Nilpotent matrix

Definition

Let $R$ be a ring and let $N$ be an $n \times n$ square matrix 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.