Eigenvalues and Singular Values of Dual Quaternion Matrices

By Liqun Qi, Ziyan Luo. 2021

63 citations recorded by the source corpus.

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.

Subfields: Computer Vision and Pattern Recognition, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/6a78376e376fffb7387a43c5ecf605416955ebee