Rational matrix structure

We bring together in this paper, in a unified way, certain results developed by various people regarding the pole/zero structure of a rational matrix and the structure of the vector-space generated by its columns. Pole and zero structure, at finite and infinite arguments, is compactly described by using elementary ideas from the language of valuation theory. The concept of column-reducedness of a rational matrix at some argument is introduced, and shown to determine when its pole/zero structure is simply that of its columns taken separately. We describe a procedure that operates on the Laurent expansion of a given rational matrix at the argument of interest in order to transform the matrix to one that is column-reduced at this argument but has the same pole/zero structure. The occurrence of such a "structure-extraction" algorithm in various contexts in system theory is pointed out. Special properties of rational bases (for a rational vector space) that are column-reduced at all arguments are noted, somewhat extending what is already well-known for minimal polynomial bases for such a space.