{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T19:52:44Z","timestamp":1775073164132,"version":"3.50.1"},"reference-count":48,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,1]],"date-time":"2022-07-01T00:00:00Z","timestamp":1656633600000},"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":["62176112"],"award-info":[{"award-number":["62176112"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["ZR2020MA053"],"award-info":[{"award-number":["ZR2020MA053"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["62176112"],"award-info":[{"award-number":["62176112"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["ZR2020MA053"],"award-info":[{"award-number":["ZR2020MA053"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we use semi-tensor product of quaternion matrices, L-representation of quaternion matrices, and GH-representation of special quaternion matrices such as quaternion (anti)-centrosymmetric matrices to solve the special solutions of quaternion matrix equation. Based on semi-tensor product of quaternion matrices and the structure matrix of the multiplication of quaternion, we propose the vector representation operation conclusion of quaternion matrices, and study the different matrix representations of quaternion matrices. Then the problem of the quaternion matrix equation is transformed into the corresponding problem in the real number fields by using vector representation and L-representation of quaternion matrices, combined with the special structure of (anti)-centrosymmetric matrices, the independent elements are extracted by GH-representation method, so as to reduce the number of variables to be calculated and improve the calculation accuracy. Finally, the effectiveness of the method is verified by numerical examples, and the time comparison with the two existing algorithms is carried out. The algorithm in this paper is also applied in a centrosymmetric color digital image restoration model.<\/jats:p>","DOI":"10.3390\/sym14071359","type":"journal-article","created":{"date-parts":[[2022,7,4]],"date-time":"2022-07-04T23:38:55Z","timestamp":1656977935000},"page":"1359","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0312-7755","authenticated-orcid":false,"given":"Xueling","family":"Fan","sequence":"first","affiliation":[{"name":"College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China"},{"name":"Research Center of Semi-Tensor Product of Matrices: Theory and Applications, Liaocheng 252000, China"}]},{"given":"Ying","family":"Li","sequence":"additional","affiliation":[{"name":"College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China"},{"name":"Research Center of Semi-Tensor Product of Matrices: Theory and Applications, Liaocheng 252000, China"}]},{"given":"Zhihong","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China"},{"name":"Research Center of Semi-Tensor Product of Matrices: Theory and Applications, Liaocheng 252000, China"}]},{"given":"Jianli","family":"Zhao","sequence":"additional","affiliation":[{"name":"College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China"},{"name":"Research Center of Semi-Tensor Product of Matrices: Theory and Applications, Liaocheng 252000, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cheng, D.Z. 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