Two novel numerical methods for the diagonalisation of a reduced biquaternion matrix in the reduced biquaternionic algebra

Abstract

Unlike quaternions and split quaternions, reduced biquaternions satisfy the multiplication commutative rule and are commonly used in image processing, fuzzy recognition, image compression, Hopfield neural networks, and digital signal processing. However, although algebraic techniques have been developed for the diagonalisation of quaternion and split quaternion matrices, the diagonalisation of a reduced biquaternion matrix is yet to be studied. In this study, we derive sufficient and necessary conditions for the diagonalisation of a reduced biquaternion matrix and devise two numerical methods for the diagonalisation of a reduced biquaternion matrix. These methods were derived using complex and real representations in the reduced biquaternionic algebra.

Keywords:

• Reduced biquaternion
• complex representation
• real representation
• diagonalisation

Acknowledgements

The authors are grateful to the referees for their valuable comments and suggestions, which helped improve the quality of this manuscript.

Disclosure statement

The authors report there are no competing interests to declare.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

The research of T. Jiang is supported by the Shandong Natural Science Foundation (No. ZR201709250116). The research of D. Zhang is supported by the Chinese Government Scholarship (CSC No. 202108370086). The research of V. I. Vasil'ev is supported by Russian Government Project Science and Universities (No. FSRG-2021-0015). The research of G. Wang is supported by the Chinese Government Scholarship (CSC No. 202008370340).