Eigenvalues and Singular Values of Dual Quaternion Matrices

TL;DR

This paper develops a spectral theory for dual quaternion matrices, including eigenvalues, eigenvectors, and singular value decomposition, with applications to robotics pose representation.

Contribution

It introduces the concepts of eigenvalues and singular values for dual quaternion matrices, extending spectral theory to this algebraic structure.

Findings

Right eigenvalues of dual quaternion Hermitian matrices are dual numbers.

Dual quaternion Hermitian matrices have exactly n eigenvalues.

The paper provides a unitary decomposition and singular value decomposition for dual quaternion matrices.

Abstract

The poses of $m$ robotics in $n$ time points may be represented by an $m \times n$ 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 $n \times n$ 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.