{"title":"A Classical System of Matrix Equations Over the Split Quaternion Algebra","authors":"Kai-Wen Si, Qing-Wen Wang, Lv-Ming Xie","doi":"10.1007/s00006-024-01348-5","DOIUrl":null,"url":null,"abstract":"<p>We design several real representations of split quaternion matrices with the primary objective of establishing both necessary and sufficient conditions for the existence of solutions within a system of split quaternion matrix equations. This includes conditions for the general solution without any constraints, as well as <span>\\(X=\\pm X^{\\eta }\\)</span> solutions and <span>\\(\\eta \\)</span>-(anti-)Hermitian solutions. Furthermore, we derive the expressions for the general solutions when it is solvable. As an application, we investigate the solutions to a system of five split quaternion matrix equations involving <span>\\(X^\\star \\)</span>. Finally, we present several algorithms and numerical examples to demonstrate the results of this paper.</p>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Applied Clifford Algebras","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00006-024-01348-5","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We design several real representations of split quaternion matrices with the primary objective of establishing both necessary and sufficient conditions for the existence of solutions within a system of split quaternion matrix equations. This includes conditions for the general solution without any constraints, as well as \(X=\pm X^{\eta }\) solutions and \(\eta \)-(anti-)Hermitian solutions. Furthermore, we derive the expressions for the general solutions when it is solvable. As an application, we investigate the solutions to a system of five split quaternion matrix equations involving \(X^\star \). Finally, we present several algorithms and numerical examples to demonstrate the results of this paper.
期刊介绍：
Advances in Applied Clifford Algebras (AACA) publishes high-quality peer-reviewed research papers as well as expository and survey articles in the area of Clifford algebras and their applications to other branches of mathematics, physics, engineering, and related fields. The journal ensures rapid publication and is organized in six sections: Analysis, Differential Geometry and Dirac Operators, Mathematical Structures, Theoretical and Mathematical Physics, Applications, and Book Reviews.
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