The General Solution to a System of Tensor Equations over the Split Quaternion Algebra with Applications
This paper presents a systematic investigation into the solvability and the general solution of a tensor equation system within the split quaternion algebra framework. As an extension of classical quaternions with distinctive pseudo-Euclidean properties, split quaternions offer unique advantages in...
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| Format: | Article |
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MDPI AG
2025-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/13/4/644 |
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| author | Zong-Ru Jia Qing-Wen Wang |
| author_facet | Zong-Ru Jia Qing-Wen Wang |
| author_sort | Zong-Ru Jia |
| collection | DOAJ |
| description | This paper presents a systematic investigation into the solvability and the general solution of a tensor equation system within the split quaternion algebra framework. As an extension of classical quaternions with distinctive pseudo-Euclidean properties, split quaternions offer unique advantages in multidimensional signal processing applications. We establish rigorous necessary and sufficient conditions for the existence of solutions to the proposed tensor equation system, accompanied by explicit formulations for general solutions when solvability criteria are satisfied. The theoretical framework is further strengthened by the development of computational algorithms and numerical validations through concrete examples. Notably, we demonstrate the practical implementation of our theoretical findings through encryption/decryption algorithms for color video data. |
| format | Article |
| id | doaj-art-f60e4efb7c8547e6a0b463fda04a9de1 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-f60e4efb7c8547e6a0b463fda04a9de12025-08-20T02:03:31ZengMDPI AGMathematics2227-73902025-02-0113464410.3390/math13040644The General Solution to a System of Tensor Equations over the Split Quaternion Algebra with ApplicationsZong-Ru Jia0Qing-Wen Wang1Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, ChinaDepartment of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, ChinaThis paper presents a systematic investigation into the solvability and the general solution of a tensor equation system within the split quaternion algebra framework. As an extension of classical quaternions with distinctive pseudo-Euclidean properties, split quaternions offer unique advantages in multidimensional signal processing applications. We establish rigorous necessary and sufficient conditions for the existence of solutions to the proposed tensor equation system, accompanied by explicit formulations for general solutions when solvability criteria are satisfied. The theoretical framework is further strengthened by the development of computational algorithms and numerical validations through concrete examples. Notably, we demonstrate the practical implementation of our theoretical findings through encryption/decryption algorithms for color video data.https://www.mdpi.com/2227-7390/13/4/644split quaternion algebratensor equationsEinstein productgeneral solutionvideo encryption |
| spellingShingle | Zong-Ru Jia Qing-Wen Wang The General Solution to a System of Tensor Equations over the Split Quaternion Algebra with Applications Mathematics split quaternion algebra tensor equations Einstein product general solution video encryption |
| title | The General Solution to a System of Tensor Equations over the Split Quaternion Algebra with Applications |
| title_full | The General Solution to a System of Tensor Equations over the Split Quaternion Algebra with Applications |
| title_fullStr | The General Solution to a System of Tensor Equations over the Split Quaternion Algebra with Applications |
| title_full_unstemmed | The General Solution to a System of Tensor Equations over the Split Quaternion Algebra with Applications |
| title_short | The General Solution to a System of Tensor Equations over the Split Quaternion Algebra with Applications |
| title_sort | general solution to a system of tensor equations over the split quaternion algebra with applications |
| topic | split quaternion algebra tensor equations Einstein product general solution video encryption |
| url | https://www.mdpi.com/2227-7390/13/4/644 |
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