nilpotent operator

A nilpotent operator is one that vanishes when raised to a high enough power: there is some positive integer k with N^k = 0, while smaller powers need not be zero. The smallest such k is the index of nilpotency. Intuitively N keeps pushing vectors toward zero, and after at most k applications everything has been pushed all the way out.

The clean spectral fact: the only eigenvalue of a nilpotent operator is 0. If N v = lambda v with v nonzero, then N^k v = lambda^k v = 0 forces lambda^k = 0, so lambda = 0. Consequently the characteristic polynomial is x^(dim V) and the minimal polynomial is x^k. A nilpotent operator is never invertible (for nonzero V) and is diagonalizable only in the trivial case N = 0, since a diagonalizable operator with sole eigenvalue 0 must already be the zero operator.

The model example is the shift that sends each basis vector to the previous one and the first one to zero: e_k -> e_(k-1) -> ... -> e_1 -> 0. Apply it k times and every basis vector reaches 0. Its matrix has 1s on the superdiagonal and 0s elsewhere — a single Jordan block for eigenvalue 0. Every nilpotent operator is a direct sum of such shift blocks of various sizes.

Nilpotents are the building blocks of Jordan theory. Any operator on a space over an algebraically closed field splits, after subtracting its semisimple (diagonalizable) part, into a nilpotent part on each generalized eigenspace. Understanding nilpotents in their own right — their chains, their Jordan blocks — is therefore the last technical hurdle before the Jordan canonical form.