Minimax principle for right eigenvalues of dual quaternion matrices and their generalized inverses

Abstract

Dual quaternions can represent rigid body motion in 3D spaces, and have found wide applications in robotics, 3D motion modelling and control, and computer graphics. In this paper, we introduce three different right linear independency for a set of dual quaternion vectors, and study some related basic properties for the set of dual quaternion vectors and dual quaternion matrices. We present a minimax principle for right eigenvalues of dual quaternion Hermitian matrices. Based upon a newly established Cauchy-Schwarz inequality for dual quaternion vectors and singular value decomposition of dual quaternion matrices, we propose an important inequality for singular values of dual quaternion matrices. We finally introduce the concept of generalized inverse of dual quaternion matrices, and present the necessary and sufficient conditions for a dual quaternion matrix to be one of four types of generalized inverses of another dual quaternion matrix.