6. Jordanization for its own sake

6.1 Practice and proof

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 v ₁ , ..., 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...