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.

中文翻译：

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一种用于双四元数矩阵奇异值分解的非结构化算法


通过在多维信号处理中对双四元数代数进行建模，可以解决数据分离、去噪和异常检测等问题。本文研究了双四元数矩阵 （SVDBQ） 的奇异值分解，建立了一个 SVDBQ 定理，该定理确保由左右奇异向量形成的幺正矩阵，同时还引入了奇异值的新形式。此外，SVDBQ 的非唯一性得到了证明，扩展了双四元数代数的理论框架。在此基础上，本文提出了一种基于双四元数矩阵同构表示矩阵的新颖、快速、非结构化算法。与通常复杂且计算成本高昂的现有方法不同，所提出的算法结构简单且速度明显加快，使其成为实时信号处理的理想选择。数值实验验证了这种新算法的效率和有效性，证明了它有可能推动信号处理的研究和实际应用。