**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:**
$$A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, \quad A^2 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad A^3 = 0$$
**Properties:**
- All eigenvalues of a nilpotent matrix are 0
- $\det(A) = 0$ and $\text{tr}(A) = 0$
- A nilpotent matrix is never invertible
- $I - A$ is always invertible with inverse $I + A + A^2 + \cdots + A^{k-1}$
- Every strictly triangular matrix is nilpotent
Nilpotent matrices appear in the Jordan canonical form decomposition.