ABSTRACT The widespread application of dual quaternion across various fields has propelled the study of its matrix theory into recent focus. We develop a structure‐preserving algorithm for dual quaternion matrix LU decomposition (DQLUD). Theoretically, we rely on the dual matrix representation of dual quaternion matrices to transform the operations in the DQLUD and reduce the computational complexity based on the special structure of the dual matrices. Furthermore, we not only design a structure‐preserving algorithm for the partial pivoting DQLUD by utilizing the order relation of dual numbers but also present a structure‐preserving algorithm for the Cholesky decomposition of dual quaternion Hermitian positive definite matrices. Numerically, comparisons with other available algorithms are carried out, which demonstrate that our proposed structure‐preserving algorithm incurs less computational cost. Finally, we translate the idea of designing structure‐preserving algorithms to the LU decomposition of dual complex matrices and successfully apply it to designing a strict authentication scheme for color images and the process of color image watermarking.
Ding et al. (Thu,) studied this question.