# Real matrix and eigenvalues

A is real matrix from order nXn. We know that A gives : $<Av,v>=0$ for v vector in $R^n$.

So what must exist?

1. Every eigenvalue of A is real.
2. A is not Invertible
3. A is Hermitian
4. A is not Hermitian
5. all answers are incorrect

I think 2 is the right answer ( can be more than 1 answer that are true) but im not sure

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## 1 Answer

$\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ is a counter-example to 1 and 3. The zero matrix is a counter example to 4.

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This is also a counter-example to 2. So 5 is the correct answer. –  1015 Feb 4 '13 at 0:16
Sure, the identity matrix is a counter example to 4. –  Damien L Feb 4 '13 at 7:53
No, the identity matrix does not fulfill the condition $<Av,v>=0$ for all $v$. –  1015 Feb 4 '13 at 13:09
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