{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T15:05:21Z","timestamp":1769094321577,"version":"3.49.0"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2019,3,5]],"date-time":"2019-03-05T00:00:00Z","timestamp":1551744000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100003453","name":"the Natural Science Foundation of GuangDong","doi-asserted-by":"crossref","award":["2014A030313625"],"award-info":[{"award-number":["2014A030313625"]}],"id":[{"id":"10.13039\/501100003453","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2019,6]]},"DOI":"10.1007\/s40314-019-0803-y","type":"journal-article","created":{"date-parts":[[2019,3,5]],"date-time":"2019-03-05T18:02:07Z","timestamp":1551808927000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["A note on the forward order law for least square g-inverse of three matrix products"],"prefix":"10.1007","volume":"38","author":[{"given":"Zhongshan","family":"Liu","sequence":"first","affiliation":[]},{"given":"Zhiping","family":"Xiong","sequence":"additional","affiliation":[]},{"given":"Yingying","family":"Qin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,3,5]]},"reference":[{"key":"803_CR1","volume-title":"Generalized Inverse: Theory and Applications","author":"A Ben-Israel","year":"2003","unstructured":"Ben-Israel A, Greville TNE (2003) Generalized Inverse: Theory and Applications, 2nd edn. Springer, New York","edition":"2"},{"key":"803_CR2","doi-asserted-by":"publisher","first-page":"1388","DOI":"10.1016\/j.laa.2010.11.022","volume":"434","author":"D Cvetkovi\u0107-IIi\u0107","year":"2011","unstructured":"Cvetkovi\u0107-IIi\u0107 D, Harte R (2011) Reverse order laws in $$C^*$$ C \u2217 -algebras. Linear Algebra Appl. 434:1388\u20131394","journal-title":"Linear Algebra Appl."},{"issue":"3","key":"803_CR3","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1080\/03081087.2018.1430119","volume":"67","author":"Dragana Cvetkovi\u0107-Ili\u0107","year":"2018","unstructured":"Cvetkovi\u0107-IIi\u0107 D, Milosevic J (2018) Reverse order laws for $$\\{1,3\\}$$ { 1 , 3 } -generalized inverses. Linear and Multilinear Algebra. https:\/\/doi.org\/10.1080\/03081087.2018.1430119","journal-title":"Linear and Multilinear Algebra"},{"issue":"15","key":"803_CR4","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.amc.2014.01.163","volume":"234","author":"D Cvetkovi\u0107-IIi\u0107","year":"2014","unstructured":"Cvetkovi\u0107-IIi\u0107 D, Nikolov J (2014) Reverse order laws for $$\\{1,2,3\\}$$ { 1 , 2 , 3 } -generalized inverses. Appl Math Comput 234(15):114\u2013117","journal-title":"Appl Math Comput"},{"key":"803_CR5","volume-title":"Generalized inverse of linear transformations","author":"SL Campbell","year":"1979","unstructured":"Campbell SL, Meyer CD (1979) Generalized inverse of linear transformations. Dover, New York"},{"key":"803_CR6","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1016\/S0024-3795(97)10068-4","volume":"277","author":"AR Depierro","year":"1996","unstructured":"Depierro AR, Wei M (1996) Reverse order laws for recive generalized inverse of products of matrices. Linear Algebra Appl 277:299\u2013311","journal-title":"Linear Algebra Appl"},{"key":"803_CR7","doi-asserted-by":"publisher","first-page":"1242","DOI":"10.1137\/050638114","volume":"29","author":"DS Djordjevic","year":"2007","unstructured":"Djordjevic DS (2007) Futher results on the reverse order law for generalized inverses. SIAM J Matrix Anal Appl 29:1242\u20131246","journal-title":"SIAM J Matrix Anal Appl"},{"key":"803_CR8","doi-asserted-by":"publisher","first-page":"518","DOI":"10.1137\/1008107","volume":"8","author":"TNE Greville","year":"1966","unstructured":"Greville TNE (1966) Note on the generalized inverses of a matrix products. SIAM Rev 8:518\u2013521","journal-title":"SIAM Rev"},{"key":"803_CR9","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1016\/0024-3795(86)90226-0","volume":"76","author":"RE Hartwing","year":"1986","unstructured":"Hartwing RE (1986) The reverse order law revisited. Linear Algebra Appl 76:241\u2013246","journal-title":"Linear Algebra Appl"},{"key":"803_CR10","doi-asserted-by":"crossref","unstructured":"Liu D, Yan H Further results on the reverse order law for $$\\{1,3\\}$$ { 1 , 3 } -inverse and $$\\{1,4\\}$$ { 1 , 4 } -inverse of a matrix product. J Inequal Appl 2010 (Article ID 312767)","DOI":"10.1155\/2010\/312767"},{"key":"803_CR11","doi-asserted-by":"crossref","first-page":"8570","DOI":"10.1016\/j.amc.2012.02.020","volume":"218","author":"XJ Liu","year":"2012","unstructured":"Liu XJ, Huang S, Cvetkovic-Ilic DS (2012) Mixed-tipe reverse-order law for $$\\{1,3\\}-inverses$$ { 1 , 3 } - i n v e r s e s over Hilbert spaces. Appl Math Comput 218:8570\u20138577","journal-title":"Appl Math Comput"},{"issue":"283","key":"803_CR12","doi-asserted-by":"publisher","first-page":"1597","DOI":"10.1090\/S0025-5718-2013-02660-9","volume":"82","author":"XJ Liu","year":"2013","unstructured":"Liu XJ, Wu S, Cvetkovic-Ilic DS (2013) New results on reverse order law for $$\\{1,2,3\\}$$ { 1 , 2 , 3 } and $$\\{1,2,4\\}$$ { 1 , 2 , 4 } -inverses of bounded operators. Math Comput 82(283):1597\u20131607","journal-title":"Math Comput"},{"key":"803_CR13","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1080\/03081087408817070","volume":"2","author":"G Marsaglia","year":"1974","unstructured":"Marsaglia G, Tyan GPHS (1974) Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2:269\u2013292","journal-title":"Linear Multilinear Algebra"},{"key":"803_CR14","doi-asserted-by":"publisher","first-page":"406","DOI":"10.1017\/S0305004100030401","volume":"51","author":"R Penrose","year":"1955","unstructured":"Penrose R (1955) A generalized for matrix. Proc Camb Philos Soc 51:406\u2013413","journal-title":"Proc Camb Philos Soc"},{"key":"803_CR15","doi-asserted-by":"publisher","first-page":"1865","DOI":"10.1080\/00207160701582077","volume":"85","author":"P Stanimirovic","year":"2008","unstructured":"Stanimirovic P, Tasic M (2008) Computing generalized inverses using $$LU$$ LU factorrization of Matrix product. Int J Comput Math 85:1865\u20131878","journal-title":"Int J Comput Math"},{"key":"803_CR16","volume-title":"Generalized inverse of matrices and its applications","author":"CR Rao","year":"1971","unstructured":"Rao CR, Mitra SK (1971) Generalized inverse of matrices and its applications. Wiley, New York"},{"key":"803_CR17","doi-asserted-by":"publisher","first-page":"772","DOI":"10.1137\/S0895479896305441","volume":"19","author":"W Sun","year":"1998","unstructured":"Sun W, Wei Y (1998) Inverse order rule for weighted generalized inverse. SIAM J Matrix Anal 19:772\u2013775","journal-title":"SIAM J Matrix Anal"},{"key":"803_CR18","first-page":"64","volume":"1","author":"Y Tian","year":"1992","unstructured":"Tian Y (1992) The Moore\u2013Penrose inverse order of a triple matrix product. Math Pract Theory 1:64\u201370","journal-title":"Math Pract Theory"},{"key":"803_CR19","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1016\/S0096-3003(03)00585-X","volume":"152","author":"Y Tian","year":"2004","unstructured":"Tian Y (2004) More on maximal and minimal ranks of Schur complements with applications. Appl Math Comput 152:675\u2013692","journal-title":"Appl Math Comput"},{"key":"803_CR20","first-page":"531","volume-title":"$$G$$ G","author":"HT Werner","year":"1992","unstructured":"Werner HT (1992) $$G$$ G -inverse of matrix products date analysis statistical inference. Eul-Verlag, Bergisch-Gladbach, pp 531\u2013546"},{"key":"803_CR21","volume-title":"Generalized inverse: theory and computations","author":"G Wang","year":"2004","unstructured":"Wang G, Wei Y, Qiao S (2004) Generalized inverse: theory and computations. Science Press, Beijing"},{"key":"803_CR22","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1016\/S0024-3795(01)00460-8","volume":"342","author":"M Wei","year":"2002","unstructured":"Wei M, Gao W (2002) Reverse order laws for least squares $$g$$ g -inverses and minimum-norm $$g$$ g -inverses of products of two matrices. Linear Algebra Appl 342:117\u2013132","journal-title":"Linear Algebra Appl"},{"key":"803_CR23","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1016\/S0024-3795(99)00053-1","volume":"293","author":"M Wei","year":"1999","unstructured":"Wei M (1999) Reverse order laws for generalized inverse of multiple matrix products. Linear Algebra Appl 293:273\u2013288","journal-title":"Linear Algebra Appl"},{"issue":"1\u20132","key":"803_CR24","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1007\/BF02832366","volume":"25","author":"Z Xiong","year":"2007","unstructured":"Xiong Z, Zheng B (2007) Forward order law for the generalized inverses of multiple matrix products. J Appl Math Comput 25(1\u20132):415\u2013424","journal-title":"J Appl Math Comput"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-019-0803-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s40314-019-0803-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-019-0803-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,13]],"date-time":"2022-09-13T01:01:43Z","timestamp":1663030903000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s40314-019-0803-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,5]]},"references-count":24,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,6]]}},"alternative-id":["803"],"URL":"https:\/\/doi.org\/10.1007\/s40314-019-0803-y","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,5]]},"assertion":[{"value":"12 July 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 December 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 January 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 March 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"48"}}