A genuine extension of the Moore–Penrose inverse to dual matrices

Abstract

The Moore–Penrose (MP) inverse is a genuine extension of the matrix inverse. In the last one and a half decades, in the study of approximate synthesis in kinematic, two generalizations of the MP inverse appeared for dual real matrices. One generalization, dual MP generalized inverse (DMPGI), satisfies the four MP conditions, but does not exist for uncountably many dual real matrices. Another generalization, MP dual generalized inverse (MPDGI), always exists, but does not satisfy the four MP conditions even if DMPGI exists. Recently, a new dual MP inverse (NDMPI) appeared. In this paper, based on the singular value decomposition of dual matrices given in this article, we show that the NDMPI can be regarded as a combination of the DMPGI and MPDGI. It coincides with the DMPGI when the DMPGI exists. When the DMPGI does not exist, it is an MPDGI extension of the DMPGI. It always satisfies the last three MP conditions, and minimizes the error of the first MP condition when that MP condition is impossible to be satisfied. Given a dual matrix, its NDMPI is the unique dual matrix that satisfies the extended version of the first MP condition, and the other three original MP conditions. If the original matrix is complex, its NDMPI is exactly the MP inverse of that matrix. Furthermore, NDMPI may be used to generate the minimum-norm solution to the dual least squares problem. Numerical results on kinematic analysis and dual linear regression are reported.