Resolvent and Rational Canonical Forms of Matrices

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 don't pretend that this approach is more efficient than other algorithms classicaly used to build these forms, but we transform theoritical and non rational methods used in the context of pertubation matrices into effective rational algorithms well adapted to Computer Algebra. 1 Resolvent Let us collect some basic facts about the resolvent of a linear operator acting on a finite-dimensional vector space (the general and classical reference for the first paragraph is [7]). A will be a matrix belonging to the algebra Mn (k) of square matrices of order n with coefficients in the zero characteristic field k. We will note I the identity matrix of Mn (k) and K an algebraic closure of k. Let oe(A) be the s..