---
tag: 0M8MHYERTY
kind: definition
name: Nilpotent endomorphism
msc: [15A04, 15A21]
defines: 063D5Z2C70
uses: [0PTF1PTJK0]
sources:
  - url: https://en.wikipedia.org/wiki/Nilpotent_matrix
    locator: lead section
---

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The map $T$ is *nilpotent* when $T^k = 0$ for some integer $k \ge 1$.
The least such $k$ is the *index* of $T$.

## Remarks

If $\dim V = n$ is finite and $T$ is nilpotent, then $T^n = 0$.
In finite dimensions, $T$ is nilpotent if and only if its characteristic polynomial is $t^n$.
Over an algebraically closed field, this means that $0$ is its only eigenvalue.
A matrix is nilpotent when its map is nilpotent, see [[nilpotent matrix|07BSPZPSMF]].
The derivative on the space of polynomials of degree less than $n$ is nilpotent.

## Explanations

### L1 human

A nilpotent map destroys everything after a few steps.
Each application pushes vectors further towards zero, until they vanish.
The shift $e_1 \mapsto 0$, $e_2 \mapsto e_1$, $e_3 \mapsto e_2$ in $F^3$ is nilpotent of index $3$.