{"id":"https://openalex.org/W4404820919","doi":"https://doi.org/10.3390/axioms13120839","title":"On Schur Forms for Matrices with Simple Eigenvalues","display_name":"On Schur Forms for Matrices with Simple Eigenvalues","publication_year":2024,"publication_date":"2024-11-28","ids":{"openalex":"https://openalex.org/W4404820919","doi":"https://doi.org/10.3390/axioms13120839"},"language":"en","primary_location":{"id":"doi:10.3390/axioms13120839","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13120839","pdf_url":"https://www.mdpi.com/2075-1680/13/12/839/pdf?version=1732803016","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/13/12/839/pdf?version=1732803016","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5012776351","display_name":"\u039c.\u039c. Konstantinov","orcid":"https://orcid.org/0000-0003-3708-516X"},"institutions":[{"id":"https://openalex.org/I206451062","display_name":"University of Architecture, Civil Engineering and Geodesy","ror":"https://ror.org/033cz7157","country_code":"BG","type":"education","lineage":["https://openalex.org/I206451062"]}],"countries":["BG"],"is_corresponding":false,"raw_author_name":"Mihail Mihaylov Konstantinov","raw_affiliation_strings":["Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, 1064 Sofia, Bulgaria"],"raw_orcid":"https://orcid.org/0000-0003-3708-516X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, 1064 Sofia, Bulgaria","institution_ids":["https://openalex.org/I206451062"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5065184308","display_name":"Petko H. Petkov","orcid":"https://orcid.org/0000-0003-0334-0900"},"institutions":[{"id":"https://openalex.org/I24768866","display_name":"Bulgarian Academy of Sciences","ror":"https://ror.org/01x8hew03","country_code":"BG","type":"government","lineage":["https://openalex.org/I24768866"]}],"countries":["BG"],"is_corresponding":true,"raw_author_name":"Petko Hristov Petkov","raw_affiliation_strings":["Bulgarian Academy of Sciences, 1040 Sofia, Bulgaria"],"raw_orcid":"https://orcid.org/0000-0003-0334-0900","affiliations":[{"raw_affiliation_string":"Bulgarian Academy of Sciences, 1040 Sofia, Bulgaria","institution_ids":["https://openalex.org/I24768866"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5065184308"],"corresponding_institution_ids":["https://openalex.org/I24768866"],"apc_list":{"value":1600,"currency":"CHF","value_usd":1782},"apc_paid":{"value":1600,"currency":"CHF","value_usd":1782},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.21677344,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"13","issue":"12","first_page":"839","last_page":"839"},"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/T11673","display_name":"Advanced Topics in Algebra","score":0.9969000220298767,"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/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.9965000152587891,"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/simple","display_name":"Simple (philosophy)","score":0.7445061802864075},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.709848165512085},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5632611513137817},{"id":"https://openalex.org/keywords/schur-decomposition","display_name":"Schur decomposition","score":0.46484559774398804},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.45673561096191406},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.4300704598426819},{"id":"https://openalex.org/keywords/schur-complement","display_name":"Schur complement","score":0.31539738178253174},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10081613063812256},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.05150878429412842},{"id":"https://openalex.org/keywords/epistemology","display_name":"Epistemology","score":0.045888543128967285}],"concepts":[{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.7445061802864075},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.709848165512085},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5632611513137817},{"id":"https://openalex.org/C99383703","wikidata":"https://www.wikidata.org/wiki/Q1064218","display_name":"Schur decomposition","level":4,"score":0.46484559774398804},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.45673561096191406},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4300704598426819},{"id":"https://openalex.org/C2731191","wikidata":"https://www.wikidata.org/wiki/Q1778169","display_name":"Schur complement","level":3,"score":0.31539738178253174},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10081613063812256},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.05150878429412842},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.045888543128967285},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms13120839","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13120839","pdf_url":"https://www.mdpi.com/2075-1680/13/12/839/pdf?version=1732803016","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:2201328c8e6f492aa7ee0f21fa8b0c2e","is_oa":false,"landing_page_url":"https://doaj.org/article/2201328c8e6f492aa7ee0f21fa8b0c2e","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 13, Iss 12, p 839 (2024)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms13120839","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13120839","pdf_url":"https://www.mdpi.com/2075-1680/13/12/839/pdf?version=1732803016","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4404820919.pdf","grobid_xml":"https://content.openalex.org/works/W4404820919.grobid-xml"},"referenced_works_count":29,"referenced_works":["https://openalex.org/W1483804921","https://openalex.org/W1488435683","https://openalex.org/W1966536333","https://openalex.org/W1975945992","https://openalex.org/W1995028404","https://openalex.org/W1996023440","https://openalex.org/W2001225792","https://openalex.org/W2004475116","https://openalex.org/W2019157832","https://openalex.org/W2021382145","https://openalex.org/W2027923267","https://openalex.org/W2064411146","https://openalex.org/W2067843337","https://openalex.org/W2068873705","https://openalex.org/W2082598533","https://openalex.org/W2150921527","https://openalex.org/W2160835412","https://openalex.org/W2406183274","https://openalex.org/W3170392540","https://openalex.org/W3208958271","https://openalex.org/W4229706427","https://openalex.org/W4248269519","https://openalex.org/W4255333822","https://openalex.org/W4302550330","https://openalex.org/W4312258136","https://openalex.org/W4377246852","https://openalex.org/W4403680470","https://openalex.org/W6661911136","https://openalex.org/W6713793862"],"related_works":["https://openalex.org/W1975201729","https://openalex.org/W2074771309","https://openalex.org/W2551678916","https://openalex.org/W3092306907","https://openalex.org/W1510582995","https://openalex.org/W3115487577","https://openalex.org/W1985400098","https://openalex.org/W1986380039","https://openalex.org/W2064598248","https://openalex.org/W4296911718"],"abstract_inverted_index":{"In":[0],"this":[1,46],"paper,":[2],"we":[3],"consider":[4,88],"various":[5],"aspects":[6],"of":[7,22,31,40,57,76],"the":[8,18,29,37,53,63,73,77,84],"Schur":[9,42,64,78,93],"problem":[10,65],"for":[11],"a":[12,49],"square":[13],"complex":[14],"matrix":[15,85],"A,":[16],"namely":[17],"similarity":[19],"unitary":[20],"transformation":[21],"A":[23,32],"into":[24],"upper":[25],"triangular":[26],"form":[27],"containing":[28],"eigenvalues":[30],"on":[33],"its":[34],"diagonal.":[35],"Since":[36],"profound":[38],"work":[39],"I.":[41],"published":[43],"in":[44,52,70,83],"1909,":[45],"has":[47],"become":[48],"fundamental":[50],"issue":[51],"theory":[54],"and":[55,91,111,118],"applications":[56],"matrices.":[58],"Nevertheless,":[59],"certain":[60],"details":[61],"concerning":[62],"need":[66],"further":[67],"clarification,":[68],"especially":[69],"connection":[71],"with":[72,101],"perturbation":[74],"analysis":[75],"decomposition":[79],"relative":[80],"to":[81,99],"perturbations":[82],"A.":[86],"We":[87],"both":[89],"canonical":[90],"condensed":[92],"forms.":[94],"Special":[95],"attention":[96],"is":[97],"paid":[98],"matrices":[100],"simple":[102],"eigenvalues.":[103],"Some":[104],"new":[105],"concepts,":[106],"such":[107],"as":[108],"quasi-Schur":[109],"forms":[110],"diagonally":[112],"spectral":[113],"matrices,":[114],"are":[115],"also":[116],"introduced":[117],"studied.":[119]},"counts_by_year":[],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-10-10T00:00:00"}
