{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:29:48Z","timestamp":1760228988282,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,29]],"date-time":"2022-05-29T00:00:00Z","timestamp":1653782400000},"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","ZR2020MA053"],"award-info":[{"award-number":["62176112","ZR2020MA053"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Shandong Province","award":["62176112","ZR2020MA053"],"award-info":[{"award-number":["62176112","ZR2020MA053"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we study the bisymmetric and skew bisymmetric solutions of quaternion generalized Lyapunov equation. With the help of semi-tensor product of matrices, some new conclusions on the expansion rules of row and column of matrix product on quaternion matrices are proposed and applied to the calculation of quaternion matrix equation. Using the H-representation method, the independent elements are extracted according to the structural characteristics of bisymmetric matrix and skew bisymmetric matrix, so as to simplify the operation process. Finally, it is compared with the real vector representation method of quaternion matrix equation to illustrate the effectiveness and superiority of the proposed method.<\/jats:p>","DOI":"10.3390\/sym14061120","type":"journal-article","created":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T02:30:06Z","timestamp":1653964206000},"page":"1120","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A New Method of Solving Special Solutions of Quaternion Generalized Lyapunov Matrix Equation"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0153-9206","authenticated-orcid":false,"given":"Zhihong","family":"Liu","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":"Xueling","family":"Fan","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":"Wenxv","family":"Ding","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":[[2022,5,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1131","DOI":"10.1109\/9.863597","article-title":"Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls","volume":"45","author":"Rami","year":"2000","journal-title":"IEEE Trans. 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