Some geometric questions in the theory of linear systems

In this paper we discuss certain geometrical aspects of linear systems which, even though they arise in the case of single-input/single-output systems, do not seen to have been explicitly recognized and studied before. We show, among other things, that the set of minimal, single-input/single-output, linear systems of degree n, when topologized in the obvious way, consists of n+1 connected components. The Cauchy index (equivalently, the signature of the Hankel matrix) characterizes the components and the geometry of each component is investigated. We also study the effect of various constraints such as asking that the system be stable or minimum phase.