# Thread: I need a counter example

1. ## I need a counter example

Hello!

Any kind soul can provide me with a counter example for this:

False statement: Every square matrix has an eigenvalue.

Thanks

2. Do you mean have no REAL eigenvalues? Or none at all?

Here is an example of a matrix with no real eigenvalues.

$\left(\begin{array}{cc}0&1\\-1&0\end{array}\right)$

3. Originally Posted by noob mathematician
Hello!

Any kind soul can provide me with a counter example for this:

False statement: Every square matrix has an eigenvalue.

Thanks
${\bf{0}}_{2 \times 2}$

CB

4. Sorry for posting under calculus

Hm.. I think that $\left(\begin{array}{cc}0&0\\0&0\end{array}\right)$ has an eigenvalue of 0? Am i wrong?

Ya I guess the question is asking about real and non-real eigenvalues.

Thanks

5. $Det|M-\lambda I|=0$

$\begin{vmatrix}
0-\lambda&0\\
0&0-\lambda
\end{vmatrix}=0$

$\lambda^2=0 \Rightarrow \ \lambda=0$

But when $\lambda=0$, we get an eigenvalue of $\begin{pmatrix}
0\\
0 \end{pmatrix}$
.

The problem is that this 0 vector has no direction which contradicts the definition of an eigenvalue.

(ie. eigenvectors have the same direction when they undergo a transformation. They're only scaled by the eigenvalue. A 0 vector does not have a direction so cannot be an eigenvector).

6. Originally Posted by noob mathematician
Sorry for posting under calculus

Hm.. I think that $\left(\begin{array}{cc}0&0\\0&0\end{array}\right)$ has an eigenvalue of 0? Am i wrong?

Ya I guess the question is asking about real and non-real eigenvalues.

Thanks
An eigen vector $x$ of $A$ is a non-zero vector such that:

$Ax= \lambda x$

for some scalar $\lambda$ (there is usually no requirement that $\lambda$ be non-zero here). So every non-zero vector is an eigen vector of the zero matrix, and $0$ is the corresponding eigenvalue.

So yes, I was wrong the zero matrix does have an eigen value (I had what had to be non-zero confused).

CB