2021/11/24 by Liqun Qi, Qi, Liqun, Ziyan Luo +1 · 3 citations
Engineering · Computer Science · #Control and Dynamics of Mobile Robots #Robotic Mechanisms and Dynamics #Digital Image Processing Techniques
paper · pdf · doi:10.48550/arxiv.2111.12211
The poses of m robotics in n time points may be represented by an m × n dual quaternion matrix. In this paper, we study the spectral theory of dual quaternion matrices. We introduce right and left eigenvalues for square dual quaternion matrices. If a right eigenvalue is a dual number, then it is also a left eigenvalue. In this case, this dual number is called an eigenvalue of that dual quaternion matrix. We show that the right eigenvalues of a dual quaternion Hermitian matrix are dual numbers. Thus, they are eigenvalues. An n × n dual quaternion Hermitian matrix is shown to have exactly n eigenvalues. It is positive semidefinite, or positive definite, if and only if all of its eigenvalues are nonnegative, or positive and appreciable, dual numbers, respectively. We present a unitary decomposition of a dual quaternion Hermitian matrix, and the singular value decomposition for a general dual quaternion matrix. The singular values of a dual quaternion matrix are nonnegative dual numbers.