Nilpotent matrices

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.