{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:15:20Z","timestamp":1760058920939,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T00:00:00Z","timestamp":1747094400000},"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>This paper proposes a class of randomized Kaczmarz and Gauss\u2013Seidel-type methods for solving the matrix equation AXB=C, where the matrices A and B may be either full-rank or rank deficient and the system may be consistent or inconsistent. These iterative methods offer high computational efficiency and low memory requirements, as they avoid costly matrix\u2013matrix multiplications. We rigorously establish theoretical convergence guarantees, proving that the generated sequences converge to the minimal Frobenius-norm solution (for consistent systems) or the minimal Frobenius-norm least squares solution (for inconsistent systems). Numerical experiments demonstrate the superiority of these methods over conventional matrix multiplication-based iterative approaches, particularly for high-dimensional problems.<\/jats:p>","DOI":"10.3390\/axioms14050367","type":"journal-article","created":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T09:26:48Z","timestamp":1747128408000},"page":"367","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Kaczmarz-Type Methods for Solving Matrix Equation AXB = C"],"prefix":"10.3390","volume":"14","author":[{"given":"Wei","family":"Zheng","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Science, Chuxiong Normal University, Chuxiong 675099, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5713-0349","authenticated-orcid":false,"given":"Lili","family":"Xing","sequence":"additional","affiliation":[{"name":"College of Science, China University of Petroleum, Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wendi","family":"Bao","sequence":"additional","affiliation":[{"name":"College of Science, China University of Petroleum, Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7057-972X","authenticated-orcid":false,"given":"Weiguo","family":"Li","sequence":"additional","affiliation":[{"name":"College of Science, China University of Petroleum, Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,13]]},"reference":[{"key":"ref_1","first-page":"687","article-title":"A note on the numerical approximate solutions for generalized Sylvester matrix equations with applications","volume":"206","author":"Bouhamidi","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/j.sysconle.2007.08.010","article-title":"On the generalized Sylvester mapping and matrix equations","volume":"57","author":"Zhou","year":"2008","journal-title":"Syst. Control Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/0024-3795(89)90067-0","article-title":"Symmetric solutions of linear matrix equations by matrix decompositions","volume":"119","author":"Chu","year":"1989","journal-title":"Linear Algebra Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.sysconle.2004.06.008","article-title":"Iterative least-squares solutions of coupled sylvester matrix equations","volume":"54","author":"Ding","year":"2005","journal-title":"Syst. Control Lett."},{"key":"ref_5","first-page":"41","article-title":"Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle","volume":"197","author":"Ding","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1016\/j.camwa.2012.11.010","article-title":"On Hermitian and skew-Hermitian splitting iteration methods for the linear matrix equation AXB = C","volume":"65","author":"Wang","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_7","first-page":"63","article-title":"The Jacobi and Gauss-Seidel-type iteration methods for the matrix equation AXB = C","volume":"292","author":"Tian","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1137\/S0895479891222106","article-title":"Large least squaress problems involving Kronecker products","volume":"15","author":"Fausett","year":"1994","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1172","DOI":"10.1137\/S0895479894265009","article-title":"Comments on large least squaress problems involving Kronecker products","volume":"16","author":"Zha","year":"1995","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_10","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_11","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1017\/S1446788708000207","article-title":"Re-nnd solutions of the matrix equation AXB = C","volume":"84","author":"Cvetkovic","year":"2008","journal-title":"J. Aust. Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1820","DOI":"10.1080\/00207160802516875","article-title":"A matrix LSQR iterative method to solve matrix equation AXB = C","volume":"87","author":"Peng","year":"2010","journal-title":"Int. J. Comput. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"8991","DOI":"10.1016\/j.jfranklin.2022.09.028","article-title":"Developing Kaczmarz method for solving Sylvester matrix equations","volume":"359","author":"Shafiei","year":"2022","journal-title":"J. Frankl. Inst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107689","DOI":"10.1016\/j.aml.2021.107689","article-title":"On the convergence of a randomized block coordinate descent algorithm for a matrix least squaress problem","volume":"124","author":"Du","year":"2022","journal-title":"Appl. Math. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"114374","DOI":"10.1016\/j.cam.2022.114374","article-title":"On the Kaczmarz methods based on relaxed greedy selection for solving matrix equation AXB = C","volume":"413","author":"Wu","year":"2022","journal-title":"J. Comput. Appl. Math."},{"unstructured":"Niu, Y., and Zheng, B. (2022). On global randomized block Kaczmarz algorithm for solving large-scale matrix equations. arXiv.","key":"ref_16"},{"doi-asserted-by":"crossref","unstructured":"Xing, L., Bao, W., and Li, W. (2023). On the convergence of the randomized block Kaczmarz algorithm for solving a matrix equation. Mathematics, 11.","key":"ref_17","DOI":"10.20944\/preprints202310.0686.v1"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"708","DOI":"10.1080\/00207160.2024.2372420","article-title":"Kaczmarz-type methods for solving matrix equations","volume":"101","author":"Li","year":"2024","journal-title":"Int. J. Comput. Math."},{"key":"ref_19","first-page":"335","article-title":"Angena\u00a8herte auflo\u00a8sung von systemen linearer gleichungen","volume":"32","author":"Kaczmarz","year":"1937","journal-title":"Bull. Internat. Acad. Polon. Sci. Lett. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1007\/s00041-008-9030-4","article-title":"A randomized Kaczmarz algorithm with exponential convergence","volume":"15","author":"Strohmer","year":"2009","journal-title":"J. Fourier Anal. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1287\/moor.1100.0456","article-title":"Randomized methods for linear constraints: Convergence rates and conditioning","volume":"35","author":"Leventhal","year":"2010","journal-title":"Math. Oper. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s10543-010-0265-5","article-title":"Randomized Kaczmarz solver for noisy linear systems","volume":"50","author":"Needell","year":"2010","journal-title":"BIT Numer. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1137\/120889897","article-title":"Randomized extended Kaczmarz for solving least squares","volume":"34","author":"Zouzias","year":"2013","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"e2233","DOI":"10.1002\/nla.2233","article-title":"Tight upper bounds for the convergence of the randomized extended Kaczmarz and Gauss-Seidel algorithms","volume":"26","author":"Du","year":"2019","journal-title":"Numer. Linear Algebra Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1590","DOI":"10.1137\/15M1014425","article-title":"Convergence properties of the randomized extended Gauss-Seidel and Kaczmarz methods","volume":"36","author":"Ma","year":"2015","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_26","first-page":"1","article-title":"The University of Florida sparse matrix collection","volume":"38","author":"Davis","year":"2011","journal-title":"ACM Trans. Math. Softw."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/367\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:31:52Z","timestamp":1760031112000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/367"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,13]]},"references-count":26,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["axioms14050367"],"URL":"https:\/\/doi.org\/10.3390\/axioms14050367","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,5,13]]}}}