This paper presents an algorithm for computing the Jordan form of a square matrix with coefficients in a field K using the computer algebra system Axiom [FLJ+85] [BBD+91].
This system presents the advantage of allowing generic programming. That is to say, we can first implement our algorithm for matrices with rational coefficients and then generalize it to matrices with coefficients in any field. Therefore we shall present the general method which is essentially band on the use of the Frobenius form of a matrix in order to compute its Jordan form; and then restrict our attention to matrices with rational coefficients.
On the one hand we streamline the algorithm froben which computes the Frobenius form of a matrix, and on the other we examine in some detail the transformation from the Frobenius form to ihe Jordan form, and give the so called algorithm Jordfonn. We study in particular, the complexity of this algorithm and prove that it is polynomial when the coefficients of the matrix are rational. Finally we give some experiments and a conclusion.