Real symmetric spectral orthogonality
= Real symmetric spectral orthogonality
{title2=$Au=\lambda u,\quad Av=\mu v,\quad\lambda\ne\mu\ \Longrightarrow\ u^*v=0$}
A real <symmetric matrix> is a <Hermitian matrix>, so $u^*Au$ is real for every complex vector $u$. If $u$ is an <eigenvector>, its <eigenvalue> is $(u^*Au)/(u^*u)$ and is real. Then $u^*Av=(Au)^*v$ gives $(\mu-\lambda)u^*v=0$. Reality is used in changing conjugate transpose to transpose; complex symmetry alone does not imply real eigenvalues.