{"id":"https://openalex.org/W1968730170","doi":"https://doi.org/10.1137/0709013","title":"An Analysis of Equality in Certain Matrix Inequalities. II","display_name":"An Analysis of Equality in Certain Matrix Inequalities. II","publication_year":1972,"publication_date":"1972-03-01","ids":{"openalex":"https://openalex.org/W1968730170","doi":"https://doi.org/10.1137/0709013","mag":"1968730170"},"language":"en","primary_location":{"id":"doi:10.1137/0709013","doi":"https://doi.org/10.1137/0709013","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0709013","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-a","jufo-3","norway-2","poland-140","russia-white-list-1"],"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 Numerical Analysis","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"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/A5110127941","display_name":"Marvin D. Marcus","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Marvin Marcus","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5071998762","display_name":"William R. Gordon","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"William R. Gordon","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.11543048,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"9","issue":"1","first_page":"130","last_page":"136"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.8895999789237976,"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"}},"topics":[{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.8895999789237976,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.10899999737739563,"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.00019999999494757503,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/matrix-inequalities","display_name":"matrix inequalities","score":0.8149999976158142},{"id":"https://openalex.org/keywords/equality-conditions","display_name":"equality conditions","score":0.7369999885559082},{"id":"https://openalex.org/keywords/eigenvalues","display_name":"eigenvalues","score":0.9779999852180481},{"id":"https://openalex.org/keywords/singular-values","display_name":"singular values","score":1.0},{"id":"https://openalex.org/keywords/complex-matrices","display_name":"complex matrices","score":0.8949999809265137},{"id":"https://openalex.org/keywords/symmetric-polynomials","display_name":"symmetric polynomials","score":0.949999988079071},{"id":"https://openalex.org/keywords/unitary-similarity","display_name":"unitary similarity","score":0.7149999737739563},{"id":"https://openalex.org/keywords/upper-triangular-matrices","display_name":"upper triangular matrices","score":0.6869999766349792}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8407976627349854},{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.7941749095916748},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6421782970428467},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6082029938697815},{"id":"https://openalex.org/C104528550","wikidata":"https://www.wikidata.org/wiki/Q506265","display_name":"Triangular matrix","level":3,"score":0.592428982257843},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5732888579368591},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1843767762184143},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08201107382774353},{"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/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.0},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","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},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/0709013","doi":"https://doi.org/10.1137/0709013","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0709013","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-a","jufo-3","norway-2","poland-140","russia-white-list-1"],"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 Numerical Analysis","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":5,"referenced_works":["https://openalex.org/W1975172027","https://openalex.org/W1988442902","https://openalex.org/W1995847820","https://openalex.org/W2062757540","https://openalex.org/W2066434958"],"related_works":["https://openalex.org/W2082117320","https://openalex.org/W2053172998","https://openalex.org/W2044123734","https://openalex.org/W2064464944","https://openalex.org/W2043478332","https://openalex.org/W2044912247","https://openalex.org/W2351317322","https://openalex.org/W4403229258","https://openalex.org/W2050748768","https://openalex.org/W1971554223"],"abstract_inverted_index":{"In":[0],"this":[1],"paper":[2,38],"the":[3,6,17,34,37,132],"authors":[4],"investigate":[5],"cases":[7],"of":[8,13,22,131],"equality":[9,149],"in":[10,36,91],"some":[11],"generalizations":[12],"Schur\u2019s":[14],"inequality":[15],"relating":[16],"eigenvalues":[18,50],"and":[19,68,79],"singular":[20,69],"values":[21,70],"an":[23,43,178],"$n":[24,44],"\\times":[25,45],"n$":[26,46],"complex":[27,47],"matrix.":[28,184],"For":[29],"example,":[30],"it":[31],"follows":[32],"from":[33],"results":[35],"that":[39,160],"if":[40,80,123,152],"A":[41],"is":[42,86,129,154,177],"matrix":[48,157],"with":[49],"$\\lambda":[51],"_1":[52,60,72,107,167],",":[53,82,93,109,118,125,168],"\\cdots":[54,63,74,83,94,110,119,126,169],",\\lambda":[55,170],"_n":[56,66,77],"$":[57,78],"(with":[58],"$|\\lambda":[59],"|":[61],"\\geqq":[62,64,73,75,146],"|\\lambda":[65],"$)":[67],"$\\alpha":[71],"\\alpha":[76],"$h(x_1":[81,124],",x_k":[84,95,127],")$":[85,128],"a":[87,155],"nonzero":[88],"symmetric":[89],"polynomial":[90],"$x_1":[92],",1":[96],"\\leqq":[97,99,115],"k":[98],"n$,":[100],"then":[101,148],"for":[102],"$q":[103],">":[104],"0$,":[105,147],"\\[h(|\\lambda":[106],"|^q":[108,113],",|\\lambda":[111],"_k":[112,171],")":[114,172],"h(\\alpha":[116],"_1^q":[117],",\\alpha":[120],"_k^q":[121],");\\]":[122],"not":[130],"form":[133],"$\\Sigma":[134],"_{j":[135],"=":[136,141,164],"0}^r":[137],"d_j":[138],"(\\Pi":[139],"_{t":[140],"1}^k":[142],"x_t":[143],")^j":[144],",d_j":[145],"holds":[150],"only":[151],"there":[153],"unitary":[156],"U":[158],"such":[159],"$U^":[161],"*":[162],"AU":[163],"{\\operatorname{diag}}":[165],"(\\lambda":[166],"+":[173],"T$,":[174],"where":[175],"T":[176],"$(n":[179],"-":[180],"k)$-square":[181],"upper":[182],"triangular":[183]},"counts_by_year":[],"updated_date":"2026-09-22T06:50:58.402377","created_date":"2016-06-24T00:00:00"}