A Genuine Extension of The Moore-Penrose Inverse to Dual Matrices

TL;DR

This paper introduces a new genuine extension of the Moore-Penrose inverse to dual matrices, called GMPI, which satisfies all Moore-Penrose conditions and generalizes previous concepts.

Contribution

The paper proposes GMPI, a unique extension of the Moore-Penrose inverse to dual matrices, unifying and extending existing generalized inverses.

Findings

GMPI coincides with the classical Moore-Penrose inverse for complex matrices.

GMPI matches the DMPGI when it exists for dual real matrices.

The extension is based on singular value decomposition of dual matrices.

Abstract

The Moore-Penrose inverse is a genuine extension of the matrix inverse. Given a complex matrix, there uniquely exists another complex matrix satisfying the four Moore-Penrose conditions, and if the original matrix is nonsingular, it is exactly the inverse of that matrix. In the last one and half decade, in the study of approximate synthesis in kinematic, two generalizations of the Moore-Penrose inverse appeared for dual real matrices,including Moore-Penrose dual generalized inverse and dual Moore-Penrose generalized inverse (DMPGI). DMPGI satisfies the four Moore-Penrose conditions, but does not exist for uncountably many dual real matrices. %Another generalization, called Moore-Penrose dual generalized inverse (MPDGI), is different from DMPGI even if DMPGI exists. In this paper, based upon the singular value decomposition of dual matrices, we extend the first Moore-Penrose condition to dual matrices and introduce a genuine extension of the Moore-Penrose inverse to dual matrices, referred to as GMPI. Given a dual complex matrix, its GMPI is the unique dual complex matrix which satisfy the first extended and the other three Moore-Penrose conditions. If the original matrix is a complex matrix, its GMPI is exactly the Moore-Penrose inverse of that matrix. And if the original matrix is a dual real matrix and its DMPGI exists, its GMPI coincides with its DMPGI.