{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:30:32Z","timestamp":1760243432020,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,30]],"date-time":"2022-08-30T00:00:00Z","timestamp":1661817600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Shandong Provincial Natural Science Foundation","award":["#ZR2019MA065"],"award-info":[{"award-number":["#ZR2019MA065"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Given the linear matrix equation AXB=C, we partition it into the form A1X11B1+A1X12B2+A2X21B1+A2X22B2=C, and then pre- and post-multiply both sides of the equation by the four orthogonal projectors generated from the coefficient matrices A1, A1, B1, and B2 to obtain four reduced linear matrix equations. In this situation, each of the four reduced equations involves just one of the four unknown submatrices X11, X12, X21, and X22, respectively. In this paper, we study the relationships between the general solution of AXB=C and the general solutions of the four reduced equations using some highly selective matrix analysis tools in relation to ranks, ranges, and generalized inverses of matrices.<\/jats:p>","DOI":"10.3390\/axioms11090440","type":"journal-article","created":{"date-parts":[[2022,8,30]],"date-time":"2022-08-30T21:25:18Z","timestamp":1661894718000},"page":"440","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Relationships between a Linear Matrix Equation and Its Four Reduced Equations"],"prefix":"10.3390","volume":"11","author":[{"given":"Bo","family":"Jiang","sequence":"first","affiliation":[{"name":"College of Mathematics and Information Science, Shandong Technology and Business University, Yantai 264005, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2967-4044","authenticated-orcid":false,"given":"Yongge","family":"Tian","sequence":"additional","affiliation":[{"name":"Shanghai Business School, College of Business and Economics, Shanghai 201400, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ruixia","family":"Yuan","sequence":"additional","affiliation":[{"name":"Shanghai Business School, College of Business and Economics, Shanghai 201400, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,30]]},"reference":[{"key":"ref_1","unstructured":"Ben\u2013Israel, A., and Greville, T.N.E. (2003). Generalized Inverses: Theory and Applications, Springer. [2nd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Bernstein, D.S. (2018). Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas Revised and Expanded Edition, Princeton University Press. [3rd ed.].","DOI":"10.1515\/9781400888252"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Campbell, S.L., and Meyer, C.D. (2009). Generalized Inverses of Linear Transformations, SIAM.","DOI":"10.1137\/1.9780898719048"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Puntanen, S., Styan, G.P.H., and Isotalo, J. (2011). Matrix Tricks for Linear Statistical Models, Our Personal Top Twenty, Springer.","DOI":"10.1007\/978-3-642-10473-2"},{"key":"ref_5","unstructured":"Rao, C.R., and Mitra, S.K. (1971). Generalized Inverse of Matrices and Its Applications, Wiley."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1017\/S0305004100030401","article-title":"A generalized inverse for matrices","volume":"51","author":"Penrose","year":"1955","journal-title":"Proc. Camb. Phil. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1194","DOI":"10.1016\/j.laa.2010.05.001","article-title":"Positive solutions to operator equations AXB=C","volume":"433","author":"Arias","year":"2010","journal-title":"Linear Algebra Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1017\/S1446788708000207","article-title":"Re-nnd solutions of the matrix equation AXB=C","volume":"84","year":"2008","journal-title":"J. Austral. Math. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1270","DOI":"10.1016\/j.camwa.2007.06.023","article-title":"Ranks of least squares solutions of the matrix equation AXB=C","volume":"55","author":"Liu","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_10","first-page":"726","article-title":"New matrix iterative methods for constraint solutions of the matrix equation AXB=C","volume":"230","author":"Peng","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1016\/j.jfranklin.2009.02.013","article-title":"Some properties of submatrices in a solution to the matrix equation AXB=C with applications","volume":"346","author":"Tian","year":"2009","journal-title":"J. Franklin Instit."},{"key":"ref_12","first-page":"126189","article-title":"Some relaxed iteration methods for solving matrix equation AXB=C","volume":"403","author":"Tian","year":"2001","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6594","DOI":"10.3934\/math.2020424","article-title":"A unified treatment for the restricted solutions of the matrix equation AXB=C","volume":"5","author":"Xu","year":"2020","journal-title":"AIMS Math."},{"key":"ref_14","first-page":"10049","article-title":"Least squares solutions with special structure to the linear matrix equation AXB=C","volume":"217","author":"Zhang","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1080\/03081087408817070","article-title":"Equalities and inequalities for ranks of matrices","volume":"2","author":"Marsaglia","year":"1974","journal-title":"Linear Multilinear Algebra"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1137\/0612044","article-title":"The matrix equation AXB+CYD=E over a principal ideal domain","volume":"12","year":"1991","journal-title":"SIAM J. Matrix. Anal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1007\/s00010-018-0610-3","article-title":"Necessary and sufficient conditions for nonlinear matrix identities to always hold","volume":"93","author":"Jiang","year":"2019","journal-title":"Aequat. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/S0024-3795(02)00345-2","article-title":"Upper and lower bounds for ranks of matrix expressions using generalized inverses","volume":"355","author":"Tian","year":"2002","journal-title":"Linear Algebra Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1493","DOI":"10.1016\/j.camwa.2010.12.068","article-title":"Relations between matrix sets generated from linear matrix expressions and their applications","volume":"61","author":"Tian","year":"2001","journal-title":"Comput. Math. Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/9\/440\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:20:07Z","timestamp":1760142007000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/9\/440"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,30]]},"references-count":19,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["axioms11090440"],"URL":"https:\/\/doi.org\/10.3390\/axioms11090440","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,8,30]]}}}