# Definition of index of nilpotency

Put content here**Definition.** The *index of nilpotency* of a nilpotent matrix $A$ is the smallest positive integer $k$ such that $A^k = 0$ but $A^{k-1} \neq 0$.
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**Properties:**
- For an $n \times n$ nilpotent matrix, the index of nilpotency is at most $n$
- The index equals the size of the largest Jordan block in the Jordan canonical form of $A$
- The index equals the degree of the minimal polynomial of $A$
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**Example.** $A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ has index 2 since $A^2 = 0$ but $A \neq 0$.
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**Example.** The $n \times n$ shift matrix with 1 on the superdiagonal has index $n$.

# Parents

* Nilpotent matrices