Do all dual matrices have dual Moore–Penrose generalized inverses ?

Abstract

This paper investigates the question of whether all dual matrices have dual Moore–Penrose generalized inverses. It shows that there are uncountably many dual matrices that do not have them. The proof is constructive and results in the construction of large sets of matrices whose members do not have such an inverse. Various types of generalized inverses of dual matrices are discussed, and the necessary and sufficient conditions for a dual matrix to be a Moore–Penrose generalized inverse of another dual matrix are provided. Necessary and sufficient conditions for other types of generalized inverses of dual matrices are also provided. A necessary condition, which can be easily computed, for a matrix to be a {1,2}-generalized inverse or a Moore–Penrose Inverse of a dual matrix is given. Dual matrices that have no generalized inverses arise in practical situations. This is shown by considering a simple example in kinematics. The paper points out that in other areas of science and engineering where dual matrices are also commonly used, formulations and computations that involve their generalized inverses need to be handled with considerable care. This is because unlike generalized inverses of ordinary matrices, generalized inverses of dual matrices do not always exist.