Frédéric Brechenmacher - Centre Alexandre Koyré

Encart 2

Le théorème de décomposition matricielle dans le traité de Mac Duffee de 1943.

THEOREM 65. Every non derogatory matrix whose minimum function is [l(x)]^h where l(x) is irreductible of degree j is similar to a matrix of the type (35) with h blocks along the diagonal.

(35)

The case where F is the complex field, or in fact any algebraically closed field, deserves special mention. In this case we can write

so that A is similar to a direct sum of matrices of the form

of h_i rows and columns. This the familiar Jordan normal form of a matrix with complex elements.

[Mac Duffee, 1943, 114-121]