{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T01:26:01Z","timestamp":1782955561350,"version":"3.54.5"},"reference-count":42,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,18]],"date-time":"2025-04-18T00:00:00Z","timestamp":1744934400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"key scientific research projects of universities in Anhui province","award":["2023AH050476"],"award-info":[{"award-number":["2023AH050476"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we study the necessary and sufficient conditions for a system of matrix equations to have a solution and a Hermitian solution. As an application, we establish the necessary and sufficient conditions for a classical matrix system to have a reducible solution. Finally, we present an algorithm, along with two concrete examples to validate the main conclusions.<\/jats:p>","DOI":"10.3390\/sym17040619","type":"journal-article","created":{"date-parts":[[2025,4,20]],"date-time":"2025-04-20T20:31:36Z","timestamp":1745181096000},"page":"619","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A System of Coupled Matrix Equations with an Application over the Commutative Quaternion Ring"],"prefix":"10.3390","volume":"17","author":[{"given":"Xiao-Quan","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics and Computation, Anqing Normal University, Anqing 246011, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Long-Sheng","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computation, Anqing Normal University, Anqing 246011, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiao-Xiao","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computation, Anqing Normal University, Anqing 246011, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qian-Wen","family":"Long","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computation, Anqing Normal University, Anqing 246011, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/BF01443559","article-title":"The real representations of complex elements and extension to bicomplex systems","volume":"40","author":"Segre","year":"1892","journal-title":"Math. 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Proceedings of the 2010 International Joint Conference on Neural Networks (IJCNN), Barcelona, Spain.","DOI":"10.1109\/IJCNN.2010.5596736"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1016\/j.neucom.2017.06.013","article-title":"Quaternionic Hopfield neural networks with twin-multistate activation function","volume":"267","author":"Kobayashi","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2012","DOI":"10.1109\/TSP.2004.828901","article-title":"Commutative reduced biquaternions and their Fourier transform for signal and image processing applications","volume":"52","author":"Pei","year":"2004","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2673","DOI":"10.1109\/TCSI.2008.920068","article-title":"Eigenvalues and singular value decompositions of reduced biquaternion matrices","volume":"55","author":"Pei","year":"2008","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"6339","DOI":"10.1007\/s11063-022-11141-9","article-title":"Global Exponential Stability Analysis of Commutative Quaternion-Valued Neural Networks with Time Delays on Time Scales","volume":"55","author":"Xia","year":"2023","journal-title":"Neural Process Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"128291","DOI":"10.1016\/j.amc.2023.128291","article-title":"On singular value decomposition and generalized inverse of a commutative quaternion matrix and applications","volume":"460","author":"Zhang","year":"2024","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1109\/9.210155","article-title":"Output feedback eigenstructure assignment using two Sylvester equations","volume":"38","author":"Syrmors","year":"1993","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1109\/TNN.2002.1031938","article-title":"A recurrent neural network for solving Sylvester equation with time-varying coefficients","volume":"13","author":"Zhang","year":"2002","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.sysconle.2004.07.002","article-title":"On the solution of a Sylvester equation appearing in descriptor systems control theory","volume":"54","author":"Castelan","year":"2005","journal-title":"Syst. Control Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.1007\/s11464-020-0865-6","article-title":"Reducible solution to a quaternion tensor equation","volume":"15","author":"Xie","year":"2020","journal-title":"Front. Math. China"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4069","DOI":"10.1007\/s12190-023-01916-1","article-title":"A coupled quaternion matrix equations with applications","volume":"69","author":"Liu","year":"2023","journal-title":"J. Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2017","DOI":"10.1080\/03081087.2014.896361","article-title":"The general solutions to some systems of matrix equations","volume":"63","author":"He","year":"2015","journal-title":"Linear Multilinear Algebra"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1007\/s43037-023-00262-5","article-title":"The \u03b7-(anti-)Hermitian solution to a constrained Sylvester-type generalized commutative quaternion matrix equation","volume":"17","author":"Chen","year":"2023","journal-title":"Banach J. Math. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1007\/s00006-019-0972-1","article-title":"Cramer\u2019s rules of \u03b7-(skew-) Hermitian solutions to the quaternion Sylvester-type matrix equations","volume":"29","author":"Kyrchei","year":"2019","journal-title":"Adv. Appl. Clifford Algebras."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"26371","DOI":"10.3934\/math.20241284","article-title":"Systems of quaternionic linear matrix equations: Solution, computation, algorithm, and applications","volume":"9","author":"Rehman","year":"2024","journal-title":"AIMS. Math."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Si, K.W., and Wang, Q.W. (2024). The General Solution to a Classical Matrix Equation AXB = C over the Dual Split Quaternion Algebra. Symmetry, 16.","DOI":"10.20944\/preprints202403.1535.v1"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"5029","DOI":"10.3934\/math.2022280","article-title":"Three special kinds of least squares solutions for the quaternion generalized Sylvester matrix equation","volume":"7","author":"Wei","year":"2022","journal-title":"AIMS.Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/j.automatica.2018.12.001","article-title":"Constrained two-side coupled Sylverster-type quaternion matrix equations","volume":"101","author":"Wang","year":"2019","journal-title":"Automatica"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s43034-023-00276-y","article-title":"The consistency and the general common solution to some quaternion matrix equations","volume":"14","author":"Xu","year":"2023","journal-title":"Ann. Funct. Anal."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.laa.2006.06.003","article-title":"Transition matrices for well-conditioned Markov chains","volume":"424","author":"Kirkland","year":"2007","journal-title":"Linear Algebra Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"881","DOI":"10.1016\/j.compbiomed.2008.05.004","article-title":"On the reducibility of compartmental matrices","volume":"38","author":"Lei","year":"2008","journal-title":"Comput. Biol. Med."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1142\/S100538671700013X","article-title":"A system of matrix equations over the quaternion algebra with applications","volume":"24","author":"Nie","year":"2017","journal-title":"Algebra Colloq."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1007\/s41980-023-00815-2","article-title":"B\u00e9zier type Kantorovich q-Baskakov operators via wavelets and some approximation properties","volume":"49","author":"Mursaleen","year":"2023","journal-title":"Bull. Iran. Math. Soc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/j.laa.2007.01.014","article-title":"On the zero pattern properties and asymptoti behavior of continuous-time positive system trajectories","volume":"425","author":"Santesso","year":"2007","journal-title":"Linear Algebra Appl."},{"key":"ref_28","first-page":"57","article-title":"A note on the general Hermitian solution to AXA* = B","volume":"21","year":"1998","journal-title":"Bull. Malays. Math. Soc."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/S0024-3795(00)00033-1","article-title":"Nonnegative-definite and positive-definite solutions to the matrix equation AXA* = B revisited","volume":"321","year":"2000","journal-title":"Linear Algebra Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"13896","DOI":"10.1002\/mma.10245","article-title":"The (anti-)\u03b7-Hermitian solution to a novel system of matrix equations over the split quaternion algebra","volume":"47","author":"Gao","year":"2024","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1137\/0131050","article-title":"Hermitian and nonnegative definite solutions of linear matrix equations","volume":"31","author":"Khatri","year":"1976","journal-title":"SIAM J. Appl. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.laa.2008.01.030","article-title":"Common hermitian and positive solutions to the adjointable operator equations AX = C, XB = D","volume":"429","author":"Xu","year":"2008","journal-title":"Linear Algebra Appl."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Ren, B.-Y., Wang, Q.-W., and Chen, X.-Y. (2023). The \u03b7-Anti-Hermitian Solution to a System of Constrained Matrix Equations over the Generalized Segre Quaternion Algebra. Symmetry, 15.","DOI":"10.3390\/sym15030592"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Wang, Q.W., and Xie, L.M. (2024). The Hermitian solution to a new system of commutative quaternion matrix equations. Symmetry, 16.","DOI":"10.20944\/preprints202402.1320.v1"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1007\/s00006-014-0449-1","article-title":"Commutative Quaternion Matrices","volume":"24","author":"Tosun","year":"2014","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Zhang, F.Z. (2011). Matrix Theory: Basic Result and Techniques, Springer Science & Business Media.","DOI":"10.1007\/978-1-4614-1099-7"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3235","DOI":"10.1007\/s00006-017-0806-y","article-title":"On hermitian solutions of the split quaternion matrix equation axb + cxd = e","volume":"27","author":"Yuan","year":"2017","journal-title":"Adv. Appl. Clifford Algebras"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1080\/03081087.2015.1037302","article-title":"L-structured quaternion matrices and quaternion linear matrix equations","volume":"64","author":"Yuan","year":"2016","journal-title":"Linear Multilinear Algebra"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1355","DOI":"10.1080\/03081087.2018.1543383","article-title":"On Hermitian solutions of the reduced biquaternion matrix equation (AXB, CXD) = (E, G)","volume":"68","author":"Yuan","year":"2018","journal-title":"Linear Multilinear Algebra"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"97","DOI":"10.2298\/FIL2301097X","article-title":"A system of matrix equations over the commutative quaternion ring","volume":"37","author":"Xie","year":"2023","journal-title":"Filomat"},{"key":"ref_41","unstructured":"Ben-Israel, A., and Greville, T.N.E. (2003). Generalized Inverses: Theory and Applications, Springer Science & Business Media. [2nd ed.]."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1080\/03081088308817543","article-title":"L-structured matrices and linear matrix equations","volume":"14","author":"Magnus","year":"1983","journal-title":"Linear Multilinear Algebra"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/619\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:17:41Z","timestamp":1760030261000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/619"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,18]]},"references-count":42,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["sym17040619"],"URL":"https:\/\/doi.org\/10.3390\/sym17040619","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,18]]}}}