Definition of nilpotent matrix

Definition. A square matrix $A$ is nilpotent if there exists a positive integer $k$ such that $A^k = 0$.

The smallest such $k$ is called the index of nilpotency.

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

Example. $A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$ is nilpotent with index 3.

Properties:

• The only eigenvalue of a nilpotent matrix is 0
• $\det(A) = 0$ and $\text{tr}(A) = 0$
• The product of commuting nilpotent matrices is nilpotent
• A nilpotent matrix is never invertible (except the $0 \times 0$ case)