Generalized proper inverse of polynomial matrices and the existence of infinite decoupling zeros

A necessary and sufficient condition is presented for the existence of a generalized proper rational inverse for nonsquare polynomial matrices. The condition is then proved to be equivalent to the absence of infinite decoupling zeros. This condition provides a simple test for the absence of infinite decoupling zeros in the polynomial fraction form. It can also be used to remove the infinite decoupling zeros in order to achieve a strongly irreducible system, and to provide a new look of the unimodular matrix operation effect at infinity, for the left and right polynomial fraction forms.<>