{"id":"https://openalex.org/W1981435579","doi":"https://doi.org/10.1155/2012/504650","title":"The Group Involutory Matrix of the Combinations of Two Idempotent Matrices","display_name":"The Group Involutory Matrix of the Combinations of Two Idempotent Matrices","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W1981435579","doi":"https://doi.org/10.1155/2012/504650","mag":"1981435579"},"language":"uk","primary_location":{"id":"doi:10.1155/2012/504650","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2012/504650","pdf_url":"https://downloads.hindawi.com/journals/jam/2012/504650.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/jam/2012/504650.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020008474","display_name":"Lingling Wu","orcid":"https://orcid.org/0000-0002-8787-1284"},"institutions":[{"id":"https://openalex.org/I125068653","display_name":"IE University","ror":"https://ror.org/02jjdwm75","country_code":"ES","type":"education","lineage":["https://openalex.org/I125068653"]},{"id":"https://openalex.org/I167274908","display_name":"Guangxi University for Nationalities","ror":"https://ror.org/0495efn48","country_code":"CN","type":"education","lineage":["https://openalex.org/I167274908"]}],"countries":["CN","ES"],"is_corresponding":false,"raw_author_name":"Lingling Wu","raw_affiliation_strings":["College of Mathematics and Computer Science, Bijie University, Guizhou 551700, China","College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006, China","College of Mathematics and Computer Science, Bijie University, Guizhou 551700","College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006"],"affiliations":[{"raw_affiliation_string":"College of Mathematics and Computer Science, Bijie University, Guizhou 551700, China","institution_ids":[]},{"raw_affiliation_string":"College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006, China","institution_ids":["https://openalex.org/I167274908"]},{"raw_affiliation_string":"College of Mathematics and Computer Science, Bijie University, Guizhou 551700","institution_ids":["https://openalex.org/I125068653"]},{"raw_affiliation_string":"College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006","institution_ids":["https://openalex.org/I167274908"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102964204","display_name":"Xiaoji Liu","orcid":"https://orcid.org/0000-0003-3499-3947"},"institutions":[{"id":"https://openalex.org/I167274908","display_name":"Guangxi University for Nationalities","ror":"https://ror.org/0495efn48","country_code":"CN","type":"education","lineage":["https://openalex.org/I167274908"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Xiaoji Liu","raw_affiliation_strings":["College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006, China","College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006"],"affiliations":[{"raw_affiliation_string":"College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006, China","institution_ids":["https://openalex.org/I167274908"]},{"raw_affiliation_string":"College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006","institution_ids":["https://openalex.org/I167274908"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5055419952","display_name":"Yaoming Yu","orcid":null},"institutions":[{"id":"https://openalex.org/I56590836","display_name":"Monash University","ror":"https://ror.org/02bfwt286","country_code":"AU","type":"education","lineage":["https://openalex.org/I56590836"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Yaoming Yu","raw_affiliation_strings":["School of Mathematical Sciences, Monash University, Clayton Cambus, VIC 3800, Australia","School of Mathematical Sciences, Monash University, Clayton Cambus, VIC 3800"],"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Monash University, Clayton Cambus, VIC 3800, Australia","institution_ids":["https://openalex.org/I56590836"]},{"raw_affiliation_string":"School of Mathematical Sciences, Monash University, Clayton Cambus, VIC 3800","institution_ids":["https://openalex.org/I56590836"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5102964204"],"corresponding_institution_ids":["https://openalex.org/I167274908"],"apc_list":{"value":1025,"currency":"USD","value_usd":1025},"apc_paid":{"value":1025,"currency":"USD","value_usd":1025},"fwci":0.0,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.07283742,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"2012","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9958000183105469,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9764999747276306,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic 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/idempotent-matrix","display_name":"Idempotent matrix","score":0.727624237537384},{"id":"https://openalex.org/keywords/involutory-matrix","display_name":"Involutory matrix","score":0.7208824157714844},{"id":"https://openalex.org/keywords/idempotence","display_name":"Idempotence","score":0.7205115556716919},{"id":"https://openalex.org/keywords/group","display_name":"Group (periodic table)","score":0.6089985370635986},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5724163055419922},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5016710758209229},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4706275463104248},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.39175835251808167},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3852021098136902},{"id":"https://openalex.org/keywords/nonnegative-matrix","display_name":"Nonnegative matrix","score":0.18946662545204163},{"id":"https://openalex.org/keywords/symmetric-matrix","display_name":"Symmetric matrix","score":0.1562378704547882},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.10453543066978455},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.10389867424964905},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10282549262046814}],"concepts":[{"id":"https://openalex.org/C33375987","wikidata":"https://www.wikidata.org/wiki/Q3751720","display_name":"Idempotent matrix","level":3,"score":0.727624237537384},{"id":"https://openalex.org/C62767890","wikidata":"https://www.wikidata.org/wiki/Q1052516","display_name":"Involutory matrix","level":5,"score":0.7208824157714844},{"id":"https://openalex.org/C172252984","wikidata":"https://www.wikidata.org/wiki/Q368988","display_name":"Idempotence","level":2,"score":0.7205115556716919},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.6089985370635986},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5724163055419922},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5016710758209229},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4706275463104248},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.39175835251808167},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3852021098136902},{"id":"https://openalex.org/C139018669","wikidata":"https://www.wikidata.org/wiki/Q6961560","display_name":"Nonnegative matrix","level":4,"score":0.18946662545204163},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.1562378704547882},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.10453543066978455},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.10389867424964905},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10282549262046814},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C43617362","wikidata":"https://www.wikidata.org/wiki/Q170050","display_name":"Chromatography","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}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1155/2012/504650","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2012/504650","pdf_url":"https://downloads.hindawi.com/journals/jam/2012/504650.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jam/1355495142","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jam/1355495142","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},{"id":"pmh:oai:RePEc:hin:jnljam:504650","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/JAM/2012/504650.xml","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:doaj.org/article:a79f53fe8de246f4a3745ed57de5e29a","is_oa":true,"landing_page_url":"https://doaj.org/article/a79f53fe8de246f4a3745ed57de5e29a","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Journal of Applied Mathematics, Vol 2012 (2012)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2012/504650","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2012/504650","pdf_url":"https://downloads.hindawi.com/journals/jam/2012/504650.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2685896101","display_name":null,"funder_award_id":"11061005","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G285574548","display_name":null,"funder_award_id":"210164","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3293442493","display_name":null,"funder_award_id":"HCIC20110","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3317480652","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3890006155","display_name":null,"funder_award_id":"HCIC201103","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4020255992","display_name":null,"funder_award_id":"Project","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5939423041","display_name":null,"funder_award_id":"Technology","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5994120800","display_name":null,"funder_award_id":"Natural","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7033253288","display_name":null,"funder_award_id":"Grants","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G96548948","display_name":null,"funder_award_id":"201103","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"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/W1981435579.pdf","grobid_xml":"https://content.openalex.org/works/W1981435579.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W1979565872","https://openalex.org/W1980603302","https://openalex.org/W1984819577","https://openalex.org/W1999485964","https://openalex.org/W2007686302","https://openalex.org/W2038474216","https://openalex.org/W2043103277","https://openalex.org/W2055576996","https://openalex.org/W2058304700","https://openalex.org/W2061483962","https://openalex.org/W2085337896","https://openalex.org/W2092437860","https://openalex.org/W2093592728","https://openalex.org/W2121474135","https://openalex.org/W2171260444","https://openalex.org/W4243028848"],"related_works":["https://openalex.org/W2038474216","https://openalex.org/W2404563124","https://openalex.org/W2350953973","https://openalex.org/W1982314691","https://openalex.org/W2140229391","https://openalex.org/W2085337896","https://openalex.org/W2363422815","https://openalex.org/W3018972575","https://openalex.org/W2159877348","https://openalex.org/W324866729"],"abstract_inverted_index":{"We":[0],"discuss":[1],"the":[2],"following":[3],"problem:":[4],"when":[5],"a":[6,43,59],"P":[7,13,18,21,23,27,31,33,38],"+":[8,11,15,19,24,29],"b":[9,45,62],"Q":[10,14,17,22,26,28,32,34,40],"c":[12,47],"d":[16,49],"e":[20,51],"f":[25,53],"g":[30,55],"of":[35],"idempotent":[36],"matrices":[37],"and":[39,58],",":[41,44,46,48,50,52,54],"where":[42],"\u2208":[56],"\u2102":[57],"\u2260":[60,63],"0,":[61,64],"is":[65],"group":[66],"involutory.":[67]},"counts_by_year":[{"year":2019,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
