An unstructured algorithm for the singular value decomposition of biquaternion matrices

Abstract

With the modeling of the biquaternion algebra in multidimensional signal processing, it has become possible to address issues such as data separation, denoising, and anomaly detection. This paper investigates the singular value decomposition of biquaternion matrices (SVDBQ), establishing an SVDBQ theorem that ensures unitary matrices formed by the left and right singular vectors, while also introducing a new form for singular values. Additionally, the non-uniqueness of SVDBQ is proven, expanding the theoretical framework of the biquaternion algebra. Building on this foundation, the paper presents a novel, fast, unstructured algorithm based on the isomorphic representation matrices of biquaternion matrices. Unlike existing methods, which are often complex and computationally expensive, the proposed algorithm is structurally simple and significantly faster, making it ideal for real-time signal processing. Numerical experiments validate the efficiency and effectiveness of this new algorithm, demonstrating its potential to advance both research and practical applications in signal processing.