Nilpotent matrix

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

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.

The matrix

is nilpotent with index 2, since ${\displaystyle A^{2}=0}$.

More generally, any ${\displaystyle n}$-dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index ${\displaystyle \leq n}$ ^{[citation needed]}. For example, the matrix

is nilpotent, with

The index of ${\displaystyle B}$ is therefore 4.