{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T09:01:57Z","timestamp":1776762117027,"version":"3.51.2"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T00:00:00Z","timestamp":1776729600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T00:00:00Z","timestamp":1776729600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12471129"],"award-info":[{"award-number":["12471129"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2026,9]]},"DOI":"10.1007\/s40314-026-03739-9","type":"journal-article","created":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T08:22:14Z","timestamp":1776759734000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Characteristics and perturbation analysis for the core inverse of tensors over the quaternion skew field"],"prefix":"10.1007","volume":"45","author":[{"given":"Ning","family":"Zhang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tianjuan","family":"Wu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4920-3370","authenticated-orcid":false,"given":"Haifeng","family":"Ma","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,4,21]]},"reference":[{"key":"3739_CR1","volume-title":"Quaternionic quantum mechanics and quantum fields","author":"SL Adler","year":"1995","unstructured":"Adler SL (1995) Quaternionic quantum mechanics and quantum fields. Oxford University Press, New York"},{"issue":"6","key":"3739_CR2","doi-asserted-by":"publisher","first-page":"681","DOI":"10.1080\/03081080902778222","volume":"58","author":"OM Baksalary","year":"2010","unstructured":"Baksalary OM, Trenkler G (2010) Core inverse of matrices. Linear Multilinear Algebra 58(6):681\u2013697","journal-title":"Linear Multilinear Algebra"},{"key":"3739_CR3","first-page":"450","volume":"236","author":"OM Baksalary","year":"2014","unstructured":"Baksalary OM, Trenkler G (2014) On a generalized core inverse. Appl Math Comput 236:450\u2013457","journal-title":"Appl Math Comput"},{"key":"3739_CR4","doi-asserted-by":"publisher","first-page":"104326","DOI":"10.1109\/ACCESS.2021.3097756","volume":"9","author":"E Bayro-Corrochano","year":"2021","unstructured":"Bayro-Corrochano E (2021) A survey on quaternion algebra and geometric algebra applications in engineering and computer science 1995\u20132020. IEEE Access 9:104326\u2013104355","journal-title":"IEEE Access"},{"key":"3739_CR5","volume-title":"Generalized inverses: theory and applications","author":"A Ben-Israel","year":"2003","unstructured":"Ben-Israel A, Greville TNE (2003) Generalized inverses: theory and applications, 2nd edn. Springer, New York","edition":"2"},{"issue":"1","key":"3739_CR6","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1146\/annurev-statistics-042720-020816","volume":"8","author":"X Bi","year":"2021","unstructured":"Bi X, Tang X, Yuan Y, Zhang A, Qu A (2021) Tensors in statistics. Annu Rev Stat Appl 8(1):345\u2013368","journal-title":"Annu Rev Stat Appl"},{"key":"3739_CR7","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719048","volume-title":"Generalized inverse of linear transformations","author":"SL Campbell","year":"2009","unstructured":"Campbell SL, Meyer CD (2009) Generalized inverse of linear transformations. SIAM, Philadelphia"},{"issue":"10","key":"3739_CR8","doi-asserted-by":"publisher","first-page":"1150","DOI":"10.1080\/01630563.2022.2087676","volume":"43","author":"Z Cong","year":"2022","unstructured":"Cong Z, Ma H (2022) Characterizations and perturbations of the core-EP inverse of tensors based on the T-product. Numer Funct Anal Optim 43(10):1150\u20131200","journal-title":"Numer Funct Anal Optim"},{"issue":"5","key":"3739_CR9","doi-asserted-by":"publisher","first-page":"1565","DOI":"10.2298\/FIL2105565C","volume":"35","author":"Y Cui","year":"2021","unstructured":"Cui Y, Ma H (2021) The perturbation bound for the T-Drazin inverse of tensor and its application. Filomat 35(5):1565\u20131587","journal-title":"Filomat"},{"issue":"7","key":"3739_CR10","doi-asserted-by":"publisher","first-page":"506","DOI":"10.1080\/00029890.1958.11991949","volume":"65","author":"MP Drazin","year":"1958","unstructured":"Drazin MP (1958) Pseudo-inverses in associative rings and semigroups. Am Math Mon 65(7):506\u2013514","journal-title":"Am Math Mon"},{"issue":"1","key":"3739_CR11","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1016\/0022-247X(67)90169-2","volume":"17","author":"I Erd\u00e9lyi","year":"1967","unstructured":"Erd\u00e9lyi I (1967) On the matrix equation $$ax = \\lambda bx$$. J Math Anal Appl 17(1):119\u2013132","journal-title":"J Math Anal Appl"},{"key":"3739_CR12","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1007\/BF02421317","volume":"27","author":"I Fredholm","year":"1903","unstructured":"Fredholm I (1903) Sur une classe d\u2019\u00e9quations fonctionnelles. Acta Math 27:365\u2013390","journal-title":"Acta Math"},{"issue":"163","key":"3739_CR13","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1080\/14786444408644923","volume":"25","author":"WR Hamilton","year":"1844","unstructured":"Hamilton WR (1844) On quaternions; or on a new system of imaginaries in algebra. Lond Edinb Dublin Philos Mag Jo Sci 25(163):10\u201313","journal-title":"Lond Edinb Dublin Philos Mag Jo Sci"},{"issue":"3","key":"3739_CR14","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1080\/03081088308817561","volume":"14","author":"RE Hartwig","year":"1983","unstructured":"Hartwig RE, Spindelb\u00f6ck K (1983) Matrices for which $$a^{*}$$ and $$a^\\dagger $$ commute. Linear Multilinear Algebra 14(3):241\u2013256","journal-title":"Linear Multilinear Algebra"},{"issue":"3","key":"3739_CR15","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1016\/j.laa.2010.09.020","volume":"435","author":"ME Kilmer","year":"2011","unstructured":"Kilmer ME, Martin CD (2011) Factorization strategies for third-order tensors. Linear Algebra Appl 435(3):641\u2013658","journal-title":"Linear Algebra Appl"},{"key":"3739_CR16","doi-asserted-by":"crossref","unstructured":"Kilmer ME, Braman NH, Hoover R (2013) Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging. SIAM J Matrix Anal Appl 34:148\u2013172","DOI":"10.1137\/110837711"},{"issue":"1","key":"3739_CR17","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1007\/s10958-011-0626-x","volume":"180","author":"II Kyrchei","year":"2012","unstructured":"Kyrchei II (2012) Determinantal representation of the Moore\u2013Penrose inverse matrix over the quaternion skew field. J Math Sci 180(1):23\u201333","journal-title":"J Math Sci"},{"key":"3739_CR18","first-page":"193","volume":"238","author":"II Kyrchei","year":"2014","unstructured":"Kyrchei II (2014) Determinantal representations of the Drazin inverse over the quaternion skew field with applications to some matrix equations. Appl Math Comput 238:193\u2013207","journal-title":"Appl Math Comput"},{"issue":"5","key":"3739_CR19","doi-asserted-by":"publisher","first-page":"104","DOI":"10.1007\/s00006-019-1024-6","volume":"29","author":"II Kyrchei","year":"2019","unstructured":"Kyrchei II (2019) The quaternion core inverse and its generalizations. Adv Appl Cliff Algebras 29(5):104","journal-title":"Adv Appl Cliff Algebras"},{"issue":"10","key":"3739_CR20","doi-asserted-by":"publisher","first-page":"954","DOI":"10.3390\/axioms12100954","volume":"12","author":"L Liao","year":"2023","unstructured":"Liao L, Guo Z, Gao Q, Wang G (2023) Color image recovery using generalized matrix completion over higher-order finite dimensional algebra. Axioms 12(10):954","journal-title":"Axioms"},{"key":"3739_CR21","unstructured":"Lim L-H (2005) Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE international workshop on computational advances in multi-sensor adaptive processing, Puerto Vallarta, Mexico, pp 129\u2013132"},{"issue":"18","key":"3739_CR22","doi-asserted-by":"publisher","first-page":"6403","DOI":"10.2298\/FIL2218403L","volume":"36","author":"Y Liu","year":"2022","unstructured":"Liu Y, Ma H (2022) Weighted generalized tensor functions based on the tensor-product and their applications. Filomat 36(18):6403\u20136426","journal-title":"Filomat"},{"issue":"1","key":"3739_CR23","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1080\/03081087.2019.1588847","volume":"69","author":"H Ma","year":"2021","unstructured":"Ma H, Li T (2021) Characterizations and representations of the core inverse and its applications. Linear Multilinear Algebra 69(1):93\u2013103","journal-title":"Linear Multilinear Algebra"},{"key":"3739_CR24","doi-asserted-by":"publisher","first-page":"258","DOI":"10.1016\/j.laa.2019.12.035","volume":"590","author":"Y Miao","year":"2020","unstructured":"Miao Y, Qi L, Wei Y (2020) Generalized tensor function via the tensor singular value decomposition based on the T-product. Linear Algebra Appl 590:258\u2013303","journal-title":"Linear Algebra Appl"},{"key":"3739_CR25","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s42967-019-00055-4","volume":"3","author":"Y Miao","year":"2021","unstructured":"Miao Y, Qi L, Wei Y (2021) T-Jordan canonical form and T-Drazin inverse based on the T-Product. Commun Appl Math Comput 3:201\u2013220","journal-title":"Commun Appl Math Comput"},{"key":"3739_CR26","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1090\/S0002-9904-1920-03308-2","volume":"26","author":"EH Moore","year":"1920","unstructured":"Moore EH (1920) On the reciprocal of the general algebraic matrix. Bull Am Math Soc 26:294\u2013295","journal-title":"Bull Am Math Soc"},{"issue":"5","key":"3739_CR27","doi-asserted-by":"publisher","first-page":"863","DOI":"10.1109\/JPROC.2021.3074329","volume":"109","author":"Y Panagakis","year":"2021","unstructured":"Panagakis Y, Kossaifi J, Chrysos GG, Oldfield J, Moschidis MA, Anandkumar A, Zafeiriou S (2021) Tensor methods in computer vision and deep learning. Proc IEEE 109(5):863\u2013890","journal-title":"Proc IEEE"},{"issue":"3","key":"3739_CR28","doi-asserted-by":"publisher","first-page":"406","DOI":"10.1017\/S0305004100030401","volume":"51","author":"R Penrose","year":"1955","unstructured":"Penrose R (1955) A generalized inverse for matrices. Math Proc Camb Philos Soc 51(3):406\u2013413","journal-title":"Math Proc Camb Philos Soc"},{"issue":"6","key":"3739_CR29","doi-asserted-by":"publisher","first-page":"1302","DOI":"10.1016\/j.jsc.2005.05.007","volume":"40","author":"L Qi","year":"2005","unstructured":"Qi L (2005) Eigenvalues of a real supersymmetric tensor. J Symb Comput 40(6):1302\u20131324","journal-title":"J Symb Comput"},{"key":"3739_CR30","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2021.107597","volume":"123","author":"Z Qin","year":"2022","unstructured":"Qin Z, Ming Z, Zhang L (2022) Singular value decomposition of third order quaternion tensors. Appl Math Lett 123:107597","journal-title":"Appl Math Lett"},{"issue":"355","key":"3739_CR31","doi-asserted-by":"publisher","first-page":"2409","DOI":"10.1090\/mcom\/4025","volume":"94","author":"Z Qin","year":"2025","unstructured":"Qin Z, Ming Z, Sun D, Zhang L (2025) Low-rank quaternion tensor completion for color video inpainting via a novel factorization strategy. Math Comput 94(355):2409\u20132456","journal-title":"Math Comput"},{"issue":"4","key":"3739_CR32","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1080\/03081087.2015.1083933","volume":"64","author":"L Sun","year":"2016","unstructured":"Sun L, Zheng B, Bu C, Wei Y (2016) Moore-Penrose inverse of tensors via Einstein product. Linear Multilinear Algebra 64(4):686\u2013698","journal-title":"Linear Multilinear Algebra"},{"issue":"1","key":"3739_CR33","doi-asserted-by":"publisher","first-page":"20220592","DOI":"10.1515\/math-2022-0592","volume":"21","author":"H Wang","year":"2023","unstructured":"Wang H, Gao J (2023) The dual index and dual core generalized inverse. Open Math 21(1):20220592","journal-title":"Open Math"},{"issue":"9","key":"3739_CR34","doi-asserted-by":"publisher","first-page":"1829","DOI":"10.1080\/03081087.2014.975702","volume":"63","author":"H Wang","year":"2015","unstructured":"Wang H, Liu X (2015) Characterizations of the core inverse and the core partial ordering. Linear Multilinear Algebra 63(9):1829\u20131836","journal-title":"Linear Multilinear Algebra"},{"key":"3739_CR35","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/0024-3795(95)00543-9","volume":"251","author":"F Zhang","year":"1997","unstructured":"Zhang F (1997) Quaternions and matrices of quaternions. Linear Algebra Appl 251:21\u201357","journal-title":"Linear Algebra Appl"},{"key":"3739_CR36","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-1099-7","volume-title":"Matrix theory: basic results and techniques","author":"F Zhang","year":"2011","unstructured":"Zhang F (2011) Matrix theory: basic results and techniques, 2nd edn. Springer, New York","edition":"2"},{"issue":"6","key":"3739_CR37","doi-asserted-by":"publisher","first-page":"2363","DOI":"10.1081\/AGB-100002394","volume":"29","author":"F Zhang","year":"2001","unstructured":"Zhang F, Wei Y (2001) Jordan canonical form of a partitioned complex matrix and its application to real quaternion matrices. Commun Algebra 29(6):2363\u20132375","journal-title":"Commun Algebra"},{"issue":"3","key":"3739_CR38","first-page":"267","volume":"43","author":"Z Zhang","year":"2021","unstructured":"Zhang Z, Ma H (2021) Tensor-core inverse and its perturbation bounds based on the T-product. Numer Math A J Chin Univ 43(3):267\u2013291","journal-title":"Numer Math A J Chin Univ"},{"issue":"5","key":"3739_CR39","doi-asserted-by":"publisher","first-page":"3937","DOI":"10.1007\/s12190-023-01909-0","volume":"69","author":"S Zuo","year":"2023","unstructured":"Zuo S, Ma H (2023) The perturbation of the Moore\u2013Penrose inverse of quaternion tensor via the QT-product. J Appl Math Comput 69(5):3937\u20133967","journal-title":"J Appl Math Comput"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03739-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-026-03739-9","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03739-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T08:22:25Z","timestamp":1776759745000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-026-03739-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,21]]},"references-count":39,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2026,9]]}},"alternative-id":["3739"],"URL":"https:\/\/doi.org\/10.1007\/s40314-026-03739-9","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,21]]},"assertion":[{"value":"31 January 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 March 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 March 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 April 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"343"}}