A Fast Structure-Preserving Method for Dual Quaternion Singular Value Decomposition

In this paper, we establish a novel dual matrix representation for dual quaternion matrices, which forms the foundation for a fast and innovative dual structure-preserving algorithm for dual quaternion singular value decomposition (DQSVD). By leveraging the dual quaternion Householder transformation and exploiting the existing properties of dual quaternions, we design a structure-preserving algorithm. This algorithm has a remarkable advantage in that it can convert quaternion operations in the process of bidiagonalizing the dual quaternion matrix into a dual matrix during DQSVD into real operations. As a result, computational efficiency is significantly enhanced. To verify the effectiveness of our proposed algorithm, we present a series of numerical examples. In these examples, we construct the dual complex matrix representation of color images and apply the concept of the structure-preserving algorithm to the dual complex singular value decomposition (DCSVD). This has been successfully employed in the watermark design of color images.