openalex.org/keywords/hermitian-matrix","display_name":"Hermitian matrix","score":0.6197625994682312},{"id":"https://openalex.org/keywords/sesquilinear-form","display_name":"Sesquilinear form","score":0.5820394158363342},{"id":"https://openalex.org/keywords/isometry","display_name":"Isometry (Riemannian geometry)","score":0.5568592548370361},{"id":"https://openalex.org/keywords/bilinear-form","display_name":"Bilinear form","score":0.5555239915847778},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.5488557815551758},{"id":"https://openalex.org/keywords/scalar","display_name":"Scalar (mathematics)","score":0.5170534253120422},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4679625928401947},{"id":"https://openalex.org/keywords/polar","display_name":"Polar","score":0.45259618759155273},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4476338326931},{"id":"https://openalex.org/keywords/canonical-form","display_name":"Canonical form","score":0.4267123341560364},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.41687777638435364},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.41371122002601624},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.41243818402290344},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.22613561153411865},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07668420672416687},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07467272877693176}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.813228964805603},{"id":"https://openalex.org/C130956294","wikidata":"https://www.wikidata.org/wiki/Q2101158","display_name":"Polar decomposition","level":3,"score":0.8125585317611694},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.6374192237854004},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.6197625994682312},{"id":"https://openalex.org/C163353815","wikidata":"https://www.wikidata.org/wiki/Q1931224","display_name":"Sesquilinear form","level":3,"score":0.5820394158363342},{"id":"https://openalex.org/C82457910","wikidata":"https://www.wikidata.org/wiki/Q740207","display_name":"Isometry (Riemannian geometry)","level":2,"score":0.5568592548370361},{"id":"https://openalex.org/C8828549","wikidata":"https://www.wikidata.org/wiki/Q837924","display_name":"Bilinear form","level":2,"score":0.5555239915847778},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5488557815551758},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.5170534253120422},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4679625928401947},{"id":"https://openalex.org/C29705727","wikidata":"https://www.wikidata.org/wiki/Q294562","display_name":"Polar","level":2,"score":0.45259618759155273},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4476338326931},{"id":"https://openalex.org/C204707403","wikidata":"https://www.wikidata.org/wiki/Q1152398","display_name":"Canonical form","level":2,"score":0.4267123341560364},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.41687777638435364},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.41371122002601624},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.41243818402290344},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.22613561153411865},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07668420672416687},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07467272877693176},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C1276947","wikidata":"https://www.wikidata.org/wiki/Q333","display_name":"Astronomy","level":1,"score":0.0}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1137/090765018","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090765018","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:pure.atira.dk:openaire_cris_publications/87272ac4-a4c5-4d77-979a-3e6ddc134fb9","is_oa":false,"landing_page_url":"https://research.manchester.ac.uk/en/publications/87272ac4-a4c5-4d77-979a-3e6ddc134fb9","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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Higham, N J, Mehl, C & Tisseur, F 2009, 'The canonical generalized polar decomposition', SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 4, pp. 2163-2180. https://doi.org/10.1137/090765018","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1293","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306400452","display_name":"MIMS EPrints (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"MIMS Preprint"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.151.8194","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.151.8194","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://eprints.ma.man.ac.uk/1293/01/covered/MIMS_ep2009_52.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.230.6516","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.230.6516","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://eprints.ma.man.ac.uk/1490/01/covered/MIMS_ep2009_52.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.332.7186","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.332.7186","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://page.math.tu-berlin.de/~mehl/papers/hmt1.pdf","raw_type":"text"},{"id":"pmh:oai:pure.atira.dk:publications/87272ac4-a4c5-4d77-979a-3e6ddc134fb9","is_oa":false,"landing_page_url":"https://www.research.manchester.ac.uk/portal/en/publications/the-canonical-generalized-polar-decomposition(87272ac4-a4c5-4d77-979a-3e6ddc134fb9).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":"Higham, N J, Mehl, C & Tisseur, F 2009, 'The canonical generalized polar decomposition', SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 4, pp. 2163-2180. https://doi.org/10.1137/090765018","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W135515345","https://openalex.org/W1483804921","https://openalex.org/W1976734466","https://openalex.org/W1981467185","https://openalex.org/W1995343004","https://openalex.org/W2020054862","https://openalex.org/W2022846189","https://openalex.org/W2053064610","https://openalex.org/W2069202710","https://openalex.org/W2088307891","https://openalex.org/W2089243963","https://openalex.org/W2090379260","https://openalex.org/W2095960807","https://openalex.org/W2102621489","https://openalex.org/W2104914078","https://openalex.org/W2112145895","https://openalex.org/W2137634211","https://openalex.org/W2150157601","https://openalex.org/W2610857016","https://openalex.org/W4206092473","https://openalex.org/W4237067400"],"related_works":["https://openalex.org/W2254382463","https://openalex.org/W1981503134","https://openalex.org/W2764379562","https://openalex.org/W2058040513","https://openalex.org/W100702267","https://openalex.org/W2544247039","https://openalex.org/W3110913837","https://openalex.org/W2951968373","https://openalex.org/W2974458704","https://openalex.org/W2025514573"],"abstract_inverted_index":{"The":[0],"polar":[1,52,151,165],"decomposition":[2,53,105,113,152,166],"of":[3,35,42,124,130,144,157,190],"a":[4,22,43,67,103,162],"square":[5,36,173],"matrix":[6,115],"has":[7,29],"been":[8],"generalized":[9,51,138,150,164],"by":[10,21,114],"several":[11],"authors":[12],"to":[13,110,153],"scalar":[14,45,91,131],"products":[15,92],"on":[16,32,93],"$\\mathbb{R}^n$":[17],"or":[18,24],"$\\mathbb{C}^n$":[19],"given":[20],"bilinear":[23],"sesquilinear":[25],"form.":[26],"Previous":[27],"work":[28],"focused":[30],"mainly":[31],"the":[33,40,49,79,84,112,145,148,188,191],"case":[34],"matrices,":[37],"sometimes":[38],"with":[39,74],"assumption":[41],"Hermitian":[44],"product.":[46],"We":[47,98,141,159],"introduce":[48],"canonical":[50,149],"$A":[54,167],"=":[55,168],"WS$,":[56,169],"defined":[57,170],"for":[58,172],"general":[59],"$m\\times":[60],"n$":[61],"matrices":[62,86,174],"A,":[63],"where":[64],"W":[65,179],"is":[66,72,180,196],"partial":[68,125],"$(M,N)$-isometry":[69],"and":[70,83,88,95,107,120,127,133,176,187],"S":[71],"N-selfadjoint":[73],"nonzero":[75],"eigenvalues":[76],"lying":[77],"in":[78,147,177],"open":[80],"right":[81],"half-plane,":[82],"nonsingular":[85],"M":[87],"N":[89],"define":[90],"$\\mathbb{C}^m$":[94],"$\\mathbb{C}^n$,":[96],"respectively.":[97],"derive":[99],"conditions":[100],"under":[101],"which":[102,178],"unique":[104],"exists":[106],"show":[108],"how":[109],"compute":[111],"iterations.":[116],"Our":[117],"treatment":[118],"derives":[119],"exploits":[121],"key":[122],"properties":[123],"$(M,N)$-isometries":[126],"orthosymmetric":[128],"pairs":[129],"products,":[132],"also":[134,160],"employs":[135],"an":[136,154,181],"appropriate":[137,155],"Moore\u2013Penrose":[139],"pseudoinverse.":[140],"relate":[142],"commutativity":[143],"factors":[146],"definition":[156],"normality.":[158],"consider":[161],"related":[163],"only":[171],"A":[175,195],"automorphism;":[182],"we":[183],"analyze":[184],"its":[185],"existence":[186],"uniqueness":[189],"selfadjoint":[192],"factor":[193],"when":[194],"singular.":[197]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":4},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":3}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
