Rational matrices: counting the poles and zeros

The authors introduce finite-dimensional vector spaces which measure generic zeros which arise when a transfer function fails to be injective or subjective. An exact sequence relates the global spaces of zeros, the global spaces of poles, and the generic zero spaces. This sequence gives a structural result which can be described by the statement: the number of zeros of any transfer function is equal to the number of poles (when everything is counted appropriately). The same result unifies and extends a number of results of geometric control theory by relating global poles and zeros of general (possible improper) transfer functions to controlled invariant and controllability subspaces (including such spaces at infinity)