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Les matrices :
formes de représentation et pratiques opératoires (1850-1930).

Frédéric Brechenmacher - Centre Alexandre Koyré

# 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,…,λs are any constants, not necessarily all different and ν1, ν2,…,νs are positive integers whose sum is n, and if ai is the array of νi rows and columns given by

 (10)

where each coordinate on the main diagonal equals λi, 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,…, en) the basis (Pe1, Pe2,…, Pen), […] (11) gives the form of A relative to the new basis. This form is called the canonical form of A.