{"title":"The \\(\\mathcal {L_C}\\)-Structure-Preserving Algorithms of Quaternion \\(LDL^H\\) Decomposition and Cholesky Decomposition","authors":"Mingcui Zhang, Ying Li, Jianhua Sun, Wenxv Ding","doi":"10.1007/s00006-023-01298-4","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, the <span>\\(\\mathcal {L_C}\\)</span>-structure-preserving algorithms of <span>\\(LDL^H\\)</span> decomposition and Cholesky decomposition of quaternion Hermitian positive definite matrices based on the semi-tensor product of matrices are studied. We first propose <span>\\(\\mathcal {L_C}\\)</span>-representation by using the semi-tensor product of matries and the structure matrix of the product of the quaternion. Then, <span>\\(\\mathcal {L_C}\\)</span>-structure-preserving algorithms of <span>\\(LDL^H\\)</span> decomposition and Cholesky decomposition of quaternion Hermitian positive definite matrices are proposed by using <span>\\(\\mathcal {L_C}\\)</span>-representation, and the advantages of our method are obtained by comparing the operation time and error with the real structure-preserving algorithms in Wei et al. (Quaternion matrix computations. Nova Science Publishers, Hauppauge, 2018). Finally, we apply the <span>\\(\\mathcal {L_C}\\)</span>-structure-preserving algorithm of Cholesky decomposition to strict authentication of color images.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 5","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2023-10-16","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/s00006-023-01298-4","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
In this paper, the \(\mathcal {L_C}\)-structure-preserving algorithms of \(LDL^H\) decomposition and Cholesky decomposition of quaternion Hermitian positive definite matrices based on the semi-tensor product of matrices are studied. We first propose \(\mathcal {L_C}\)-representation by using the semi-tensor product of matries and the structure matrix of the product of the quaternion. Then, \(\mathcal {L_C}\)-structure-preserving algorithms of \(LDL^H\) decomposition and Cholesky decomposition of quaternion Hermitian positive definite matrices are proposed by using \(\mathcal {L_C}\)-representation, and the advantages of our method are obtained by comparing the operation time and error with the real structure-preserving algorithms in Wei et al. (Quaternion matrix computations. Nova Science Publishers, Hauppauge, 2018). Finally, we apply the \(\mathcal {L_C}\)-structure-preserving algorithm of Cholesky decomposition to strict authentication of color images.
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.