{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:59:56Z","timestamp":1760151596542,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,3,23]],"date-time":"2022-03-23T00:00:00Z","timestamp":1647993600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>For two n\u00d7m real matrices X and Y, X is said to be majorized by Y, written as X\u227aY if X=SY for some doubly stochastic matrix of order n. Matrix majorization has several applications in statistics, wireless communications and other fields of science and engineering. Hwang and Park obtained the necessary and sufficient conditions for X,Y to satisfy X\u227aY for the cases where the rank of Y=n\u22121 and the rank of Y=n. In this paper, we obtain some necessary and sufficient conditions for X,Y to satisfy X\u227aY for the cases where the rank of Y=n\u22122 and in general for rank of Y=n\u2212k, where 1\u2264k\u2264n\u22121. We obtain some necessary and sufficient conditions for X to be majorized by Y with some conditions on X and Y. The matrix X is said to be doubly stochastic majorized by Y if there is S\u2208\u03a9m such that X=YS. In this paper, we obtain some necessary and sufficient conditions for X to be doubly stochastic majorized by Y. We introduced a new concept of column stochastic majorization in this paper. A matrix X is said to be column stochastic majorized by Y, denoted as X\u2aafcY, if there exists a column stochastic matrix S such that X=SY. We give characterizations of column stochastic majorization and doubly stochastic majorization for (0,1) matrices.<\/jats:p>","DOI":"10.3390\/axioms11040146","type":"journal-article","created":{"date-parts":[[2022,3,23]],"date-time":"2022-03-23T12:20:25Z","timestamp":1648038025000},"page":"146","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some Results on Majorization of Matrices"],"prefix":"10.3390","volume":"11","author":[{"given":"Divya K.","family":"Udayan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Amrita School of Engineering, Amrita Vishwavidyapeetham, Coimbatore 641112, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2226-1845","authenticated-orcid":false,"given":"Kanagasabapathi","family":"Somasundaram","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Amrita School of Engineering, Amrita Vishwavidyapeetham, Coimbatore 641112, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/0024-3795(84)90027-2","article-title":"The doubly stochastic matrices of a vector majorization","volume":"61","author":"Brualdi","year":"1984","journal-title":"Linear Algebra Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.laa.2005.02.003","article-title":"Weak matrix majorization","volume":"403","author":"Massey","year":"2005","journal-title":"Linear Algebra Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/S0024-3795(99)00148-2","article-title":"Majorization Polytopes","volume":"297","author":"Dahl","year":"1999","journal-title":"Linear Algebra Appl."},{"key":"ref_4","unstructured":"Marshall, A.W., and Olkin, I. 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(2007). Majorization and Matrix-Monotone Functions in Wireless Communications, Now Publishers Inc.","DOI":"10.1561\/9781601980410"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1080\/14697680400016182","article-title":"Rank reduction of correlation matrices by Majorization","volume":"4","author":"Pietersz","year":"2004","journal-title":"Quant. Financ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/0024-3795(89)90580-6","article-title":"Majorization, Doubly stochastic matices, and comperison of eigenvalues","volume":"118","author":"Ando","year":"1989","journal-title":"Linear ALgebra Appl."},{"key":"ref_13","first-page":"479","article-title":"A note on multivariate majorization","volume":"14","author":"Hwang","year":"1999","journal-title":"Comm. Korean Math. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.laa.2019.09.038","article-title":"Alexander Guterman, and Pavel Shteyner, Majorization for (0,1)-matrices","volume":"585","author":"Dahl","year":"2020","journal-title":"Linear Algebra Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/4\/146\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:41:38Z","timestamp":1760136098000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/4\/146"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,23]]},"references-count":14,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["axioms11040146"],"URL":"https:\/\/doi.org\/10.3390\/axioms11040146","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,3,23]]}}}