{"id":"https://openalex.org/W2487411778","doi":"https://doi.org/10.1137/15m1041031","title":"On Special Cases of the Generalized Max-Plus Eigenproblem","display_name":"On Special Cases of the Generalized Max-Plus Eigenproblem","publication_year":2016,"publication_date":"2016-01-01","ids":{"openalex":"https://openalex.org/W2487411778","doi":"https://doi.org/10.1137/15m1041031","mag":"2487411778"},"language":"en","primary_location":{"id":"doi:10.1137/15m1041031","is_oa":true,"landing_page_url":"https://doi.org/10.1137/15m1041031","pdf_url":"https://epubs.siam.org/doi/pdf/10.1137/15M1041031","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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":true,"oa_status":"hybrid","oa_url":"https://epubs.siam.org/doi/pdf/10.1137/15M1041031","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057034818","display_name":"Peter Butkovi\u010d","orcid":"https://orcid.org/0000-0003-3912-9250"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Peter Butkovi\u010d","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5085129713","display_name":"Daniel Jones","orcid":"https://orcid.org/0000-0002-5901-5122"},"institutions":[{"id":"https://openalex.org/I79619799","display_name":"University of Birmingham","ror":"https://ror.org/03angcq70","country_code":"GB","type":"education","lineage":["https://openalex.org/I79619799"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Daniel Jones","raw_affiliation_strings":["School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK"],"affiliations":[{"raw_affiliation_string":"School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK","institution_ids":["https://openalex.org/I79619799"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5057034818"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":{"value":2578,"currency":"EUR","value_usd":2780},"fwci":1.9168,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.83433686,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"37","issue":"3","first_page":"1002","last_page":"1021"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11673","display_name":"Advanced Topics in Algebra","score":0.9991000294685364,"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"}},"topics":[{"id":"https://openalex.org/T11673","display_name":"Advanced Topics in Algebra","score":0.9991000294685364,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9944999814033508,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.9883000254631042,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8662488460540771},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7322428226470947},{"id":"https://openalex.org/keywords/lambda","display_name":"Lambda","score":0.6997098922729492},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.6579089760780334},{"id":"https://openalex.org/keywords/spectrum","display_name":"Spectrum (functional analysis)","score":0.5621185302734375},{"id":"https://openalex.org/keywords/symmetric-function","display_name":"Symmetric function","score":0.45394447445869446},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.4472118020057678},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.4152790606021881},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.16125431656837463},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.0654926598072052}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8662488460540771},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7322428226470947},{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.6997098922729492},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6579089760780334},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.5621185302734375},{"id":"https://openalex.org/C3746008","wikidata":"https://www.wikidata.org/wiki/Q981351","display_name":"Symmetric function","level":2,"score":0.45394447445869446},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.4472118020057678},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4152790606021881},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.16125431656837463},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0654926598072052},{"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/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer 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/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/15m1041031","is_oa":true,"landing_page_url":"https://doi.org/10.1137/15m1041031","pdf_url":"https://epubs.siam.org/doi/pdf/10.1137/15M1041031","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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:Publications/284e8d6c-7220-46b1-b5c5-b0a63c81688d","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306402634","display_name":"University of Birmingham Research Portal (University of Birmingham)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79619799","host_organization_name":"University of Birmingham","host_organization_lineage":["https://openalex.org/I79619799"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":""},{"id":"pmh:oai:pure.atira.dk:openaire_cris_publications/284e8d6c-7220-46b1-b5c5-b0a63c81688d","is_oa":true,"landing_page_url":"https://research.birmingham.ac.uk/en/publications/284e8d6c-7220-46b1-b5c5-b0a63c81688d","pdf_url":null,"source":{"id":"https://openalex.org/S4306402634","display_name":"University of Birmingham Research Portal (University of Birmingham)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79619799","host_organization_name":"University of Birmingham","host_organization_lineage":["https://openalex.org/I79619799"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Butkovic, P & Jones, D 2016, 'On special cases of the generalized max-plus eigenproblem', S I A M Journal on Matrix Analysis and Applications, vol. 37, no. 3, pp. 1002\u20131021. https://doi.org/10.1137/15M1041031","raw_type":"article"},{"id":"pmh:oai:pure.atira.dk:publications/284e8d6c-7220-46b1-b5c5-b0a63c81688d","is_oa":true,"landing_page_url":"https://research.birmingham.ac.uk/portal/en/publications/on-special-cases-of-the-generalized-maxplus-eigenproblem(284e8d6c-7220-46b1-b5c5-b0a63c81688d).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306402634","display_name":"University of Birmingham Research Portal (University of Birmingham)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79619799","host_organization_name":"University of Birmingham","host_organization_lineage":["https://openalex.org/I79619799"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":""}],"best_oa_location":{"id":"doi:10.1137/15m1041031","is_oa":true,"landing_page_url":"https://doi.org/10.1137/15m1041031","pdf_url":"https://epubs.siam.org/doi/pdf/10.1137/15M1041031","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1807818597","display_name":null,"funder_award_id":"EP/J00829X/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G3036096617","display_name":"Perron-Frobenius theory and max-algebraic combinatorics of nonnegative matrices","funder_award_id":"EP/J00829X/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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2487411778.pdf","grobid_xml":"https://content.openalex.org/works/W2487411778.grobid-xml"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W303090291","https://openalex.org/W2004777406","https://openalex.org/W2005348468","https://openalex.org/W2039294930","https://openalex.org/W2074493271","https://openalex.org/W2088418904","https://openalex.org/W2089186297","https://openalex.org/W2109513024","https://openalex.org/W2127017400","https://openalex.org/W2151838128","https://openalex.org/W2152366635","https://openalex.org/W2154682386"],"related_works":["https://openalex.org/W2951695430","https://openalex.org/W4300891849","https://openalex.org/W2318687609","https://openalex.org/W91590591","https://openalex.org/W159772521","https://openalex.org/W2950100995","https://openalex.org/W2806574890","https://openalex.org/W2952745759","https://openalex.org/W2062050236","https://openalex.org/W2069948527"],"abstract_inverted_index":{"We":[0,46,71],"study":[1],"the":[2,14,54,66,75,79,98,105,117],"generalized":[3,33],"eigenproblem":[4],"$A\\otimes":[5],"x=\\lambda\\otimes":[6],"B\\otimes":[7],"x,":[8],"$":[9],"where":[10],"$A,B\\in\\mathbb{R}^{m\\times":[11],"n}$":[12],"in":[13,44,78],"max-plus":[15],"algebra.":[16],"It":[17,87],"is":[18,29,42,51,65,83,92,96,104],"known":[19,43],"that":[20,48,89],"if":[21,49],"$A$":[22,91,103],"and":[23,94],"$B$":[24,82,95],"are":[25],"symmetric,":[26,52],"then":[27,53],"there":[28],"at":[30],"most":[31],"one":[32],"eigenvalue,":[34],"but":[35],"no":[36],"description":[37,115],"of":[38,57,61,102,116],"this":[39],"unique":[40,67,106],"candidate":[41,68,107],"general.":[45],"prove":[47],"$C=A-B$":[50],"common":[55],"value":[56],"all":[58],"saddle":[59],"points":[60],"$C$":[62],"(if":[63],"any)":[64],"for":[69,108],"$\\lambda.$":[70,109],"also":[72],"explicitly":[73],"describe":[74],"whole":[76],"spectrum":[77,118],"case":[80],"when":[81,90,119],"an":[84],"outer":[85],"product.":[86],"follows":[88],"symmetric":[93],"constant,":[97],"smallest":[99],"column":[100],"maximum":[101],"Finally,":[110],"we":[111],"provide":[112],"a":[113],"complete":[114],"$n=2$.":[120]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2026-04-17T18:11:37.981687","created_date":"2025-10-10T00:00:00"}