When Does a Dual Matrix Have a Dual Generalized Inverse?

Firdaus E Udwadia

doi:10.3390/sym13081386

Abstract

This paper deals with the existence of various types of dual generalized inverses of dual matrices. New and foundational results on the necessary and sufficient conditions for various types of dual generalized inverses to exist are obtained. It is shown that unlike real matrices, dual matrices may not have {1}-dual generalized inverses. A necessary and sufficient condition for a dual matrix to have a {1}-dual generalized inverse is obtained. It is shown that a dual matrix always has a {1}-, {1,3}-, {1,4}-, {1,2,3}-, {1,2,4}-dual generalized inverse if and only if it has a {1}-dual generalized inverse and that every dual matrix has a {2}- and a {2,4}-dual generalized inverse. Explicit expressions, which have not been reported to date in the literature, for all these dual inverses are provided. It is shown that the Moore–Penrose dual generalized inverse of a dual matrix exists if and only if the dual matrix has a {1}-dual generalized inverse; an explicit expression for this dual inverse, when it exists, is obtained irrespective of the rank of its real part. Explicit expressions for the Moore–Penrose dual inverse of a dual matrix, in terms of {1}-dual generalized inverses of products, are also obtained. Several new results related to the determination of dual Moore-Penrose inverses using less restrictive dual inverses are also provided.

Highlights

The use of dual matrices has become common in various areas of science and engineering, such as the kinematic analysis and synthesis of machines and mechanisms, robotics and machine vision
It is shown that, while the {1,2,3,4}-dual generalized inverse of a dual matrix, when it exists, is unique, the {1,3}- and {1,4}-dual generalized inverses are not unique. This is because {1,3}- and {1,4}-dual generalized inverses are less restrictive than the {1,2,3,4}-dual generalized inverse, in that they are defined as dual matrices that need to satisfy only two of the four dual MP conditions, while an MP-dual generalized inverse (MPDGI) is defined as a matrix that must satisfy all four of them
This paper deals with the question of when a dual matrix Ahas a specified kind of dual generalized inverse

Summary

Introduction

The use of dual matrices has become common in various areas of science and engineering, such as the kinematic analysis and synthesis of machines and mechanisms, robotics and machine vision (see, for example, [1,2,3]). It is only recently that the question of the existence of the Moore–Penrose dual generalized inverse has been raised and, in reference [4], it is shown that unlike the assured existence of a Moore–Penrose inverse of a real matrix, all dual matrices may not have Moore–Penrose (MP) dual inverses. This is demonstrated by the construction of an uncountably infinite set of dual matrices that are guaranteed to have no Moore–Penrose dual inverses. A recent finding shows that the reason why the Moore–Penrose dual generalized inverses of the dual matrices constructed in reference [4] do not exist is that these dual matrices do not have a {1}-dual generalized inverse [5]

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Conclusion