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# Definition of nilpotent matrix

Put content here.**Definition.** A square matrix $A$ is *nilpotent* if there exists a positive integer $k$ such that $A^k = 0$.
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The smallest such $k$ is called the *index of nilpotency*.
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**Example.** $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ is nilpotent with index 2, since $A^2 = 0$ but $A \neq 0$.
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**Example.** $A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$ is nilpotent with index 3.
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**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)

# Parents

* Nilpotent matrices