The probability that an operator is nilpotent

TL;DR

This paper investigates the probability that a random linear operator on a finite-dimensional vector space is nilpotent, showing it equals 1 divided by the space's dimension, and provides a new, algebraic proof of this fact.

Contribution

The paper offers a novel, algebraic proof of the probability that a linear operator is nilpotent, avoiding calculations and inspired by combinatorial methods.

Findings

Probability that a random linear operator is nilpotent is 1/N.

Provides a new proof using linear algebra techniques.

Connects nilpotent operators to combinatorial structures.

Abstract

Choose a random linear operator on a vector space of finite cardinality N: then the probability that it is nilpotent is 1/N. This is a linear analogue of the fact that for a random self-map of a set of cardinality N, the probability that some iterate is constant is 1/N. The first result is due to Fine, Herstein and Hall, and the second is essentially Cayley's tree formula. We give a new proof of the result on nilpotents, analogous to Joyal's beautiful proof of Cayley's formula. It uses only general linear algebra and avoids calculation entirely.