{"id":"https://openalex.org/W2101137939","doi":"https://doi.org/10.1137/080721406","title":"Definite Matrix Polynomials and their Linearization by Definite Pencils","display_name":"Definite Matrix Polynomials and their Linearization by Definite Pencils","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2101137939","doi":"https://doi.org/10.1137/080721406","mag":"2101137939"},"language":"en","primary_location":{"id":"doi:10.1137/080721406","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080721406","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/A5014969883","display_name":"D. Steven Mackey","orcid":"https://orcid.org/0000-0001-7906-6317"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"D. Steven Mackey","raw_affiliation_strings":["steve.mackey@wmich.edu#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"steve.mackey@wmich.edu#TAB#","institution_ids":[]}]},{"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":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.5453,"has_fulltext":false,"cited_by_count":35,"citation_normalized_percentile":{"value":0.92811888,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"31","issue":"2","first_page":"478","last_page":"502"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9998999834060669,"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":0.9998999834060669,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9911999702453613,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.8931032419204712},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8336296081542969},{"id":"https://openalex.org/keywords/positive-definiteness","display_name":"Positive definiteness","score":0.7077156901359558},{"id":"https://openalex.org/keywords/hermitian-matrix","display_name":"Hermitian matrix","score":0.6912558078765869},{"id":"https://openalex.org/keywords/matrix-polynomial","display_name":"Matrix polynomial","score":0.688134491443634},{"id":"https://openalex.org/keywords/polynomial-matrix","display_name":"Polynomial matrix","score":0.6659424304962158},{"id":"https://openalex.org/keywords/linearization","display_name":"Linearization","score":0.5938011407852173},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.49055641889572144},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4838339388370514},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4608108699321747},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.44834983348846436},{"id":"https://openalex.org/keywords/interlacing","display_name":"Interlacing","score":0.4224308729171753},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.25525373220443726},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.09820112586021423}],"concepts":[{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.8931032419204712},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8336296081542969},{"id":"https://openalex.org/C2778265155","wikidata":"https://www.wikidata.org/wiki/Q7233276","display_name":"Positive definiteness","level":4,"score":0.7077156901359558},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.6912558078765869},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.688134491443634},{"id":"https://openalex.org/C126352355","wikidata":"https://www.wikidata.org/wiki/Q2562273","display_name":"Polynomial matrix","level":4,"score":0.6659424304962158},{"id":"https://openalex.org/C11210021","wikidata":"https://www.wikidata.org/wiki/Q1520713","display_name":"Linearization","level":3,"score":0.5938011407852173},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.49055641889572144},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4838339388370514},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4608108699321747},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.44834983348846436},{"id":"https://openalex.org/C95439218","wikidata":"https://www.wikidata.org/wiki/Q1666173","display_name":"Interlacing","level":2,"score":0.4224308729171753},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.25525373220443726},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.09820112586021423},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1137/080721406","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080721406","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/abc56e57-8368-441d-a47f-b807df923f58","is_oa":false,"landing_page_url":"https://research.manchester.ac.uk/en/publications/abc56e57-8368-441d-a47f-b807df923f58","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, MacKey, D S & Tisseur, F 2009, 'Definite matrix polynomials and their linearization by definite pencils', SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 2, pp. 478-502. https://doi.org/10.1137/080721406","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1084","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:10.1.1.140.218","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.140.218","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/1159/01/covered/MIMS_ep2007_97.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.151.4834","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.151.4834","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/1266/01/covered/MIMS_ep2007_97.pdf","raw_type":"text"},{"id":"pmh:oai:pure.atira.dk:publications/abc56e57-8368-441d-a47f-b807df923f58","is_oa":false,"landing_page_url":"https://www.research.manchester.ac.uk/portal/en/publications/definite-matrix-polynomials-and-their-linearization-by-definite-pencils(abc56e57-8368-441d-a47f-b807df923f58).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, MacKey, D S & Tisseur, F 2009, 'Definite matrix polynomials and their linearization by definite pencils', SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 2, pp. 478-502. https://doi.org/10.1137/080721406","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5654735135","display_name":null,"funder_award_id":"EP/D079403/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":15,"referenced_works":["https://openalex.org/W573934613","https://openalex.org/W658559791","https://openalex.org/W1580390795","https://openalex.org/W1965573117","https://openalex.org/W1999834714","https://openalex.org/W2020804487","https://openalex.org/W2022838696","https://openalex.org/W2024893327","https://openalex.org/W2032875563","https://openalex.org/W2053529276","https://openalex.org/W2056193169","https://openalex.org/W2069705790","https://openalex.org/W2104250481","https://openalex.org/W2117870622","https://openalex.org/W2128978199"],"related_works":["https://openalex.org/W2369793817","https://openalex.org/W2364489971","https://openalex.org/W2361979571","https://openalex.org/W4205226845","https://openalex.org/W3208377712","https://openalex.org/W2043819621","https://openalex.org/W2063005030","https://openalex.org/W2147922812","https://openalex.org/W2101137939","https://openalex.org/W2327741030"],"abstract_inverted_index":{"Hyperbolic":[0],"matrix":[1,9,40,100],"polynomials":[2,10,68],"are":[3,30,157],"an":[4,70],"important":[5,161],"class":[6,66],"of":[7,38,49,51,67,75,138,150,164],"Hermitian":[8,139],"that":[11,27,45,64,84,112,156,188],"contain":[12],"overdamped":[13],"quadratics":[14],"as":[15,89],"a":[16,43,90,98,107,129,134,147,169,177,181],"special":[17,91,162],"case.":[18,92],"They":[19],"share":[20],"with":[21,180],"definite":[22,60,87,99,122,130,143,170,178,183],"pencils":[23,88],"the":[24,36,47,52,79,113,152,160],"spectral":[25],"property":[26],"their":[28],"eigenvalues":[29],"real":[31,81],"and":[32,83,124,141],"semisimple.":[33],"We":[34,62,110],"extend":[35],"definition":[37],"hyperbolic":[39],"polynomial":[41,101,172],"in":[42,73,106,132,154],"way":[44],"relaxes":[46],"requirement":[48],"definiteness":[50,76],"leading":[53,184],"coefficient":[54,185],"matrix,":[55],"yielding":[56],"what":[57],"we":[58,145,166],"call":[59],"polynomials.":[61],"show":[63,111,167],"this":[65,116],"has":[69,128],"elegant":[71],"characterization":[72,149],"terms":[74],"intervals":[77],"on":[78],"extended":[80],"line":[82],"it":[85,127],"includes":[86],"A":[93],"fundamental":[94],"question":[95,117],"is":[96,118,121,189],"whether":[97],"P":[102,120,144],"can":[103,173],"be":[104,174],"linearized":[105],"structure-preserving":[108],"way.":[109],"answer":[114],"to":[115],"affirmative:":[119],"if":[123,126],"only":[125],"linearization":[131,179],"$\\mathbb{H}(P)$,":[133],"certain":[135],"vector":[136],"space":[137],"pencils;":[140],"for":[142],"give":[146],"complete":[148],"all":[151],"linearizations":[153],"$\\mathbb{H}(P)$":[155],"definite.":[158],"For":[159],"case":[163],"quadratics,":[165],"how":[168],"quadratic":[171],"transformed":[175],"into":[176],"positive":[182],"matrix\u2014a":[186],"form":[187],"particularly":[190],"attractive":[191],"numerically.":[192]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
