I am given the following task:
"For which $a$, $b \in \mathbb{R}$ there exists a scalar product, such that $A = \left( \begin{matrix} 0 & 0 & a b \\ 1 & 0 & a \\ 0 & 1 & b\end{matrix}\right) $ is a self-adjoint matrix".
A hint says, the task breaks down in finding $a$, $b$ such that $A$ is diagonalizable. But I can't figure out why.
As far as I got is, that Self-adjoint means that $\left< A v, w\right> = \left< v, A w\right> \forall v, w \in V$. So we are looking for $a,b$ such that there is a scalar product on $V$ such that this equality holds. I know that for every Bilinearform $f$ there is a Matrix $B$ such that $f(v,w) =
Thanks for any help