On the Algebraic Structure of the Moore Penrose Inverse of Full Row or Full Column Rank Polynomial Matrices

Abstract

This work establishes the connection between the finite and infinite algebraic structure of a full row or column rank polynomial matrix and its Moore Penrose inverse. For non-singular matrices, it is known that the inverse has no finite zeros and its finite poles are the finite zeros of the original matrix, while the infinite poles and zeros of the original matrix are the infinite zeros and poles of its inverse respectively. The fact that the Moore Penrose inverse is unique for every matrix along with the fact that for non-singular matrices it coincides with the normal inverse leads to the assumption that a similar connection holds for the structure, both finite and infinite, of the Moore Penrose inverse of a polynomial matrix. We prove that the Moore Penrose inverse of a full row or column rank polynomial matrix has no finite zeros, and that extra poles appear, compared to the non-singular case. Furthermore, we prove that the result regarding the infinite structure of nonsingular matrices is extended to the case of the Moore Penrose inverse of a full rank polynomial matrix. Finally we determine a connection between the minimal indices of the polynomial matrix and its Moore Penrose inverse. The results are also presented through an illustrative example.