Real symmetric spectral orthogonality ($Au=\lambda u,Av=\mu v,\lambda \ne \mu ⟹u^{\ast }v=0$)

A real symmetric matrix is a Hermitian matrix, so $u^{\ast }Au$ is real for every complex vector $u$. If $u$ is an eigenvector, its eigenvalue is $(u^{\ast }Au)/(u^{\ast }u)$ and is real. Then $u^{\ast }Av=(Au{)}^{\ast }v$ gives $(\mu -\lambda )u^{\ast }v=0$. Reality is used in changing conjugate transpose to transpose; complex symmetry alone does not imply real eigenvalues.