# 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
$\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ is a counter-example to 1 and 3. The zero matrix is a counter example to 4.
No, the identity matrix does not fulfill the condition $<Av,v>=0$ for all $v$. –  1015 Feb 4 '13 at 13:09