---
tag: 0419GJC84P
kind: definition
name: Nilpotent matrix
msc: [15A99]
defines: 07BSPZPSMF
uses: [0CP7YY6NAG]
sources:
  - url: https://en.wikipedia.org/wiki/Nilpotent_matrix
    locator: lead section
---

## Definition

Let $R$ be a ring and let $N$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $N$ is *nilpotent* when $N^k = 0$ for some integer $k \ge 1$.
The least such $k$ is the *index* of $N$.

## Remarks

The matrix $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ is nilpotent of index $2$.
Each strictly upper triangular matrix is nilpotent.
Over a field, an $n \times n$ matrix $N$ is nilpotent if and only if its characteristic polynomial is $x^n$; then $N^n = 0$.
The only eigenvalue of a nilpotent matrix over a field is $0$, so its trace and determinant are $0$.

## Explanations

### L1 human

A nilpotent matrix becomes the zero matrix after we multiply it by itself enough times.
A shift that moves each coordinate one step down, and drops the last one, is an example.
After $n$ shifts nothing is left.