Orthogonal Eigenvector Matrices which are Symmetric

Question

What (extra) conditions must be satisfied by a real symmetric matrix, $A$, with distinct eigenvalues, so that its orthogonal matrix of eigenvectors V, can be arranged to also be symmetric? I.e. if $A^T=A=V \Lambda V^T$, where $V^{-1}=V^T$, and $\Lambda$ is a diagonal matrix of (distinct) eigenvalues, what additional condition(s) on $A$ are required so that $V=V^T$?

Answer 1

I don't think there's any special characterization of such matrices. As a rule of thumb, the "natural" properties of a matrix are those invariant under change of basis, i.e., those that can be inferred from the eigenvalues alone.