{"id":"https://openalex.org/W2003892203","doi":"https://doi.org/10.1137/050638114","title":"Further Results on the Reverse Order Law for Generalized Inverses","display_name":"Further Results on the Reverse Order Law for Generalized Inverses","publication_year":2007,"publication_date":"2007-12-13","ids":{"openalex":"https://openalex.org/W2003892203","doi":"https://doi.org/10.1137/050638114","mag":"2003892203"},"language":"en","primary_location":{"id":"doi:10.1137/050638114","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050638114","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/A5021851290","display_name":"Dragan S. Djordjevi\u0107","orcid":"https://orcid.org/0000-0002-4687-9327"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Dragan S. Djordjevi\u0107","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5021851290"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.9886,"has_fulltext":false,"cited_by_count":60,"citation_normalized_percentile":{"value":0.90469696,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"29","issue":"4","first_page":"1242","last_page":"1246"},"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/T12037","display_name":"Algebraic and Geometric Analysis","score":0.9919000267982483,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9908000230789185,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8316079378128052},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5509827733039856},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4784255623817444},{"id":"https://openalex.org/keywords/law","display_name":"Law","score":0.37662962079048157},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.3702823519706726},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3326738476753235},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2571392059326172}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8316079378128052},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5509827733039856},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4784255623817444},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.37662962079048157},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3702823519706726},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3326738476753235},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2571392059326172},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/050638114","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050638114","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":[{"display_name":"Peace, Justice and strong institutions","score":0.7099999785423279,"id":"https://metadata.un.org/sdg/16"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":14,"referenced_works":["https://openalex.org/W1503134271","https://openalex.org/W1517447268","https://openalex.org/W1967898597","https://openalex.org/W2007077313","https://openalex.org/W2016864970","https://openalex.org/W2020491530","https://openalex.org/W2020566771","https://openalex.org/W2024443953","https://openalex.org/W2055576996","https://openalex.org/W2075066642","https://openalex.org/W2075224042","https://openalex.org/W2077264102","https://openalex.org/W2097720652","https://openalex.org/W2187494506"],"related_works":["https://openalex.org/W1973641256","https://openalex.org/W2089274067","https://openalex.org/W56994397","https://openalex.org/W1121942432","https://openalex.org/W2418069309","https://openalex.org/W2376158287","https://openalex.org/W1526165314","https://openalex.org/W2359859943","https://openalex.org/W2591547446","https://openalex.org/W578832185"],"abstract_inverted_index":{"Previous":[0,335],"article":[1,3,336,338],"Next":[2,337],"Further":[4,932,1146],"Results":[5,39,1147],"on":[6,102,343,456,597,655,733,934,1046,1148,1181],"the":[7,30,80,103,137,231,344,367,411,415,497,598,734,780,785,803,825,870,935,940,964,1051,1106,1128,1149],"Reverse":[8,227,280,345,407,623,708,799,871,914,986,1005,1124,1150],"Order":[9,346,364,408,555,872,1151],"Law":[10,347,365,409,873,1152],"for":[11,29,84,208,230,257,283,348,366,410,557,602,626,711,738,784,802,824,844,874,895,939,963,972,1008,1055,1074,1103,1127,1153,1199,1220],"Generalized":[12,238,683,876,965],"InversesDragan":[13],"S.":[14,16,75,124],"Djordjevi\u0107Dragan":[15],"Djordjevi\u0107https://doi.org/10.1137/050638114PDFBibTexSections":[17],"ToolsAdd":[18],"to":[19,41,79,254,391,649],"favoritesExport":[20],"CitationTrack":[21],"CitationsEmail":[22],"SectionsAboutAbstractThe":[23],"reverse":[24,81,138,522,735,761,781,892,936,1052,1071,1100],"order":[25,82,139,206,228,281,333,523,600,624,709,736,762,782,800,893,915,937,987,1006,1053,1072,1101,1125,1206],"rule":[26,83,207],"$(AB)^\\dag=B^\\dag":[27],"A^\\dag$":[28],"Moore\u2013Penrose":[31,412,786,805,941,1129],"inverse":[32,105,158,317,458,806,942],"is":[33,313,501],"established":[34],"in":[35,159,180,433,439,505,807,917,943,989,1012,1032,1035],"several":[36],"equivalent":[37],"forms.":[38],"related":[40],"other":[42],"generalized":[43,85,104,178,210,232,316,481,712,826],"inverses":[44,179,233,259,432,713,848,1011],"are":[45],"also":[46],"proved.[1]":[47],"Google":[48,70,72,95,120,122,170,197,307],"Scholar[2]":[49],"R.":[50,163],"H.":[51,309],"Bouldin,":[52],"The":[53,126,154,385,475,553,821,1070,1099],"pseudo-inverse":[54],"of":[55,106,128,136,234,260,291,293,318,350,371,394,417,421,459,463,479,499,577,606,644,646,652,674,682,714,743,869,878,897,967,1082,1105,1108,1131,1157,1161],"a":[56,107,315,675,1158],"product,":[57,109],"SIAM":[58,110,212],"J.":[59,143,151,152,213,310],"Appl.":[60,263],"Math.,":[61],"24":[62],"(1973),":[63],"pp.":[64,93,114,147,168,191,219,245,268,301,325],"489\u2013495.":[65],"SMJMAP":[66],"0036-1399":[67],"LinkISIGoogle":[68,118,223],"Scholar[3]":[69],"Scholar[4]":[71],"Scholar[5]":[73],"D.":[74],"Djordjevi\u0107,":[76],"Unified":[77],"approach":[78],"inverses,":[86],"Acta":[87],"Sci.":[88],"Math.":[89,161,264],"(Szeged),":[90],"67":[91],"(2001),":[92],"761\u2013776.":[94],"Scholar[6]":[96],"T.":[97],"N.":[98],"E.":[99],"Greville,":[100],"Note":[101,342],"matrix":[108,236,261,608,1084],"Rev.,":[111],"8":[112,588,686],"(1966),":[113],"518\u2013521.":[115],"SIREAD":[116],"0036-1445":[117],"Scholar[7]":[119],"Scholar[8]":[121],"Scholar[9]":[123],"Izumino,":[125],"product":[127,498,1107],"operators":[129,500],"with":[130,414,945],"closed":[131],"range":[132],"and":[133,156,177,182,201,202,276,277,287,388,435,443,477,484,508,527,567,582,610,630,659,718,746,765,789,829,853,883,901,920,948,974,992,1028,1058,1078,1086,1112,1134,1155,1163,1184,1201,1222],"an":[134,502,968],"extension":[135],"law,":[140,524],"Tohoku":[141],"Math":[142],"(2),":[144],"34":[145],"(1982),":[146],"43\u201352.":[148],"CrossrefISIGoogle":[149,195,247,272,305,329],"Scholar[10]":[150],"Koliha,":[153],"Drazin":[155,672],"Moore-Penrose":[157,258,431],"$C^*$-algebras,":[160],"Proc.":[162],"Ir.":[164],"Acad.,":[165],"99A":[166],"(1999),":[167],"17\u201327.":[169],"Scholar[11]":[171],"M.":[172,274],"Z.":[173],"Nashed,":[174],"Inner,":[175],"outer,":[176],"Banach":[181],"Hilbert":[183,506,656,850],"spaces,":[184],"Numer.":[185],"Funct.":[186],"Anal.":[187,215],"Optim.,":[188],"9":[189],"(1987),":[190],"261\u2013325.":[192],"NFAODL":[193],"0163-0563":[194],"Scholar[12]":[196],"Scholar[13]":[198],"W.":[199,278],"Sun":[200],"Y.":[203,225,249],"Wei,":[204],"Inverse":[205,370,413,673],"weighted":[209,436,804],"inverse,":[211],"Matrix":[214,373,579,880,1159,1182],"Appl.,":[216,242,298,322],"19":[217],"(1998),":[218],"772\u2013775.":[220],"SJMAEL":[221],"0895-4798":[222],"Scholar[14]":[224],"Tian,":[226,250],"laws":[229,282,625,823,843,988,1007,1073],"multiple":[235,715,1083],"products,":[237,262],"Inverses,":[239],"Linear":[240,296,320,419],"Algebra":[241,297,321,900,991,1034],"211":[243],"(1994),":[244,324],"85\u2013100.":[246],"Scholar[15]":[248],"Using":[251],"rank":[252],"formulas":[253],"characterize":[255],"equalities":[256],"Comput.,":[265],"147":[266],"(2004),":[267],"581\u2013600.":[269],"AMHCBQ":[270],"0096-3003":[271],"Scholar[16]":[273],"Wei":[275],"Guo,":[279],"least":[284,603,739],"squares":[285],"g-inverses":[286,290],"minimum":[288],"norm":[289],"products":[292,896],"two":[294,879,1109],"matrices,":[295],"342":[299],"(2002),":[300],"117\u2013132.":[302],"LAAPAW":[303,327],"0024-3795":[304,328],"Scholar[17]":[306],"Scholar[18]":[308],"Werner,":[311],"When":[312],"$B^-A^-$":[314],"$AB$?,":[319],"210":[323],"255\u2013263.":[326],"ScholarKeywordsMoore\u2013Penrose":[330],"inversegeneralized":[331,1204],"inversesreverse":[332,1205],"law":[334,601,710,737,763,783,801,894,916,938,1054,1102,1126],"FiguresRelatedReferencesCited":[339],"ByDetails":[340],"A":[341,595,731,865,1044],"g-Inverse":[349],"Operator":[351,583,647,653,660,969],"ProductAxioms,":[352],"Vol.":[353,375,396,423,445,465,487,511,543,585,613,633,662,698,721,749,768,791,811,831,855,904,922,950,976,995,1017,1060,1089,1114,1136,1165],"11,":[354,376],"No.":[355,377,398,467,513,545,587,615,635,664,700,723,751,770,813,833,857,906,924,952,978,997,1019,1062,1091,1116,1138,1167],"5":[356,618,998],"|":[357,379,400,425,447,469,489,515,547,589,617,637,666,685,702,725,753,772,793,815,835,859,885,908,926,954,980,999,1021,1038,1064,1093,1118,1140,1169],"12":[358,590,773,1117],"May":[359,704,861,1190],"2022":[360,382,403],"Cross":[361,383,404,429,451,473,493,519,551,593,621,641,670,689,706,729,757,776,797,819,839,863,889,912,930,958,984,1003,1025,1042,1068,1097,1122,1144,1173],"Ref":[362,384,405,430,452,474,494,520,552,594,622,642,671,690,707,730,758,777,798,820,840,864,890,913,931,959,985,1004,1026,1043,1069,1098,1123,1145,1174],"Forward":[363,554],"Reflexive":[368,875],"Inner":[369],"Multiple":[372,578],"ProductsAxioms,":[374],"3":[378,564,636,814,979],"10":[380,514],"March":[381,619],"\u03bb-Aluthge":[386],"transform":[387],"its":[389,902,993],"applications":[390],"some":[392,480,1047],"classes":[393],"operatorsFilomat,":[395],"36,":[397,977],"1":[399,401,448,546,560,570,665,794,836,860,886,909,927,955,1000,1022,1039,1063,1065,1094,1119,1139,1141,1168,1170],"Jan":[402,887,1040,1142,1171],"Developing":[406],"Product":[416],"Three":[418],"OperatorsJournal":[420],"Mathematics,":[422,464,612],"2021":[424,428,450],"31":[426],"Oct":[427,956],"rings":[434,944],"partial":[437],"isometries":[438],"C*\u2212":[440],"algebrasApplied":[441],"Mathematics":[442,788,828,852,919,947,1015,1057,1111,1224],"Computation,":[444,790,830,854,921,949,1059,1113],"395":[446],"Apr":[449],"Some":[453,650],"additive":[454],"results":[455,933,1049],"M-P":[457],"operator":[460,504,716,744],"productAsian-European":[461],"Journal":[462,971,1180],"14,":[466],"01":[468],"18":[470],"October":[471,668,687],"2019":[472,518,550,592,620],"structures":[476],"properties":[478],"invertible":[482],"operatorsLinear":[483,899],"Multilinear":[485,509,631,719,747,766,1087],"Algebra,":[486,510,632,720,748,767,1088],"99":[488],"2":[490,562,572,616,638,1020],"December":[491,549,1193],"2020":[492],"Conditions":[495],"that":[496],"EP":[503],"C*-moduleLinear":[507],"68,":[512],"16":[516],"January":[517],"Mixed-type":[521,841,891],"ternary":[525],"powers":[526],"functional":[528],"calculusRevista":[529],"de":[530,534],"la":[531],"Real":[532],"Academia":[533],"Ciencias":[535],"Exactas,":[536],"F\u00edsicas":[537],"y":[538],"Naturales.":[539],"Serie":[540],"A.":[541],"Matem\u00e1ticas,":[542],"114,":[544],"6":[548,701,1092],"Laws":[556,962],"$$\\{1,2,3\\}$$":[558],"{":[559,569],",":[561,563,571,573],"}":[565,575],"-":[566],"$$\\{1,2,4\\}$$":[568],"4":[574,834,953],"-Inverses":[576],"ProductsComplex":[580],"Analysis":[581,658,1133,1183],"Theory,":[584,661],"13,":[586],"April":[591,982],"note":[596,732],"forward":[599],"square":[604,740],"g-inverse":[605],"three":[607,898],"productsComputational":[609],"Applied":[611,1202,1223],"38,":[614],"{1,":[627,845,1075,1079],"3}-generalized":[628],"inversesLinear":[629],"67,":[634],"February":[639],"2018":[640],"Applications":[643,884],"Completions":[645],"Matrices":[648],"Properties":[651,681],"Products":[654],"SpacesComplex":[657],"12,":[663],"26":[667],"2016":[669,705],"$$2":[676],"\\times":[677],"2$$":[678],"Block":[679],"MatrixAlgebraic":[680],"Inverses":[684,877,966],"2017":[688],"Commuting":[691],"$${{C^*}}$$":[692],"C":[693,808],"\u2217":[694],"modular":[695],"operatorsAequationes":[696],"mathematicae,":[697,810],"90,":[699],"30":[703,816],"productLinear":[717,745],"64,":[722,750],"7":[724,752,771,925],"11":[726,907],"September":[727,755],"2015":[728,756],"g":[741],"-inverse":[742],"21":[754],"Hartwig's":[759],"triple":[760],"revisitedLinear":[764],"62,":[769],"July":[774],"2013":[775,796],"Identities":[778],"concerning":[779,1050],"inverseApplied":[787],"220":[792],"Sep":[795,1066,1095],"*-algebrasAequationes":[809],"85,":[812],"August":[817,1188],"2012":[818,838,862,888],"absorption":[822],"inversesApplied":[827],"219,":[832],"Nov":[837],"reverse-order":[842],"3,":[846],"4}-generalized":[847],"over":[849],"spacesApplied":[851],"218,":[856,923,951],"17":[858],"New":[866],"Equivalent":[867],"Condition":[868],"ProductsInformation":[881],"Engineering":[882],"Applications,":[903,994,1135,1164],"435,":[905],"Dec":[910,928],"2011":[911,929,957,983,1002,1024,1041],"C\u2217-algebrasApplied":[918],"involutionApplied":[946],"Mixed-Type":[960],"Reverse-Order":[961],"ProductArabian":[970],"Science":[973],"Engineering,":[975],"20":[981],"C\u2217-algebrasLinear":[990],"434,":[996],"Mar":[1001],"{1,3,4}":[1009],"-generalized":[1010],"C\u2217":[1013],"-algebrasApplied":[1014],"Letters,":[1016],"24,":[1018],"Feb":[1023,1120],"Regular":[1027],"Singular":[1029],"Boundary":[1030],"Problems":[1031],"MapleComputer":[1033],"Scientific":[1036],"Computing":[1037],"comment":[1045],"recent":[1048],"{1,3,4}-inversesApplied":[1056],"217,":[1061],"2010":[1067,1096,1121,1143,1172],"2,":[1076,1080],"3}-":[1077],"4}-inverses":[1081],"productsLinear":[1085],"58,":[1090],"{1,3,4}-inverse":[1104],"matricesApplied":[1110],"215,":[1115],"inverseJournal":[1130],"Mathematical":[1132],"361,":[1137],"-Inverse":[1154,1156],"ProductJournal":[1160],"Inequalities":[1162],"2010,":[1166],"Volume":[1175],"29,":[1176],"Issue":[1177],"4|":[1178],"2008SIAM":[1179],"Applications1051-1410":[1185],"History":[1186],"Submitted:12":[1187],"2005Accepted:10":[1189],"2007Published":[1191],"online:13":[1192],"2007":[1194,1197],"InformationCopyright":[1195],"\u00a9":[1196],"Society":[1198],"Industrial":[1200,1221],"MathematicsKeywordsMoore\u2013Penrose":[1203],"lawMSC":[1207],"codes47A0515A09PDF":[1208],"Download":[1209],"Article":[1210],"&":[1211],"Publication":[1212],"DataArticle":[1213],"DOI:10.1137/050638114Article":[1214],"page":[1215],"range:pp.":[1216],"1242-1246ISSN":[1217],"(print):0895-4798ISSN":[1218],"(online):1095-7162Publisher:Society":[1219]},"counts_by_year":[{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":4}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
