{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T06:32:11Z","timestamp":1771655531900,"version":"3.50.1"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2023,11,29]],"date-time":"2023-11-29T00:00:00Z","timestamp":1701216000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,11,29]],"date-time":"2023-11-29T00:00:00Z","timestamp":1701216000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001412","name":"Council of Scientific and Industrial Research, India","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001412","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007488","name":"Indian Institute of Technology Delhi","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100007488","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2023,12]]},"DOI":"10.1007\/s12190-023-01938-9","type":"journal-article","created":{"date-parts":[[2023,11,29]],"date-time":"2023-11-29T17:02:25Z","timestamp":1701277345000},"page":"4553-4585","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On generalized-Drazin inverses and GD-star matrices"],"prefix":"10.1007","volume":"69","author":[{"given":"Amit","family":"Kumar","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7540-2528","authenticated-orcid":false,"given":"Vaibhav","family":"Shekhar","sequence":"additional","affiliation":[]},{"given":"Debasisha","family":"Mishra","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,11,29]]},"reference":[{"key":"1938_CR1","doi-asserted-by":"publisher","first-page":"622","DOI":"10.3390\/math7070622","volume":"7","author":"D Ahmed","year":"2019","unstructured":"Ahmed, D., Hama, M., Jwamer, K.H.F., Shateyi, S.: A seventh-Order scheme for computing the generalized Drazin inverse. Mathematics 7, 622 (2019)","journal-title":"Mathematics"},{"key":"1938_CR2","doi-asserted-by":"publisher","first-page":"2798","DOI":"10.1016\/j.laa.2009.01.015","volume":"430","author":"OM Baksalary","year":"2009","unstructured":"Baksalary, O.M., Styan, G.P.H., Trenkler, G.: On a matrix decomposition of Hartwig and Spindelb\u00f6ck. Linear Algebra Appl. 430, 2798\u20132812 (2009)","journal-title":"Linear Algebra Appl."},{"issue":"6","key":"1938_CR3","doi-asserted-by":"publisher","first-page":"681","DOI":"10.1080\/03081080902778222","volume":"58","author":"OM Baksalary","year":"2010","unstructured":"Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58(6), 681\u2013697 (2010)","journal-title":"Linear Multilinear Algebra"},{"key":"1938_CR4","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1214\/lnms\/1215463789","volume":"24","author":"JK Baksalary","year":"1994","unstructured":"Baksalary, J.K., Hauke, J., Styan, G.P.H.: On some distributional properties of quadratic forms in normal variables and on some associated matrix partial orderings. Lecture Notes-Monograph Ser. 24, 111\u2013121 (1994)","journal-title":"Lecture Notes-Monograph Ser."},{"key":"1938_CR5","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1016\/0024-3795(90)90349-H","volume":"127","author":"JK Baksalary","year":"1990","unstructured":"Baksalary, J.K., Puntanen, S.: Characterizations of the best linear unbiased estimator in the general Gauss\u2013Markov model with the use of matrix partial orderings. Linear Algebra Appl. 127, 363\u2013370 (1990)","journal-title":"Linear Algebra Appl."},{"key":"1938_CR6","volume-title":"Generalized Inverses: Theory and Applications","author":"A Ben-Israel","year":"2003","unstructured":"Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)","edition":"2"},{"key":"1938_CR7","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511721533","volume-title":"Partially Ordered Rings and Semi-algebraic Geometry","author":"GW Brumfiel","year":"1979","unstructured":"Brumfiel, G.W.: Partially Ordered Rings and Semi-algebraic Geometry. Cambridge University Press, Cambridge (1979)"},{"issue":"1","key":"1938_CR8","doi-asserted-by":"publisher","first-page":"146","DOI":"10.1007\/s13538-011-0052-z","volume":"42","author":"JCA Barata","year":"2012","unstructured":"Barata, J.C.A., Hussein, M.S.: The Moore-Penrose pseudoinverse: a tutorial review of the theory. Braz. J. Phys. 42(1), 146\u2013165 (2012)","journal-title":"Braz. J. Phys."},{"key":"1938_CR9","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719048","volume-title":"Generalized Inverses of Linear Transformations","author":"SL Campbell","year":"2009","unstructured":"Campbell, S.L., Meyer, C.D.: Generalized Inverses of Linear Transformations. SIAM, Philadelphia (2009)"},{"key":"1938_CR10","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1016\/0024-3795(78)90048-4","volume":"20","author":"SL Campbell","year":"1978","unstructured":"Campbell, S.L., Meyer, C.D.: Weak Drazin inverses. Linear Algebra Appl. 20, 167\u2013178 (1978)","journal-title":"Linear Algebra Appl."},{"key":"1938_CR11","doi-asserted-by":"publisher","first-page":"1057","DOI":"10.1137\/0508081","volume":"8","author":"SL Campbell","year":"1977","unstructured":"Campbell, S.L.: Linear systems of differential equations with singular coefficients. SIAM J. Appl. Math. 8, 1057\u20131066 (1977)","journal-title":"SIAM J. Appl. Math."},{"key":"1938_CR12","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1016\/0024-3795(79)90100-9","volume":"23","author":"SL Campbell","year":"1979","unstructured":"Campbell, S.L.: Limit behavior of solutions of singular difference equations. Linear Algebra Appl. 23, 167\u2013178 (1979)","journal-title":"Linear Algebra Appl."},{"key":"1938_CR13","doi-asserted-by":"publisher","first-page":"514","DOI":"10.13001\/1081-3810.3365","volume":"34","author":"N Castro-Gonz\u00e1lez","year":"2018","unstructured":"Castro-Gonz\u00e1lez, N., Hartwig, R.E.: Perturbation results and the forward-order law for the Moore\u2013Penrose inverse of a product. Electron. J. Linear Algebra 34, 514\u2013525 (2018)","journal-title":"Electron. J. Linear Algebra"},{"key":"1938_CR14","first-page":"12","volume":"317","author":"C Coll","year":"2018","unstructured":"Coll, C., Lattanzi, M.B., Thome, N.: Weighted G-Drazin inverses and a new pre-order on rectangular matrices. Appl. Math. Comput. 317, 12\u201324 (2018)","journal-title":"Appl. Math. Comput."},{"key":"1938_CR15","doi-asserted-by":"publisher","first-page":"663","DOI":"10.1016\/j.jmaa.2011.04.085","volume":"382","author":"CY Deng","year":"2011","unstructured":"Deng, C.Y.: Reverse-order law for the group inverses. J. Math. Anal. Appl. 382, 663\u2013671 (2011)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"1938_CR16","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1090\/S0002-9904-1978-14442-5","volume":"84","author":"MP Drazin","year":"1978","unstructured":"Drazin, M.P.: Natural structures on semigroups with involution. Bull. Am. Math. Soc. 84(1), 139\u2013141 (1978)","journal-title":"Bull. Am. Math. Soc."},{"key":"1938_CR17","unstructured":"Gupta, V.K., Jayaram, B.: Clifford\u2019s order from fuzzy logic connectives. Diss, IIT Hyderabad (2022)"},{"key":"1938_CR18","first-page":"1749","volume":"189","author":"W Guo","year":"2007","unstructured":"Guo, W., Wei, M., Zhao, J.: Forward order law for g-inverses of the product of two matrices. Appl. Math. Comput. 189, 1749\u20131754 (2007)","journal-title":"Appl. Math. Comput."},{"key":"1938_CR19","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1016\/S0024-3795(96)00541-1","volume":"264","author":"J Gro\u00df","year":"1997","unstructured":"Gro\u00df, J., Trenkler, G.: Generalized and hypergeneralized projectors. Linear Algebra Appl. 264, 463\u2013474 (1997)","journal-title":"Linear Algebra Appl."},{"key":"1938_CR20","doi-asserted-by":"publisher","first-page":"518","DOI":"10.1137\/1008107","volume":"8","author":"TNE Greville","year":"1966","unstructured":"Greville, T.N.E.: Note on the generalized inverse of a matrix product. SIAM Rev. 8, 518\u2013521 (1966)","journal-title":"SIAM Rev."},{"key":"1938_CR21","doi-asserted-by":"publisher","first-page":"585","DOI":"10.2140\/pjm.1963.13.585","volume":"13","author":"PR Halmos","year":"1963","unstructured":"Halmos, P.R., McLaughlin, J.E.: Partial isometries. Pacific J. Math. 13, 585\u2013596 (1963)","journal-title":"Pacific J. Math."},{"key":"1938_CR22","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1080\/03081088308817561","volume":"14","author":"RE Hartwig","year":"1984","unstructured":"Hartwig, R.E., Spindleb\u00f6ck, K.: Matrices for which $$A^*$$ and $$A^{\\dagger }$$ commute. Linear Multilinear Algebra 14, 241\u2013256 (1984)","journal-title":"Linear Multilinear Algebra"},{"key":"1938_CR23","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1080\/03081088308817529","volume":"13","author":"RE Hartwig","year":"1983","unstructured":"Hartwig, R.E., Spindleb\u00f6ck, K.: Partial isometries, contractions and EP matrices. Linear Multilinear Algebra 13, 295\u2013310 (1983)","journal-title":"Linear Multilinear Algebra"},{"issue":"1","key":"1938_CR24","first-page":"1","volume":"25","author":"RE Hartwig","year":"1980","unstructured":"Hartwig, R.E.: How to partially order regular elements. Math. Japon. 25(1), 1\u201313 (1980)","journal-title":"Math. Japon."},{"key":"1938_CR25","doi-asserted-by":"publisher","DOI":"10.1080\/03081087.2020.1857678","author":"MV Hern\u00e1ndez","year":"2020","unstructured":"Hern\u00e1ndez, M.V., Lattanzi, M.B., Thome, N.: GDMP-inverses of a matrix and their duals. Linear Multilinear Algebra (2020). https:\/\/doi.org\/10.1080\/03081087.2020.1857678","journal-title":"Linear Multilinear Algebra"},{"issue":"3","key":"1938_CR26","doi-asserted-by":"publisher","first-page":"148","DOI":"10.1007\/s13398-021-01090-8","volume":"115","author":"MV Hern\u00e1ndez","year":"2021","unstructured":"Hern\u00e1ndez, M.V., Lattanzi, M.B., Thome, N.: From projectors to 1MP and MP1 generalized inverses and their induced partial orders. RACSAM 115(3), 148 (2021)","journal-title":"RACSAM"},{"key":"1938_CR27","first-page":"108","volume":"282","author":"A Hern\u00e1ndez","year":"2016","unstructured":"Hern\u00e1ndez, A., Lattanzi, M.B., Thome, N.: On some new pre-orders defined by weighted Drazin inverses. Appl. Math. Comput. 282, 108\u2013116 (2016)","journal-title":"Appl. Math. Comput."},{"key":"1938_CR28","first-page":"10669","volume":"218","author":"A Hern\u00e1ndez","year":"2012","unstructured":"Hern\u00e1ndez, A., Lattanzi, M.B., Thome, N., Urquiza, F.: The star partial order and the eigenprojection at 0 on EP matrices. Appl. Math. Comput. 218, 10669\u201310678 (2012)","journal-title":"Appl. Math. Comput."},{"key":"1938_CR29","doi-asserted-by":"publisher","first-page":"632","DOI":"10.13001\/ela.2022.7015","volume":"38","author":"A Kumar","year":"2022","unstructured":"Kumar, A., Shekhar, V., Mishra, D.: $$W$$-weighted GDMP inverse for rectangular matrices. Electron. J. Linear Algebra 38, 632\u2013657 (2022)","journal-title":"Electron. J. Linear Algebra"},{"key":"1938_CR30","first-page":"453","volume":"264","author":"II Kyrchei","year":"2015","unstructured":"Kyrchei, I.I.: Determinantal representations of the W-weighted Drazin inverse over the quaternion skew field. Appl. Math. Comput. 264, 453\u2013465 (2015)","journal-title":"Appl. Math. Comput."},{"issue":"2","key":"1938_CR31","doi-asserted-by":"publisher","first-page":"589","DOI":"10.2298\/FIL1802589L","volume":"32","author":"Z Liu","year":"2018","unstructured":"Liu, Z., Xiong, Z.: The forward-order laws for {1, 2, 3}- and {1, 2, 4}-inverses of a three matrix products. Filomat 32(2), 589\u2013598 (2018)","journal-title":"Filomat"},{"key":"1938_CR32","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1137\/0601031","volume":"1","author":"CD Meyer","year":"1980","unstructured":"Meyer, C.D.: The condition of a finite Markov chain and perturbation bounds for the limiting probabilities. SIAM J. Algebra Discrete Methods 1, 273\u2013283 (1980)","journal-title":"SIAM J. Algebra Discrete Methods"},{"key":"1938_CR33","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/0024-3795(87)90248-5","volume":"92","author":"SK Mitra","year":"1987","unstructured":"Mitra, S.K.: On group inverses and the sharp order. Linear Algebra Appl. 92, 17\u201337 (1987)","journal-title":"Linear Algebra Appl."},{"key":"1938_CR34","doi-asserted-by":"publisher","DOI":"10.1142\/7170","volume-title":"Matrix Partial Orders, Shorted Operators and Applications","author":"SK Mitra","year":"2010","unstructured":"Mitra, S.K., Bhimasankaram, P., Malik, S.B.: Matrix Partial Orders, Shorted Operators and Applications. World Scientific Publishing Company, Singapore (2010)"},{"key":"1938_CR35","doi-asserted-by":"publisher","unstructured":"Mosic, D.: Drazin-star and star-Drazin matrices. Results Math. 75, 1\u201321 (2020). https:\/\/doi.org\/10.1007\/s00025-020-01191-7","DOI":"10.1007\/s00025-020-01191-7"},{"key":"1938_CR36","doi-asserted-by":"publisher","first-page":"406","DOI":"10.1017\/S0305004100030401","volume":"51","author":"R Penrose","year":"1955","unstructured":"Penrose, R.: A generalized inverse for matrices. Camb. Philosoph. Soc. 51, 406\u2013413 (1955)","journal-title":"Camb. Philosoph. Soc."},{"key":"1938_CR37","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1137\/0117004","volume":"17","author":"GW Stewart","year":"1969","unstructured":"Stewart, G.W.: On the continuity of the generalized inverse. SIAM J. Appl. Math. 17, 33\u201345 (1969)","journal-title":"SIAM J. Appl. Math."},{"key":"1938_CR38","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1016\/0024-3795(85)90043-6","volume":"70","author":"C Stepniak","year":"1985","unstructured":"Stepniak, C.: Ordering of nonnegative definite matrices with application to comparison of linear models. Linear Algebra Appl. 70, 67\u201371 (1985)","journal-title":"Linear Algebra Appl."},{"key":"1938_CR39","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1016\/j.laa.2015.09.046","volume":"488","author":"H Wang","year":"2016","unstructured":"Wang, H., Liu, X.: Partial orders based on core-nilpotent decomposition. Linear Algebra Appl. 488, 235\u2013248 (2016)","journal-title":"Linear Algebra Appl."},{"key":"1938_CR40","doi-asserted-by":"publisher","first-page":"672","DOI":"10.1016\/j.jmaa.2016.06.026","volume":"444","author":"X Wang","year":"2016","unstructured":"Wang, X., Yu, A., Li, T., Deng, C.: Reverse-order laws for the Drazin inverses. J. Math. Anal. Appl. 444, 672\u2013689 (2016)","journal-title":"J. Math. Anal. Appl."},{"key":"1938_CR41","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1016\/j.laa.2016.08.008","volume":"508","author":"H Wang","year":"2016","unstructured":"Wang, H.: Core-EP decomposition and its applications. Linear Algebra Appl. 508, 289\u2013300 (2016)","journal-title":"Linear Algebra Appl."},{"key":"1938_CR42","doi-asserted-by":"publisher","first-page":"649","DOI":"10.1016\/j.aml.2007.07.007","volume":"21","author":"Z Xiong","year":"2008","unstructured":"Xiong, Z., Zheng, B.: The reverse-order laws for {1, 2, 3}- and {1, 2, 4}-inverses of a two-matrix product. Appl. Math. Lett. 21, 649\u2013655 (2008)","journal-title":"Appl. Math. Lett."},{"key":"1938_CR43","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1007\/BF02832366","volume":"25","author":"Z Xiong","year":"2007","unstructured":"Xiong, Z., Zheng, B.: Forward order law for the generalized inverses of multiple matrix product. J. Appl. Math. Comput. 25, 415\u2013424 (2007)","journal-title":"J. Appl. Math. Comput."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01938-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-023-01938-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01938-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,21]],"date-time":"2023-12-21T13:25:47Z","timestamp":1703165147000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-023-01938-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,29]]},"references-count":43,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["1938"],"URL":"https:\/\/doi.org\/10.1007\/s12190-023-01938-9","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,29]]},"assertion":[{"value":"16 September 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 November 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 November 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 November 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}