# Nilpotent matrices

Put content here**Definition:** A square matrix $A$ is **nilpotent** if there exists a positive integer $k$ such that:
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$$A^k = 0$$
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The smallest such $k$ is called the **index of nilpotency**.
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**Example:**
$$A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, \quad A^2 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad A^3 = 0$$
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**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
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Nilpotent matrices appear in the Jordan canonical form decomposition.

# Parents

* Particular types of matrices