On the algebraic structure of the Moore-Penrose inverse of a polynomial matrix

This work establishes the connection between the finite and infinite algebraic structure of singular polynomial matrices and their Moore-Penrose (MP) inverse. The uniqueness of the MP inverse leads to the assumption that such a relation must exist. It is proved that the MP inverse of a singular polynomial matrix has no finite zeros and its finite poles are fully determined. Furthermore, the existence of a correspondence between the infinite pole/zero structure of any singular polynomial matrix and its MP inverse is proved to exist. Finally, it is shown that the minimal indices of the two aforementioned matrices are connected as well.