Minimax Principle for Eigenvalues of Dual Quaternion Hermitian Matrices and Generalized Inverses of Dual Quaternion Matrices

Abstract

AbstractDual 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 concepts for a set of dual quaternion vectors, and study some related basic properties for dual quaternion vectors and dual quaternion matrices. We present a minimax principle for 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 inequality for singular values of dual quaternion matrices. Finally, we introduce the concept of generalized inverses of dual quaternion matrices, and present necessary and sufficient conditions for a dual quaternion matrix to be one of four types of generalized inverses of another dual quaternion matrix.KEYWORDS: Dual quaternion matrixdual quaternion vectoreigenvaluegeneralized inverselinear independenceminimax principle Additional informationFundingThis work was partially supported by Hong Kong Innovation and Technology Commission (InnoHK Project CIMDA). Chen Ling's work was supported by Natural Science Foundation of China (No. 11971138). Hong Yan's work was supported by Hong Kong Research Grants Council (Project 11204821), Hong Kong Innovation and Technology Commission (InnoHK Project CIMDA) and City University of Hong Kong (Project 9610034).