Dual generalized inverses and their use in solving systems of linear dual equations

Abstract

This paper deals with properties of dual generalized inverses and uses them in solving linear dual equations. The first part of this paper addresses several properties of dual generalized inverses of dual matrices and provides new theoretical and computational insights. In the second part necessary and sufficient conditions for a dual matrix to have a {1}-dual generalized inverse are obtained. A dual matrix with a rank deficient primal part generically (almost always) is shown to have no {1}-dual generalized inverse and hence, neither a {1,3}-dual generalized inverse or a Moore-Penrose dual generalized inverse. The necessary and sufficient condition to have a solution to a system of linear dual equations is obtained. An explicit expression for the solution, when one exists, is obtained and it is shown not to be unique, in general. The dual analog of the real least-squares problem is obtained through the {1,3}-dual generalized inverse; as with real matrices, it is shown that this least-squares dual solution is not unique, in general. The dual analog of the real minimum-length least-squares solution is obtained through the development of the Moore-Penrose dual generalized inverse. Numerical examples are provided throughout, mainly from kinematics, to illustrate the results obtained.