{"id":"https://openalex.org/W1967639198","doi":"https://doi.org/10.1137/s0895479894260900","title":"The Set of 2-by-3 Matrix Pencils \u2014 Kronecker Structures and Their Transitions under Perturbations","display_name":"The Set of 2-by-3 Matrix Pencils \u2014 Kronecker Structures and Their Transitions under Perturbations","publication_year":1996,"publication_date":"1996-01-01","ids":{"openalex":"https://openalex.org/W1967639198","doi":"https://doi.org/10.1137/s0895479894260900","mag":"1967639198"},"language":"en","primary_location":{"id":"doi:10.1137/s0895479894260900","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479894260900","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/A5070862414","display_name":"Erik Elmroth","orcid":"https://orcid.org/0000-0002-2633-6798"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Erik Elmroth","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5047473434","display_name":"Bo K\u00e5gstr\u00f6m","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bo K\u00e5gstr\u00f6m","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5070862414"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.1224,"has_fulltext":false,"cited_by_count":26,"citation_normalized_percentile":{"value":0.93594818,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"17","issue":"1","first_page":"1","last_page":"34"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9998000264167786,"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":0.9998000264167786,"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/T13126","display_name":"Scientific Research and Discoveries","score":0.9825000166893005,"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"}},{"id":"https://openalex.org/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9815999865531921,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/kronecker-delta","display_name":"Kronecker delta","score":0.8928215503692627},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7705081105232239},{"id":"https://openalex.org/keywords/canonical-form","display_name":"Canonical form","score":0.5409823656082153},{"id":"https://openalex.org/keywords/pencil","display_name":"Pencil (optics)","score":0.49265342950820923},{"id":"https://openalex.org/keywords/kronecker-product","display_name":"Kronecker product","score":0.46293964982032776},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.42453569173812866},{"id":"https://openalex.org/keywords/orbit","display_name":"Orbit (dynamics)","score":0.4202724099159241},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.37897542119026184},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3397737145423889}],"concepts":[{"id":"https://openalex.org/C39482219","wikidata":"https://www.wikidata.org/wiki/Q192826","display_name":"Kronecker delta","level":2,"score":0.8928215503692627},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7705081105232239},{"id":"https://openalex.org/C204707403","wikidata":"https://www.wikidata.org/wiki/Q1152398","display_name":"Canonical form","level":2,"score":0.5409823656082153},{"id":"https://openalex.org/C134949993","wikidata":"https://www.wikidata.org/wiki/Q2068617","display_name":"Pencil (optics)","level":2,"score":0.49265342950820923},{"id":"https://openalex.org/C46030957","wikidata":"https://www.wikidata.org/wiki/Q1238125","display_name":"Kronecker product","level":3,"score":0.46293964982032776},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.42453569173812866},{"id":"https://openalex.org/C196644772","wikidata":"https://www.wikidata.org/wiki/Q3884964","display_name":"Orbit (dynamics)","level":2,"score":0.4202724099159241},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.37897542119026184},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3397737145423889},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","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},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","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/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/s0895479894260900","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479894260900","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:CiteSeerX.psu:10.1.1.33.1936","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.1936","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.umu.se/~elmroth/papers/run2by3.ps","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":18,"referenced_works":["https://openalex.org/W30053609","https://openalex.org/W168910255","https://openalex.org/W1809022293","https://openalex.org/W1984452258","https://openalex.org/W1985910458","https://openalex.org/W2008256634","https://openalex.org/W2013474260","https://openalex.org/W2019609727","https://openalex.org/W2027833201","https://openalex.org/W2039333245","https://openalex.org/W2084830761","https://openalex.org/W2095848239","https://openalex.org/W2097178696","https://openalex.org/W2119459069","https://openalex.org/W2128217535","https://openalex.org/W2134046186","https://openalex.org/W4205696294","https://openalex.org/W4233995484"],"related_works":["https://openalex.org/W4365800530","https://openalex.org/W2120296059","https://openalex.org/W1996965151","https://openalex.org/W2095223301","https://openalex.org/W2025362408","https://openalex.org/W2016208021","https://openalex.org/W2119459069","https://openalex.org/W2770308590","https://openalex.org/W2091170701","https://openalex.org/W1985031765"],"abstract_inverted_index":{"The":[0,20,38,104],"set":[1,74],"(or":[2],"family)":[3],"of":[4,25,42,45,75,87,112,120,140,211,243,246,260,263,282,302],"2-by-3":[5,76,143],"matrix":[6],"pencils":[7],"$A":[8,145,181],"-":[9,146,157,182],"\\lambda":[10,147,158,183],"B$":[11,148,184],"comprises":[12],"18":[13],"structurally":[14],"different":[15,47],"Kronecker":[16,48,89,115,169,209,236],"structures":[17,49,62,266,285],"(canonical":[18],"forms).":[19],"algebraic":[21],"and":[22,28,52,106,117,196,288],"geometric":[23,110,121],"characteristics":[24,111,122],"the":[26,29,43,61,68,73,85,94,118,134,177,241,244,247,261,264,275,284,297,303,308],"generic":[27,142,213],"17":[30],"nongeneric":[31,168,188,223,265],"cases":[32],"are":[33,97,171,190,199,225,252,286,305],"examined":[34],"in":[35,54,67,93,99,227,270,294],"full":[36],"detail.":[37],"complete":[39],"closure":[40,56,95,102,242,298],"hierarchy":[41,96],"orbits":[44],"all":[46],"is":[50,215],"derived":[51],"presented":[53,98],"a":[55,100,141,166,186,202,207,212,221,228,235,258],"graph":[57],"that":[58,150,179,205,230,290],"shows":[59,109],"how":[60],"relate":[63],"to":[64,91,125,185,219,307],"each":[65],"other":[66],"12-dimensional":[69],"space":[70],"spanned":[71],"by":[72,254],"pencils.":[77],"Necessary":[78],"conditions":[79],"on":[80,240],"perturbations":[81,136,178,198,269,283],"for":[82,133,176,279],"transiting":[83,124],"from":[84],"orbit":[86,245],"one":[88,232],"structure":[90,116,170,210,224,291],"another":[92],"labeled":[101],"graph.":[103],"node":[105],"arc":[107],"labels":[108],"an":[113,126],"orbit\u2019s":[114],"change":[119],"when":[123],"adjacent":[127],"node,":[128],"respectively.":[129],"Computable":[130],"normwise":[131],"bounds":[132],"smallest":[135],"$(\\delta":[137],"A,\\delta":[138],"B)$":[139],"pencil":[144,214],"such":[149],"$(":[151],"A":[152,155],"+":[153,161,312],"\\delta":[154,162],")":[156],"(":[159],"B":[160,163],")$":[164,314],"has":[165],"specific":[167],"presented.":[172],"First,":[173],"explicit":[174],"expressions":[175],"transfer":[180],"specified":[187,208],"form":[189,238],"derived.":[191],"In":[192],"this":[193],"context":[194],"tractable":[195],"intractable":[197],"defined.":[200],"Second,":[201],"modified":[203],"GUPTRI":[204,276],"computes":[206],"used.":[216],"Perturbations":[217],"devised":[218],"impose":[220],"certain":[222],"computed":[226],"way":[229],"guarantees":[231],"will":[233],"find":[234],"canonical":[237],"(KCF)":[239],"intended":[248],"KCF.":[249],"Both":[250],"approaches":[251],"illustrated":[253],"computational":[255],"experiments.":[256],"Moreover,":[257],"study":[259],"behaviour":[262],"under":[267],"random":[268],"finite":[271],"precision":[272],"arithmetic":[273],"(using":[274],"software)":[277],"show":[278],"which":[280],"sizes":[281],"invariant":[287],"also":[289],"transitions":[292],"occur":[293],"accordance":[295],"with":[296],"hierarchy.":[299],"Finally,":[300],"some":[301],"results":[304],"extended":[306],"general":[309],"m-by-$(":[310],"m":[311],"1":[313],"case.":[315]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2014,"cited_by_count":6},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
