Definition of index of nilpotency

Created over 8 years ago, updated 10 days ago

Definition. The index of nilpotency of a nilpotent matrix $A$ is the smallest positive integer $k$ such that $A^k = 0$ but $A^{k-1} \neq 0$.

Properties:

For an $n \times n$ nilpotent matrix, the index of nilpotency is at most $n$
The index equals the size of the largest Jordan block in the Jordan canonical form of $A$
The index equals the degree of the minimal polynomial of $A$

Example. $A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ has index 2 since $A^2 = 0$ but $A \neq 0$.

Example. The $n \times n$ shift matrix with 1 on the superdiagonal has index $n$.