Canonical Forms and Orbit Spaces of Linear Systems

Since globally continuous canonical forms exist only for very special group actions we propose to look instead for canonical forms which yield continuous sections on appropriate cellular decompositions of the orbit space. In this expository paper we present some general results concerning these 'cellular' canonical forms for various types of group actions. In particular, lower bounds for the discontinuity of cellular forms (as measured by the cardinality of the underlying cellular decomposition) are derived in terms of the (co-)homology of the associated orbit space. Conversely, cellular canonical forms are shown to yield an efficient tool for analyzing topological properties of the orbit space. The general definitions and results are illustrated by an analysis of the similarity action of the general linear group on the space of reachable linear systems. In particular, we compute the homology groups of this space modulo similarity and prove that the Kronecker-Popov form is a cellular canonical form of minimal discontinuity.