 Solution

Choose real eigenvectors $u,v$ with distinct eigenvalues $\lambda ,\mu$. Symmetry givesTherefore $(\lambda -\mu )u^{T}v=0$, and $\boxed{u\cdot v=0}$. For complex choices, replace transpose by conjugate transpose; the reality proved in part (i) gives the same conclusion for the Hermitian inner product. Thus different eigenspaces are orthogonal.