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.

翻译：对偶四元数能够表示三维空间中的刚体运动，在机器人学、三维运动建模与控制以及计算机图形学中具有广泛应用。本文引入对偶四元数向量的三种不同右线性无关性概念，并研究对偶四元数向量集及对偶四元数矩阵的相关基本性质。针对对偶四元数埃尔米特矩阵的右特征值，提出一种极小极大原理。基于新建立的对偶四元数向量柯西-施瓦茨不等式以及对偶四元数矩阵的奇异值分解，推导出对偶四元数矩阵奇异值的重要不等式。最后引入对偶四元数矩阵广义逆的概念，给出一个对偶四元数矩阵成为另一对偶四元数矩阵四类广义逆的充要条件。