 Equal-coupling three-state perturbation

A real perturbation with zero diagonal and every off-diagonal element $t$ has one eigenvalue $2t$ on the symmetric vector and a twofold eigenvalue $-t$ on its orthogonal complement. It is $t(J_3-I)$, where $J_3$ is the all-ones matrix. Diagonal entries alone do not give first-order energies in a degenerate subspace.