{"id":"https://openalex.org/W2012593053","doi":"https://doi.org/10.1137/130915339","title":"$M$-Tensors and Some Applications","display_name":"$M$-Tensors and Some Applications","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W2012593053","doi":"https://doi.org/10.1137/130915339","mag":"2012593053"},"language":"en","primary_location":{"id":"doi:10.1137/130915339","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130915339","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":true,"oa_status":"green","oa_url":"https://espace.curtin.edu.au/bitstream/20.500.11937/35236/2/200863_133060_SIMAX-M-Tensor-ZQZ-2014.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100426746","display_name":"Liping Zhang","orcid":"https://orcid.org/0000-0002-3839-9470"},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Liping Zhang","raw_affiliation_strings":["Tsinghua University"],"affiliations":[{"raw_affiliation_string":"Tsinghua University","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5011886900","display_name":"Liqun Qi","orcid":"https://orcid.org/0000-0002-1112-5250"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Liqun Qi","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5037628561","display_name":"Guanglu Zhou","orcid":"https://orcid.org/0000-0002-8811-885X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Guanglu Zhou","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5100426746"],"corresponding_institution_ids":["https://openalex.org/I99065089"],"apc_list":null,"apc_paid":null,"fwci":13.1337,"has_fulltext":true,"cited_by_count":287,"citation_normalized_percentile":{"value":0.99573561,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":100},"biblio":{"volume":"35","issue":"2","first_page":"437","last_page":"452"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational 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/T12303","display_name":"Tensor decomposition and applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational 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/T11304","display_name":"Advanced Neuroimaging Techniques and Applications","score":0.9768000245094299,"subfield":{"id":"https://openalex.org/subfields/2741","display_name":"Radiology, Nuclear Medicine and Imaging"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9387999773025513,"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/tensor","display_name":"Tensor (intrinsic definition)","score":0.8651400804519653},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8354308605194092},{"id":"https://openalex.org/keywords/positive-definiteness","display_name":"Positive definiteness","score":0.8122367858886719},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.8121974468231201},{"id":"https://openalex.org/keywords/invariants-of-tensors","display_name":"Invariants of tensors","score":0.6830326914787292},{"id":"https://openalex.org/keywords/diagonal","display_name":"Diagonal","score":0.6300307512283325},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5241494178771973},{"id":"https://openalex.org/keywords/symmetric-tensor","display_name":"Symmetric tensor","score":0.48483431339263916},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4703013300895691},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.364324688911438},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3551281690597534},{"id":"https://openalex.org/keywords/exact-solutions-in-general-relativity","display_name":"Exact solutions in general relativity","score":0.11982724070549011},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10956811904907227},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10908046364784241},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.06781268119812012}],"concepts":[{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.8651400804519653},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8354308605194092},{"id":"https://openalex.org/C2778265155","wikidata":"https://www.wikidata.org/wiki/Q7233276","display_name":"Positive definiteness","level":4,"score":0.8122367858886719},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.8121974468231201},{"id":"https://openalex.org/C158158286","wikidata":"https://www.wikidata.org/wiki/Q9009080","display_name":"Invariants of tensors","level":3,"score":0.6830326914787292},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.6300307512283325},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5241494178771973},{"id":"https://openalex.org/C20178491","wikidata":"https://www.wikidata.org/wiki/Q2204117","display_name":"Symmetric tensor","level":3,"score":0.48483431339263916},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4703013300895691},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.364324688911438},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3551281690597534},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.11982724070549011},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10956811904907227},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10908046364784241},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.06781268119812012}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/130915339","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130915339","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"},{"id":"pmh:oai:espace.curtin.edu.au:20.500.11937/35236","is_oa":true,"landing_page_url":"http://hdl.handle.net/20.500.11937/35236","pdf_url":"https://espace.curtin.edu.au/bitstream/20.500.11937/35236/2/200863_133060_SIMAX-M-Tensor-ZQZ-2014.pdf","source":{"id":"https://openalex.org/S4306401790","display_name":"eSpace (Curtin University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205640436","host_organization_name":"Curtin University","host_organization_lineage":["https://openalex.org/I205640436"],"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":"Journal Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.726.5366","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.726.5366","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.polyu.edu.hk/ama/staff/new/qilq/SIMAX-M-Tensor-ZQZ-2014.pdf","raw_type":"text"},{"id":"pmh:oai:ira.lib.polyu.edu.hk:10397/9440","is_oa":false,"landing_page_url":"http://hdl.handle.net/10397/9440","pdf_url":null,"source":{"id":"https://openalex.org/S4306400205","display_name":"PolyU Institutional Research Archive (Hong Kong Polytechnic University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I14243506","host_organization_name":"Hong Kong Polytechnic University","host_organization_lineage":["https://openalex.org/I14243506"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Journal/Magazine Article"}],"best_oa_location":{"id":"pmh:oai:espace.curtin.edu.au:20.500.11937/35236","is_oa":true,"landing_page_url":"http://hdl.handle.net/20.500.11937/35236","pdf_url":"https://espace.curtin.edu.au/bitstream/20.500.11937/35236/2/200863_133060_SIMAX-M-Tensor-ZQZ-2014.pdf","source":{"id":"https://openalex.org/S4306401790","display_name":"eSpace (Curtin University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205640436","host_organization_name":"Curtin University","host_organization_lineage":["https://openalex.org/I205640436"],"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":"Journal Article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G114794119","display_name":null,"funder_award_id":"PolyU 501909","funder_id":"https://openalex.org/F4320322598","funder_display_name":"Hong Kong Polytechnic University"},{"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/G3328376380","display_name":null,"funder_award_id":"502111","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G37568934","display_name":null,"funder_award_id":"Grant","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G391238517","display_name":null,"funder_award_id":", and","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4478465004","display_name":null,"funder_award_id":"502111","funder_id":"https://openalex.org/F4320322598","funder_display_name":"Hong Kong Polytechnic University"},{"id":"https://openalex.org/G5410887993","display_name":null,"funder_award_id":"11171180","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5823696539","display_name":null,"funder_award_id":"501212","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/G6302506420","display_name":null,"funder_award_id":"502510","funder_id":"https://openalex.org/F4320322598","funder_display_name":"Hong Kong Polytechnic University"},{"id":"https://openalex.org/G7356035589","display_name":null,"funder_award_id":"11271221","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8304336271","display_name":null,"funder_award_id":"501212","funder_id":"https://openalex.org/F4320322598","funder_display_name":"Hong Kong Polytechnic University"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322598","display_name":"Hong Kong Polytechnic University","ror":"https://ror.org/0030zas98"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2012593053.pdf"},"referenced_works_count":27,"referenced_works":["https://openalex.org/W1500921805","https://openalex.org/W1514622556","https://openalex.org/W1525100186","https://openalex.org/W1807190763","https://openalex.org/W1971702983","https://openalex.org/W2014748986","https://openalex.org/W2015156848","https://openalex.org/W2020374088","https://openalex.org/W2024266080","https://openalex.org/W2041197939","https://openalex.org/W2044269663","https://openalex.org/W2066105650","https://openalex.org/W2075801894","https://openalex.org/W2076404366","https://openalex.org/W2086589508","https://openalex.org/W2090208105","https://openalex.org/W2095180325","https://openalex.org/W2113722075","https://openalex.org/W2118214857","https://openalex.org/W2120001551","https://openalex.org/W2121275187","https://openalex.org/W2129444930","https://openalex.org/W2150818617","https://openalex.org/W2159552694","https://openalex.org/W2610857016","https://openalex.org/W2964074814","https://openalex.org/W3035235874"],"related_works":["https://openalex.org/W2369793817","https://openalex.org/W2364489971","https://openalex.org/W2361979571","https://openalex.org/W4205226845","https://openalex.org/W3208377712","https://openalex.org/W2043819621","https://openalex.org/W2373372131","https://openalex.org/W1977297792","https://openalex.org/W2964211946","https://openalex.org/W2614099022"],"abstract_inverted_index":{"We":[0,10,20,84,109,132,150],"introduce":[1,111],"$M$-tensors.":[2],"This":[3],"concept":[4,7],"extends":[5],"the":[6,14,28,52,61,65,138,157],"of":[8,38,55,64,68,71,105,127,141,144,160],"$M$-matrices.":[9],"denote":[11],"$Z$-tensors":[12,26],"as":[13],"tensors":[15],"with":[16,148],"nonpositive":[17],"off-diagonal":[18],"entries.":[19],"show":[21,59,85,123],"that":[22,60,86,124],"$M$-tensors":[23,106,129],"must":[24,32,42],"be":[25,33,43],"and":[27,78,93,114],"maximal":[29],"diagonal":[30,36],"entry":[31],"nonnegative.":[34,44,100],"The":[35],"elements":[37],"a":[39,87,142,162],"symmetric":[40,46],"$M$-tensor":[41,47,73,91],"A":[45],"is":[48,74,80,89],"copositive.":[49],"Based":[50],"on":[51],"spectral":[53,103],"theory":[54],"nonnegative":[56],"tensors,":[57],"we":[58,122],"minimal":[62],"value":[63],"real":[66],"parts":[67],"all":[69,96,125],"eigenvalues":[70],"an":[72,90,153],"its":[75,81,97],"smallest":[76,82],"H$^+$-eigenvalue":[77],"also":[79,110,151],"H-eigenvalue.":[83],"$Z$-tensor":[88],"if":[92,95],"only":[94],"H$^+$-eigenvalues":[98],"are":[99,107,118,130],"Some":[101],"further":[102],"properties":[104],"given.":[108,119],"strong":[112,128],"$M$-tensors,":[113],"some":[115],"corresponding":[116],"conclusions":[117],"In":[120],"particular,":[121],"$H$-eigenvalues":[126],"positive.":[131],"apply":[133],"this":[134],"property":[135],"to":[136],"study":[137],"positive":[139,158],"definiteness":[140,159],"class":[143],"multivariate":[145,163],"forms":[146],"associated":[147],"$Z$-tensors.":[149],"propose":[152],"algorithm":[154],"for":[155],"testing":[156],"such":[161],"form.":[164]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":21},{"year":2024,"cited_by_count":12},{"year":2023,"cited_by_count":15},{"year":2022,"cited_by_count":16},{"year":2021,"cited_by_count":22},{"year":2020,"cited_by_count":32},{"year":2019,"cited_by_count":29},{"year":2018,"cited_by_count":27},{"year":2017,"cited_by_count":27},{"year":2016,"cited_by_count":40},{"year":2015,"cited_by_count":29},{"year":2014,"cited_by_count":16}],"updated_date":"2026-04-16T08:26:57.006410","created_date":"2025-10-10T00:00:00"}
