Jordanization for its own sake
Reduction of endomorphisms

Proving Jordan's theorem for a nilpotent matrix A of order n is easy if we take care to use the corresponding Young table, which has n cells. Denote by m the nilpotent index of A.

We begin by choosing vectors v₁, ..., v_p in E that raise a basis of the quotient E/Ker A^{m – 1}, i.e.p = dim E – dim Ker A^{m – 1}

vectors of Ker A^m which are independent modulo Ker A^{m – 1}.

These vectors are placed in the cells of the last column of the Young table of A. To each of these vectors, we successively apply the powers of A in such a way as to fill, from right to left, the line of length m leading to it.

Once these cells have been filled, we proceed in the same way with the restriction of A to...