Resolvent and rational canonical forms of matrices

Abstract

The goal of this paper is to explain how to derive from the resolvent of a matrix the following classical rational canonical forms: the Dunford decomposition, the rational spectral decomposition, the Jacobson and the rational Jordan forms (with corresponding change of basis matrices). We do not pretend that this approach is more efficient than other algorithms classically used to build these forms, but we transform theoretical and nonrational methods used in the context of perturbation matrices into effective rational algorithms well adapted to Computer Algebra.