In linear algebra, a nilpotent matrix is a square matrix N such that

${\displaystyle N^{k}=0\,}$

for some positive integer ${\displaystyle k}$. The smallest such ${\displaystyle k}$ is called the index of ${\displaystyle N}$, sometimes the degree of ${\displaystyle N}$.

More generally, a nilpotent transformation is a linear transformation ${\displaystyle L}$ of a vector space such that ${\displaystyle L^{k}=0}$ for some positive integer ${\displaystyle k}$ (and thus, ${\displaystyle L^{j}=0}$ for all ${\displaystyle j\geq k}$). Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.