I had an exercise that gave me $f:V \to V$ and a scalar product $\Phi$.
$f$ was symmetric in $\Phi$ [that is $\forall v,w\in V$, $\Phi(f(v),w)=\Phi(v,f(w))]$
The spectral theorem tells me that I can find an orthonormal base for $V$ of eigenvectors. Another previous theorem tells simply that different eigenspaces are mutually orthogonal.
But, orthogonal for which scalar product??
The eigenvectors I found in that exercise were orthogonal for both the canonical product and $\Phi$. Was it luck?