Author’s summary: The well-known theory of “rational canonical form of an operator” describes the invariant factors, or elementary divisors, as a complete set of invariants of a similarity class of an operator on a finite-dimensional vector space over a given field . A finer part of the theory is the contribution by Frobenius dealing with the structure of the centralizer of an operator. The viewpoint is that of finitely generated modules over a PID.
In this paper we approach the issue from a “dynamic” viewpoint, as explained in the author’s paper [J. Ramanujan Math. Soc. 22, No. 1, 35–56 (2007; Zbl 1181.22022)]. We also extend the theory to affine maps. The formulation is in terms of the action of the general linear group , resp. the group of invertible affine maps , on the semigroup of all linear, resp. affine, maps by conjugacy. The theory of rational canonical forms is connected with the orbits, and the Frobenius’ theory with the orbit-classes, of the action of on the semigroup of linear maps. We describe a parametrization of orbits and orbit-classes of both - and -actions, and also provide a parametrization of all affine maps themselves, which is independent of the choices of linear or affine co-ordinate systems. An important ingredient in these parametrizations is a certain flag. For a linear map on , let denote its centralizer associative -algebra, and the multiplicative group of invertible elements in . In this situation, we associate a canonical, maximal, -invariant flag, and precisely describe the orbits of on . Using this approach, we strengthen the classical theory in a number of ways.