Suppose $A$ is a symmetric matrix with complex entries. Let $c$ and $k$ be two distinct eigenvalues and $X$ and $Y$ be two eigenvectors belonging to $c$ and $k$. $AX=cX => (AX)^T=(cX)^T => X^TA^T=cX^T => X^TA=cX^T => X^TAY=cX^TY => X^TkY=cX^TY => (k-c)X^TY=0 => X^TY=0$,since $c≠k$.
My teacher gave the result for real symmetric matrices. I don't see why it won't be true for complex matrices. Isn't there a notion of orthogonality in $\Bbb C^n$?