# nilpotent matrix

The square matrix $A$ is said to be nilpotent if $A^{n}=\underbrace{AA\cdots A}_{\textrm{n times}}=\mathbf{0}$ for some positive integer $n$ (here $\mathbf{0}$ denotes the matrix where every entry is 0).

###### Theorem (Characterization of nilpotent matrices).

A matrix is nilpotent iff its eigenvalues are all 0.

###### Proof.

Let $A$ be a nilpotent matrix. Assume $A^{n}=\mathbf{0}$. Let $\lambda$ be an eigenvalue of $A$. Then $A\mathbf{x}=\lambda\mathbf{x}$ for some nonzero vector $\mathbf{x}$. By induction $\lambda^{n}\mathbf{x}=A^{n}\mathbf{x}=0$, so $\lambda=0$.

Conversely, suppose that all eigenvalues of $A$ are zero. Then the chararacteristic polynomial of $A$: $\det(\lambda I-A)=\lambda^{n}$. It now follows from the Cayley-Hamilton theorem that $A^{n}=\mathbf{0}$. ∎

Since the determinant is the product of the eigenvalues it follows that a nilpotent matrix has determinant 0. Similarly, since the trace of a square matrix is the sum of the eigenvalues, it follows that it has trace 0.

One class of nilpotent matrices are the http://planetmath.org/node/4381strictly triangular matrices (lower or upper), this follows from the fact that the eigenvalues of a triangular matrix are the diagonal elements, and thus are all zero in the case of strictly triangular matrices.

Note for $2\times 2$ matrices $A$ the theorem implies that $A$ is nilpotent iff $A=\mathbf{0}$ or $A^{2}=\mathbf{0}$.

Also it is worth noticing that any matrix that is similar to a nilpotent matrix is nilpotent.

Title nilpotent matrix NilpotentMatrix 2013-03-22 13:05:56 2013-03-22 13:05:56 jgade (861) jgade (861) 17 jgade (861) Definition msc 15-00