**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)