Why should eigenvectors be orthonormal if we start in an orthonormal basis?

Question

As part of the development of degenerate perturbation theory, I think the author uses the following fact tacitly. I'll assume we're in a finite-dimensional space for simplicity. Suppose I have an orthonormal basis $E=\{e_i\}$ and some Hermitian operator $H_1$ which is not diagonalized by $E$. When I do diagonalize $H_1$ in this basis, am I guaranteed that the basis so obtained will be orthonormal (given that I started in an orthonormal basis)? Am I guaranteed at least that they will be orthogonal (if I have degenerate eigenvalues, this is a concern).