The Cholesky and Cholesky-QR Decompositions of Dual Quaternion Matrices

In this paper, we study the Cholesky and Cholesky-QR decompositions of dual quaternion matrices. Firstly, the dual quaternion Gauss transformation matrix is given as the basic tool to realize these two decompositions. Secondly, it is proved that a dual quaternion Hermitian matrix with positive definite standard part can be performed the Cholesky decomposition, and the corresponding direct algorithm is given. Moreover, if the standard part of a dual quaternion matrix is column full rank, the Cholesky-QR decomposition can be realized. Inspired by the real structure-preserving algorithm of quaternion matrices, we design a real calculation-preserving algorithm to realize the Cholesky decomposition of dual quaternion matrices. Numerical experiments show that the real calculation-preserving algorithm is more effective than the direct algorithm. Finally, a concrete example of applying the Cholesky decomposition of dual quaternion matrices to the Cholesky-QR decomposition of dual quaternion matrices is given.