The General Solution to a System of Tensor Equations over the Split Quaternion Algebra with Applications

Zong-Ru Jia and Qing-Wen Wang ()
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Zong-Ru Jia: Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, China
Qing-Wen Wang: Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, China

Mathematics, 2025, vol. 13, issue 4, 1-21

Abstract: 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.

Keywords: split quaternion algebra; tensor equations; Einstein product; general solution; video encryption (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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