{"id":"https://openalex.org/W2056919620","doi":"https://doi.org/10.1090/s0025-5718-08-02128-5","title":"Spectral decomposition of real symmetric quadratic $\\lambda $-matrices and its applications","display_name":"Spectral decomposition of real symmetric quadratic $\\lambda $-matrices and its applications","publication_year":2008,"publication_date":"2008-10-14","ids":{"openalex":"https://openalex.org/W2056919620","doi":"https://doi.org/10.1090/s0025-5718-08-02128-5","mag":"2056919620"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-08-02128-5","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-08-02128-5","pdf_url":"https://www.ams.org/mcom/2009-78-265/S0025-5718-08-02128-5/S0025-5718-08-02128-5.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://www.ams.org/mcom/2009-78-265/S0025-5718-08-02128-5/S0025-5718-08-02128-5.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029654954","display_name":"Moody T. Chu","orcid":"https://orcid.org/0000-0001-8544-3588"},"institutions":[{"id":"https://openalex.org/I137902535","display_name":"North Carolina State University","ror":"https://ror.org/04tj63d06","country_code":"US","type":"education","lineage":["https://openalex.org/I137902535"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Moody T. Chu","raw_affiliation_strings":["Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205","Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695\u20108205"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205","institution_ids":["https://openalex.org/I137902535"]},{"raw_affiliation_string":"Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695\u20108205","institution_ids":["https://openalex.org/I137902535"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100294101","display_name":"Shufang Xu","orcid":null},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Shu-Fang Xu","raw_affiliation_strings":["LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China","LMAM, School of Mathematical Sciences,  Peking University, Beijing 100871, China"],"affiliations":[{"raw_affiliation_string":"LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China","institution_ids":["https://openalex.org/I20231570"]},{"raw_affiliation_string":"LMAM, School of Mathematical Sciences,  Peking University, Beijing 100871, China","institution_ids":["https://openalex.org/I20231570"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5029654954"],"corresponding_institution_ids":["https://openalex.org/I137902535"],"apc_list":null,"apc_paid":null,"fwci":4.9395,"has_fulltext":true,"cited_by_count":44,"citation_normalized_percentile":{"value":0.9502293,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"78","issue":"265","first_page":"293","last_page":"293"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":1.0,"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":1.0,"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/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9957000017166138,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":0.9918000102043152,"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":0.7816779613494873},{"id":"https://openalex.org/keywords/invertible-matrix","display_name":"Invertible matrix","score":0.6287919282913208},{"id":"https://openalex.org/keywords/eigenfunction","display_name":"Eigenfunction","score":0.6193400025367737},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.6049209833145142},{"id":"https://openalex.org/keywords/canonical-form","display_name":"Canonical form","score":0.5628927946090698},{"id":"https://openalex.org/keywords/matrix-decomposition","display_name":"Matrix decomposition","score":0.5552492141723633},{"id":"https://openalex.org/keywords/lambda","display_name":"Lambda","score":0.5241249203681946},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.520135223865509},{"id":"https://openalex.org/keywords/eigendecomposition-of-a-matrix","display_name":"Eigendecomposition of a matrix","score":0.4998605251312256},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.467172235250473},{"id":"https://openalex.org/keywords/decomposition","display_name":"Decomposition","score":0.43162792921066284},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.42732715606689453},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.4250315725803375},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.4132928252220154},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.37141960859298706},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.0871199369430542}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7816779613494873},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.6287919282913208},{"id":"https://openalex.org/C128803854","wikidata":"https://www.wikidata.org/wiki/Q1307821","display_name":"Eigenfunction","level":3,"score":0.6193400025367737},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6049209833145142},{"id":"https://openalex.org/C204707403","wikidata":"https://www.wikidata.org/wiki/Q1152398","display_name":"Canonical form","level":2,"score":0.5628927946090698},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.5552492141723633},{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.5241249203681946},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.520135223865509},{"id":"https://openalex.org/C169756996","wikidata":"https://www.wikidata.org/wiki/Q194919","display_name":"Eigendecomposition of a matrix","level":3,"score":0.4998605251312256},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.467172235250473},{"id":"https://openalex.org/C124681953","wikidata":"https://www.wikidata.org/wiki/Q339062","display_name":"Decomposition","level":2,"score":0.43162792921066284},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.42732715606689453},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.4250315725803375},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4132928252220154},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.37141960859298706},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0871199369430542},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"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/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"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/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"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},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"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":1,"locations":[{"id":"doi:10.1090/s0025-5718-08-02128-5","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-08-02128-5","pdf_url":"https://www.ams.org/mcom/2009-78-265/S0025-5718-08-02128-5/S0025-5718-08-02128-5.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-08-02128-5","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-08-02128-5","pdf_url":"https://www.ams.org/mcom/2009-78-265/S0025-5718-08-02128-5/S0025-5718-08-02128-5.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4997596672","display_name":"Collaborative Proposal:    Quadratic Inverse Eigenvalue Problems for Model Updating in Science and  Engineering:   Theory and Computation","funder_award_id":"0505880","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5848258319","display_name":null,"funder_award_id":"0 and","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6890617935","display_name":"MSPA-MCS: Collaborative Research: Fast Nonnegative Matrix Factorizations: Theory, Algorithms, and Applications","funder_award_id":"0732299","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2056919620.pdf","grobid_xml":"https://content.openalex.org/works/W2056919620.grobid-xml"},"referenced_works_count":30,"referenced_works":["https://openalex.org/W239613228","https://openalex.org/W626183909","https://openalex.org/W645542774","https://openalex.org/W785198034","https://openalex.org/W1573717114","https://openalex.org/W1965573117","https://openalex.org/W1970606629","https://openalex.org/W1975363339","https://openalex.org/W1986201906","https://openalex.org/W1989480656","https://openalex.org/W2004229590","https://openalex.org/W2007032864","https://openalex.org/W2015344163","https://openalex.org/W2015845899","https://openalex.org/W2028613545","https://openalex.org/W2039538606","https://openalex.org/W2052727894","https://openalex.org/W2082496628","https://openalex.org/W2086049388","https://openalex.org/W2124654249","https://openalex.org/W2128978199","https://openalex.org/W2154915620","https://openalex.org/W2160256678","https://openalex.org/W2166000224","https://openalex.org/W2316727981","https://openalex.org/W2327908882","https://openalex.org/W2610857016","https://openalex.org/W4249638744","https://openalex.org/W4251761138","https://openalex.org/W6836625313"],"related_works":["https://openalex.org/W1493283577","https://openalex.org/W1993839002","https://openalex.org/W2053448926","https://openalex.org/W2003002680","https://openalex.org/W2083381963","https://openalex.org/W2998723788","https://openalex.org/W4300915466","https://openalex.org/W2063368412","https://openalex.org/W2254382463","https://openalex.org/W3094027405"],"abstract_inverted_index":{"Spectral":[0,33],"decomposition":[1,34,117],"provides":[2],"a":[3,10,140],"canonical":[4,21],"representation":[5,104],"of":[6,15,36,50,55,113,124,142],"an":[7],"operator":[8],"over":[9],"vector":[11],"space":[12],"in":[13,39,102,129],"terms":[14],"its":[16,59],"eigenvalues":[17],"and":[18,31],"eigenfunctions.":[19],"The":[20,42],"form":[22],"often":[23],"facilitates":[24],"discussions":[25],"which,":[26,128],"otherwise,":[27],"would":[28],"be":[29,105,119,132],"complicated":[30],"involved.":[32],"is":[35,61,108],"fundamental":[37],"importance":[38],"many":[40],"applications.":[41],"well-known":[43],"GLR":[44,70],"theory":[45,71],"generalizes":[46],"the":[47,69,73,96,103,111,114,122,135],"classical":[48],"result":[49],"eigendecomposition":[51],"to":[52,76,95,139],"matrix":[53,80],"polynomials":[54],"higher":[56],"degrees,":[57],"but":[58],"development":[60],"based":[62],"on":[63],"complex":[64],"numbers.":[65],"This":[66],"paper":[67],"modifies":[68],"for":[72],"special":[74],"application":[75],"real":[77,115,125],"symmetric":[78],"quadratic":[79],"polynomials,":[81],"$\\mathcal":[82],"{Q}(\\lambda":[83],")=M":[84],"\\lambda":[85,89],"^{2}":[86],"+":[87,90],"C":[88],"K$,":[91],"$M$":[92],"nonsingular,":[93],"subject":[94],"specific":[97],"restriction":[98],"that":[99,110],"all":[100],"matrices":[101],"real-valued.":[106],"It":[107],"shown":[109],"existence":[112],"spectral":[116,136],"can":[118,131],"characterized":[120],"through":[121],"notion":[123],"standard":[126],"pair":[127],"turn,":[130],"constructed":[133],"from":[134],"data.":[137],"Applications":[138],"variety":[141],"challenging":[143],"inverse":[144],"problems":[145],"are":[146],"discussed.":[147]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":4},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":3}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}
