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.

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

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

CB

4. Sorry for posting under calculus

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

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

Thanks

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

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

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

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

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 $\displaystyle \left(\begin{array}{cc}0&0\\0&0\end{array}\right)$ 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 $\displaystyle x$ of $\displaystyle A$ is a non-zero vector such that:

$\displaystyle Ax= \lambda x$

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