{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:42:44Z","timestamp":1760143364030,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T00:00:00Z","timestamp":1707091200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12061015","GUIKE21220024","2018GXNSFDA281023"],"award-info":[{"award-number":["12061015","GUIKE21220024","2018GXNSFDA281023"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Guangxi Science and Technology Base and Talents Special Project","award":["12061015","GUIKE21220024","2018GXNSFDA281023"],"award-info":[{"award-number":["12061015","GUIKE21220024","2018GXNSFDA281023"]}]},{"DOI":"10.13039\/100012547","name":"Guangxi Natural Science Foundation","doi-asserted-by":"publisher","award":["12061015","GUIKE21220024","2018GXNSFDA281023"],"award-info":[{"award-number":["12061015","GUIKE21220024","2018GXNSFDA281023"]}],"id":[{"id":"10.13039\/100012547","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we present a new concept of the generalized core orthogonality (called the C-S orthogonality) for two generalized core invertible matrices A and B. A is said to be C-S orthogonal to B if ASB=0 and BAS=0, where AS is the generalized core inverse of A. The characterizations of C-S orthogonal matrices and the C-S additivity are also provided. And, the connection between the C-S orthogonality and C-S partial order has been given using their canonical form. Moreover, the concept of the strongly C-S orthogonality is defined and characterized.<\/jats:p>","DOI":"10.3390\/axioms13020110","type":"journal-article","created":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T08:41:43Z","timestamp":1707122503000},"page":"110","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["C-S and Strongly C-S Orthogonal Matrices"],"prefix":"10.3390","volume":"13","author":[{"given":"Xiaoji","family":"Liu","sequence":"first","affiliation":[{"name":"School of Education, Guangxi Vocational Normal University, Nanning 530007, China"}]},{"given":"Ying","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Guangxi Minzu University, Nanning 530006, China"}]},{"given":"Hongwei","family":"Jin","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Guangxi Minzu University, Nanning 530006, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"225","DOI":"10.2140\/pjm.1961.11.225","article-title":"Relative hermitian matrices","volume":"11","author":"Hestenes","year":"1961","journal-title":"Pac. J. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/0024-3795(86)90148-5","article-title":"On some characterizations of the \u201cstar\u201d partial ordering for matrices and rank subtractivity","volume":"82","author":"Hartwig","year":"1986","journal-title":"Linear Algebra Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5052","DOI":"10.1080\/03081087.2021.1902923","article-title":"Core and strongly core orthogonal matrices","volume":"70","author":"Ferreyra","year":"2021","journal-title":"Linear Multilinear Algebr."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2543","DOI":"10.1080\/03081087.2022.2111544","article-title":"Further results on strongly core orthogonal matrix","volume":"71","author":"Liu","year":"2023","journal-title":"Linear Multilinear Algebr."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Mosi\u0107, D., Dolinar, G., Kuzma, B., and Marovt, J. (2022). Core-EP orthogonal operators. Linear Multilinear Algebr., 1\u201315.","DOI":"10.1080\/03081087.2022.2033155"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1007\/s40840-023-01478-2","article-title":"The C-S inverse and its applications","volume":"46","author":"Wang","year":"2023","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_7","first-page":"394","article-title":"On the reciprocal of the general algebraic matrix","volume":"26","author":"Moore","year":"1920","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_8","first-page":"82","article-title":"Application of calculus of matrices to method of least squares: With special reference to geodetic calculations","volume":"49","author":"Bjerhammar","year":"1951","journal-title":"Trans. R. Inst. Technol. Stock. 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Jpn."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/0024-3795(87)90248-5","article-title":"On group inverses and the sharp order","volume":"92","author":"Mitra","year":"1987","journal-title":"Linear Algebra Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.laa.2016.08.008","article-title":"Core-EP decomposition and its applications","volume":"508","author":"Wang","year":"2016","journal-title":"Linear Algebra Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/2\/110\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:55:17Z","timestamp":1760104517000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/2\/110"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,5]]},"references-count":16,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["axioms13020110"],"URL":"https:\/\/doi.org\/10.3390\/axioms13020110","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,2,5]]}}}