# Orthogonal Eigenvector Matrices which are Symmetric

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 $V=V^T$?

• Are you looking for $A=VBV^T$ where $V^{-1}=V^T=V$ and V is symmetric or B is symmetric? – Sar Jul 27 '18 at 22:17