{"title":"求解四元数矩阵方程最小二乘特解的STP方法","authors":"Weihua Chen, Caiqin Song","doi":"10.1007/s00006-024-01367-2","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we apply the semi-tensor product of matrices and the real vector representation of a quaternion matrix to find the least squares lower (upper) triangular Toeplitz solution of <span>\\(AX-XB=C\\)</span>, <span>\\(AXB-CX^{T}D=E\\)</span> and (anti)centrosymmetric solution of <span>\\(AXB-CYD=E\\)</span>. And the expressions of the least squares lower (upper) triangular Toeplitz and (anti)centrosymmetric solution for the studied equations are derived. Additionally, the necessary and sufficient conditions for the existence of solutions and general expression of the studied equations are given. Eventually, some numerical examples are provided for showing the validity and superiority of our method.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"STP Method for Solving the Least Squares Special Solutions of Quaternion Matrix Equations\",\"authors\":\"Weihua Chen, Caiqin Song\",\"doi\":\"10.1007/s00006-024-01367-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we apply the semi-tensor product of matrices and the real vector representation of a quaternion matrix to find the least squares lower (upper) triangular Toeplitz solution of <span>\\\\(AX-XB=C\\\\)</span>, <span>\\\\(AXB-CX^{T}D=E\\\\)</span> and (anti)centrosymmetric solution of <span>\\\\(AXB-CYD=E\\\\)</span>. And the expressions of the least squares lower (upper) triangular Toeplitz and (anti)centrosymmetric solution for the studied equations are derived. Additionally, the necessary and sufficient conditions for the existence of solutions and general expression of the studied equations are given. Eventually, some numerical examples are provided for showing the validity and superiority of our method.</p></div>\",\"PeriodicalId\":7330,\"journal\":{\"name\":\"Advances in Applied Clifford Algebras\",\"volume\":\"35 1\",\"pages\":\"\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2024-12-02\",\"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://link.springer.com/article/10.1007/s00006-024-01367-2\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Applied Clifford Algebras","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00006-024-01367-2","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
STP Method for Solving the Least Squares Special Solutions of Quaternion Matrix Equations
In this paper, we apply the semi-tensor product of matrices and the real vector representation of a quaternion matrix to find the least squares lower (upper) triangular Toeplitz solution of \(AX-XB=C\), \(AXB-CX^{T}D=E\) and (anti)centrosymmetric solution of \(AXB-CYD=E\). And the expressions of the least squares lower (upper) triangular Toeplitz and (anti)centrosymmetric solution for the studied equations are derived. Additionally, the necessary and sufficient conditions for the existence of solutions and general expression of the studied equations are given. Eventually, some numerical examples are provided for showing the validity and superiority of our method.
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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