# Nilpotent Matrix and Eigenvalues of the Matrix

## Problem 11

An $n\times n$ matrix $A$ is called **nilpotent** if $A^k=O$, where $O$ is the $n\times n$ zero matrix.

Prove the followings.

**(a)** The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero.

**(b)** The matrix $A$ is nilpotent if and only if $A^n=O$.

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## Hint.

Hint for (a)

- $(\Rightarrow)$ Consider $A \mathbf{x}=\lambda \mathbf{x}$, where $\lambda$ is an eigenvalue of $A$ and $\mathbf{x}$ is an eigenvector corresponding to $\lambda$.
- $(\Leftarrow)$ Consider triangulation or Jordan normal/canonical form of $A$. Or use Cayley-Hamilton theorem.

## Proof of (a).

$(\Rightarrow)$

Suppose the matrix $A$ is nilpotent. Namely there exists $k \in \N$ such that $A^k=O$. Let $\lambda$ be an eigenvalue of $A$ and let $\mathbf{x}$ be the eigenvector corresponding to the eigenvalue $\lambda$.

Then they satisfy the equality $A\mathbf{x}=\lambda \mathbf{x}$. Multiplying this equality by $A$ on the left, we have

\[A^2\mathbf{x}=\lambda A\mathbf{x}=\lambda^2 \mathbf{x} .\]
Repeatedly multiplying by $A$, we obtain that $A^k \mathbf{x}=\lambda^k \mathbf{x}$. (To prove this statement, use mathematical induction.)

Now since $A^k=O$, we get $\lambda^k \mathbf{x}=0_n$, $n$-dimensional zero vector.

Since $\mathbf{x}$ is an eigenvector and hence nonzero by definition, we obtain that $\lambda^k=0$, and hence $\lambda=0$.

$(\Leftarrow)$

Now we assume that all the eigenvalues of the matrix $A$ are zero.

We prove that $A$ is nilpotent.

There exists an invertible $n\times n$ matrix $P$ such that $P^{-1} A P$ is an upper triangular matrix whose diagonal entries are eigenvalues of $A$.

(This is always possible. Study a triangularizable matrix or Jordan normal/canonical form.)

Hence we have

\[P^{-1} A P= \begin{bmatrix}

0 & * & \cdots & * \\

0 & 0 & \cdots & * \\

\vdots & \vdots & \ddots & \vdots \\

0 & 0 & \cdots & 0

\end{bmatrix}.

\]

Then we have $(P^{-1}AP)^n=O$. This implies that $P^{-1} A^n P=O$ and thus $A^n=POP^{-1}=O$.

Therefore the matrix $A$ is nilpotent.

#### Another proof of $(\Leftarrow)$ using Cayley-Hamilton theorem

Suppose that all the eigenvalues of the matrix $A$ are zero.

Then the characteristic polynomial of the matrix $A$ is

\[p(t)=\det(A-tI)=\pm t^n.\]

Hence by the Cayley-Hamilton theorem says that

\[p(A)=\pm A^n=O,\]
the zero matrix.

Thus, $A$ is nilpotent.

Note also that this method also proves the part (b).

## Proof of (b).

If $A^n=O$, then by definition the matrix $A$ is nilpotent.

On the other hand, suppose $A$ is nilpotent. Then by Part (a), the eigenvalues of $A$ are all zero. Then by the same argument of the proof of part (a) $(\Leftarrow)$, we have $A^n=O$.

## Comment.

Part (b) implies the following.

Suppose that you are given $n \times n$ matrix $B$.

You calculate the power $B^n$, and if it is not zero, then the power $B^k$ is never going to be the zero matrix $O$ no matter how large the number $k$ is.

## Related Question.

**Problem**. Prove that every diagonalizable nilpotent matrix is the zero matrix.

See the post ↴

Every Diagonalizable Nilpotent Matrix is the Zero Matrix

for a proof of this problem.

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## 3 Responses

[…] We want to show that all eigenvalues are zero. (Review Nilpotent matrix and eigenvalues of the matrix) […]

[…] $A$ is nilpotent, all the eigenvalues of $A$ are $0$. (See the post “Nilpotent matrix and eigenvalues of the matrix” for the proof.) Hence the diagonal entries of $D$ are zero, and we have $D=O$, the zero […]

[…] the post ↴ Nilpotent Matrix and Eigenvalues of the Matrix for a proof of this […]