Jordan Normal and Rational Normal Form Algorithms

X iv :c s/ 04 12 00 5v 1 [ cs .S C ] 2 D ec 2 00 4 Jordan Normal and Rational Normal Form Algorithms Bernard Parisse, Morgane Vaughan Institut Fourier CNRS-UMR 5582 100 rue des Maths Université de Grenoble I 38402 St Martin d'Hères Cédex Résumé In this paper, we present a determinist Jordan normal form algorithms based on the Fadeev formula : (λ · I − A) ·B(λ) = P (λ) · I where B(λ) is (λ · I − A)'s omatrix and P (λ) is A's hara teristi polynomial. This rational Jordan normal form algorithm di ers from usual algorithms sin e it is not based on the Frobenius/Smith normal form but rather on the idea already remarked in Gantma her that the non-zero olumn ve tors of B(λ0) are eigenve tors of A asso iated to λ0 for any root λ0 of the hara teristi al polynomial. The omplexity of the algorithm is O(n4) eld operations if we know the fa torization of the hara teristi polynomial (or O(n5 ln(n)) operations for a matrix of integers of xed size). This algorithm has been implemented using the Maple and Gia /X as omputer algebra systems. 1 Introdu tion Let's remember that the Jordan normal form of a matrix is : A = 