Encart 10

Le rôle de la représentation matricielle dans la formulation contemporaine de la relation mathématique entre diviseurs élémentaires et forme de Jordan.

Trois exemples de décompositions matricielles associées à la décomposition en diviseurs élémentaires d’un même polynôme caractéristique : |A-λI| = (λ-1)²(λ-2)³ (λ-3).

Forme de Jordan

Diviseurs

élémentaire

(λ-1), (λ-1),
(λ-2), (λ-2),(λ-2), (λ-3)

(λ-1), (λ-1),
(λ-2)³,
(λ-3)

(λ-1), (λ-1),
(λ-2), (λ-2)²,
(λ-3)

Extraits du traité de Wedderburn de 1933.

[Wedderburn, 1933] :

Theorem 6. If λ₁, λ₂,…,λ_s are any constants, not necessarily all different and ν₁, ν₂,…,ν_s are positive integers whose sum is n, and if a_i is the array of ν_i rows and columns given by

(10)

where each coordinate on the main diagonal equals λ_i, those on the parallel on its right are 1, and the remaining ones are 0, and if a is the matrix of n rows and columns given by

a =

(11)

composed of blocks of terms defined by (10) arranged so that the main diagonal of each lies on the main diagonal of a, the other coordinates being 0, then λ-a has the elementary divisors

(12)

[…]

If A is a matrix with the same elementary divisors as a, it follows from theorem 5, that there is a matrix P such that A = PaP^-1 and hence, if we choose in place of the fundamental basis (e₁, e₂,…, e_n) the basis (Pe₁, Pe_2,…,Pe_n), […] (11) gives the form of A relative to the new basis. This form is called the canonical form of A.