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.
Parents
Children
- Definition of nilpotent matrixDefinition. A square matrix $A$ is nilpotent if there exists a positive integer $k$ such that $A^k = 0$. The smallest...
- Every nilpotent matrix is similar to a sum of nilpotent Jordan blocksTheorem. Every nilpotent matrix is similar to a matrix in Jordan form with blocks of the form: $$J k(0) =...
- A complex square matrix is nilpotent iff all its eigenvalues are zeroFor a positive-size complex square matrix, nilpotence is equivalent to having only the eigenvalue zero. Nilpotence...
- Definition of index of nilpotencyDefinition. The index of nilpotency of a nilpotent matrix $A$ is the smallest positive integer $k$ such that $A^k =...
- Commuting diagonalizable and nilpotent decomposition over the complex numbersOver $\mathbb C$, every matrix is similar to a Jordan matrix $J=D+N$ with $D$ diagonal, $N$ nilpotent, and $DN=ND$....
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