{"id":"https://openalex.org/W1967301416","doi":"https://doi.org/10.1137/s0895479896313334","title":"Real Solutions of the Equation $\\Phi^t(A)=\\frac1nJ_n$\\protect\\unboldmath","display_name":"Real Solutions of the Equation $\\Phi^t(A)=\\frac1nJ_n$\\protect\\unboldmath","publication_year":2000,"publication_date":"2000-01-01","ids":{"openalex":"https://openalex.org/W1967301416","doi":"https://doi.org/10.1137/s0895479896313334","mag":"1967301416"},"language":"en","primary_location":{"id":"doi:10.1137/s0895479896313334","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479896313334","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":null,"display_name":"Zhang Xian","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Zhang Xian","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Yang Zhongpeng","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yang Zhongpeng","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5102208094","display_name":"Cao Chong-guang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cao Chongguang","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.9375,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.74385291,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"21","issue":"2","first_page":"642","last_page":"645"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9908999800682068,"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":0.9908999800682068,"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":0.9674999713897705,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.9469000101089478,"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":0.7588330507278442},{"id":"https://openalex.org/keywords/invertible-matrix","display_name":"Invertible matrix","score":0.6526864767074585},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6086946725845337},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.57182776927948},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.5575088858604431},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5245478749275208},{"id":"https://openalex.org/keywords/diagonal","display_name":"Diagonal","score":0.5188083052635193},{"id":"https://openalex.org/keywords/transpose","display_name":"Transpose","score":0.4855215549468994},{"id":"https://openalex.org/keywords/trace","display_name":"TRACE (psycholinguistics)","score":0.44489070773124695},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4304540753364563},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.4247889816761017},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.4228413999080658},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3868274688720703},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3416503071784973},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.27070608735084534},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.26135504245758057},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1855449080467224},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.13670918345451355},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.07901343703269958}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7588330507278442},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.6526864767074585},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6086946725845337},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.57182776927948},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5575088858604431},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5245478749275208},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.5188083052635193},{"id":"https://openalex.org/C200106649","wikidata":"https://www.wikidata.org/wiki/Q223683","display_name":"Transpose","level":3,"score":0.4855215549468994},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.44489070773124695},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4304540753364563},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.4247889816761017},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.4228413999080658},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3868274688720703},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3416503071784973},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.27070608735084534},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26135504245758057},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1855449080467224},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.13670918345451355},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.07901343703269958},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","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/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/s0895479896313334","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479896313334","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":1,"referenced_works":["https://openalex.org/W170003236"],"related_works":["https://openalex.org/W2003786218","https://openalex.org/W2376104713","https://openalex.org/W2356636683","https://openalex.org/W4205331489","https://openalex.org/W4385493077","https://openalex.org/W201246341","https://openalex.org/W2084928321","https://openalex.org/W1625431148","https://openalex.org/W2195730736","https://openalex.org/W3028475238"],"abstract_inverted_index":{"For":[0,51],"nonsingular":[1],"n":[2,4,80],"x":[3],"matrices":[5],"A,":[6],"$\\Phi":[7,71,105,155],"(A)=A\\circ":[8,156],"A^{-T}$":[9],"($\\circ":[10],"$":[11],"denotes":[12],"the":[13,18,43,46,59,76,99,130,153],"Hadamard":[14],"product":[15],"and":[16,36,49,68,79,84,116,138],"A-T":[17],"inverse":[19],"transpose,":[20],"(A-1":[21],")T,":[22],"of":[23],"A)":[24],"arises":[25],"in":[26,37,140],"mathematical":[27],"control":[28],"theory":[29],"associated":[30],"with":[31],"chemical":[32],"engineering":[33],"design":[34],"problems":[35],"a":[38,159],"matrix":[39],"theoretic":[40],"problem":[41,131],"involving":[42],"relation":[44],"between":[45],"diagonal":[47],"entries":[48],"eigenvalues.":[50],"any":[52],"positive":[53],"integer":[54],"t,":[55],"we":[56],"show":[57],"that":[58,85,98,132],"equation":[60,101,154],"$$":[61,64],"\\Phi":[62],"^t(A)=\\frac{1}{n}J_n":[63],"hasat":[65],"least":[66],"2t-1":[67],"2t":[69],"mutually":[70],"$-distinct":[72,106],"real":[73,107,119,160],"solutions":[74,87,120],"for":[75,114,121],"case":[77],"n=4":[78],">":[81],"4,":[82],"respectively,":[83],"these":[86],"can":[88],"be":[89],"determined":[90],"by":[91,136],"an":[92],"inverted":[93],"iteration.":[94],"We":[95],"also":[96],"prove":[97],"above":[100,127],"has":[102,117,133],"only":[103],"one":[104],"solution":[108],"{\\scriptsize":[109],"$(\\begin{array}{cc}":[110],"1&1\\\\[-2pt]":[111],"1&-1":[112],"\\end{array})$}":[113],"$t=1$":[115],"no":[118],"$t\\ge":[122],"2$":[123],"when":[124],"n=2.":[125],"The":[126],"results":[128],"answer":[129],"been":[134],"proposed":[135],"Johnson":[137],"Shapiro":[139],"[":[141],"SIAM":[142],"J.":[143],"Algebraic":[144],"Discrete":[145],"Methods,":[146],"7":[147],"(1986),":[148],"pp.":[149],"627--644].":[150],"When":[151],"does":[152],"A^{-T}=\\frac{1}{n}J_n$":[157],"have":[158],"solution?":[161]},"counts_by_year":[],"updated_date":"2026-04-16T08:26:57.006410","created_date":"2025-10-10T00:00:00"}