{"id":"https://openalex.org/W2009187315","doi":"https://doi.org/10.1080/00207720802145791","title":"Closed form solutions for matrix linear systems using double matrix exponential functions","display_name":"Closed form solutions for matrix linear systems using double matrix exponential functions","publication_year":2008,"publication_date":"2008-11-03","ids":{"openalex":"https://openalex.org/W2009187315","doi":"https://doi.org/10.1080/00207720802145791","mag":"2009187315"},"language":"en","primary_location":{"id":"doi:10.1080/00207720802145791","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207720802145791","pdf_url":null,"source":{"id":"https://openalex.org/S129640837","display_name":"International Journal of Systems Science","issn_l":"0020-7721","issn":["0020-7721","1464-5319"],"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":"International Journal of Systems Science","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/A5060424750","display_name":"Bin Zhou","orcid":"https://orcid.org/0000-0003-1272-2652"},"institutions":[{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Bin Zhou","raw_affiliation_strings":["Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, PR China","Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, PR China#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, PR China","institution_ids":["https://openalex.org/I204983213"]},{"raw_affiliation_string":"Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, PR China#TAB#","institution_ids":["https://openalex.org/I204983213"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5043623971","display_name":"Guang\u2010Ren Duan","orcid":"https://orcid.org/0000-0002-8169-2218"},"institutions":[{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Guang-Ren Duan","raw_affiliation_strings":["Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, PR China","Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, PR China#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, PR China","institution_ids":["https://openalex.org/I204983213"]},{"raw_affiliation_string":"Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, PR China#TAB#","institution_ids":["https://openalex.org/I204983213"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100852833","display_name":"Zhen Zhong","orcid":null},"institutions":[{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhen Zhong","raw_affiliation_strings":["Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, PR China","Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, PR China#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, PR China","institution_ids":["https://openalex.org/I204983213"]},{"raw_affiliation_string":"Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, PR China#TAB#","institution_ids":["https://openalex.org/I204983213"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5060424750"],"corresponding_institution_ids":["https://openalex.org/I204983213"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.0804402,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"40","issue":"1","first_page":"91","last_page":"99"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9962000250816345,"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":0.9962000250816345,"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":0.9940999746322632,"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/T11394","display_name":"Dynamics and Control of Mechanical Systems","score":0.9927999973297119,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems 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/matrix-exponential","display_name":"Matrix exponential","score":0.7311807870864868},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.6576582193374634},{"id":"https://openalex.org/keywords/matrix-function","display_name":"Matrix function","score":0.6283659934997559},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5699945092201233},{"id":"https://openalex.org/keywords/state-transition-matrix","display_name":"State-transition matrix","score":0.566048264503479},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.5336533188819885},{"id":"https://openalex.org/keywords/extension","display_name":"Extension (predicate logic)","score":0.5100510716438293},{"id":"https://openalex.org/keywords/pascal-matrix","display_name":"Pascal matrix","score":0.4734841585159302},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.4691694378852844},{"id":"https://openalex.org/keywords/square-matrix","display_name":"Square matrix","score":0.4689720869064331},{"id":"https://openalex.org/keywords/augmented-matrix","display_name":"Augmented matrix","score":0.4508192837238312},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4404241442680359},{"id":"https://openalex.org/keywords/convergent-matrix","display_name":"Convergent matrix","score":0.4158492982387543},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4126371741294861},{"id":"https://openalex.org/keywords/state-vector","display_name":"State vector","score":0.41215649247169495},{"id":"https://openalex.org/keywords/symmetric-matrix","display_name":"Symmetric matrix","score":0.26011770963668823},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.23470595479011536},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2247706949710846},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.09212812781333923},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.0869455337524414}],"concepts":[{"id":"https://openalex.org/C195906000","wikidata":"https://www.wikidata.org/wiki/Q1191722","display_name":"Matrix exponential","level":3,"score":0.7311807870864868},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.6576582193374634},{"id":"https://openalex.org/C4263655","wikidata":"https://www.wikidata.org/wiki/Q2699958","display_name":"Matrix function","level":4,"score":0.6283659934997559},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5699945092201233},{"id":"https://openalex.org/C22629506","wikidata":"https://www.wikidata.org/wiki/Q7602993","display_name":"State-transition matrix","level":4,"score":0.566048264503479},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.5336533188819885},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.5100510716438293},{"id":"https://openalex.org/C183357087","wikidata":"https://www.wikidata.org/wiki/Q7141739","display_name":"Pascal matrix","level":5,"score":0.4734841585159302},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.4691694378852844},{"id":"https://openalex.org/C69044650","wikidata":"https://www.wikidata.org/wiki/Q2739329","display_name":"Square matrix","level":4,"score":0.4689720869064331},{"id":"https://openalex.org/C105129939","wikidata":"https://www.wikidata.org/wiki/Q2750301","display_name":"Augmented matrix","level":5,"score":0.4508192837238312},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4404241442680359},{"id":"https://openalex.org/C9848182","wikidata":"https://www.wikidata.org/wiki/Q5166397","display_name":"Convergent matrix","level":5,"score":0.4158492982387543},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4126371741294861},{"id":"https://openalex.org/C2777798563","wikidata":"https://www.wikidata.org/wiki/Q7603916","display_name":"State vector","level":2,"score":0.41215649247169495},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.26011770963668823},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.23470595479011536},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2247706949710846},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.09212812781333923},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.0869455337524414},{"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","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/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00207720802145791","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207720802145791","pdf_url":null,"source":{"id":"https://openalex.org/S129640837","display_name":"International Journal of Systems Science","issn_l":"0020-7721","issn":["0020-7721","1464-5319"],"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":"International Journal of Systems Science","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":9,"referenced_works":["https://openalex.org/W1563829997","https://openalex.org/W1965854789","https://openalex.org/W1985502554","https://openalex.org/W2019208902","https://openalex.org/W2087413002","https://openalex.org/W2148704791","https://openalex.org/W2548131485","https://openalex.org/W2798707604","https://openalex.org/W2798813531"],"related_works":["https://openalex.org/W1981749077","https://openalex.org/W1984232695","https://openalex.org/W2388793872","https://openalex.org/W2767157329","https://openalex.org/W120389515","https://openalex.org/W4226165187","https://openalex.org/W2009187315","https://openalex.org/W3167249028","https://openalex.org/W2071296300","https://openalex.org/W2010774598"],"abstract_inverted_index":{"The":[0,19,35],"article":[1],"presents":[2],"closed":[3],"form":[4],"solutions":[5,37,78],"for":[6,79],"a":[7,17,52,73],"class":[8],"of":[9,25,30,66,76,99],"matrix":[10,47,59,68,100],"linear":[11,82,101],"systems":[12,40,83,102],"whose":[13],"state":[14,23],"variable":[15],"is":[16,63,72],"matrix.":[18],"formulation":[20,50],"evaluates":[21],"the":[22,26,31,77,86,97],"response":[24],"system":[27,33,105],"in":[28,85,96],"terms":[29],"original":[32,104],"matrices.":[34,106],"proposed":[36],"naturally":[38],"fit":[39],"which":[41,62],"are":[42],"most":[43],"described":[44],"conveniently":[45],"by":[46],"processes.":[48],"Its":[49],"uses":[51],"compact":[53],"notation":[54],"referred":[55],"to":[56],"as":[57],"double":[58],"exponential":[60,69],"functions,":[61],"an":[64,93],"extension":[65,75],"normal":[67],"function.":[70],"It":[71],"straightforward":[74],"ordinary":[80],"vector":[81],"studied":[84],"past":[87],"several":[88],"decades":[89],"and":[90],"will":[91],"play":[92],"important":[94],"role":[95],"design":[98],"using":[103]},"counts_by_year":[{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":1}],"updated_date":"2026-07-29T09:40:50.615796","created_date":"2025-10-10T00:00:00"}