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

Author

Listed:
  • 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
    Collaborative Innovation Center for the Marine Artificial Intelligence, Shanghai University, Shanghai 200444, China)

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.

Suggested Citation

  • Zong-Ru Jia & Qing-Wen Wang, 2025. "The General Solution to a System of Tensor Equations over the Split Quaternion Algebra with Applications," Mathematics, MDPI, vol. 13(4), pages 1-21, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:4:p:644-:d:1592346
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