Are the Lagrange multipliers used in Kohn-Sham DFT/HF-based quantum molecular dynamics real valued?

Question

I have seen it written in several places that the matrix of Lagrange multipliers used to impose orthonormality in quantum molecular dynamics schemes based on Kohn-Sham DFT or Hartree-Fock for instance, $\Lambda_{ij}$, is symmetric ($\Lambda_{ij}=\Lambda_{ji}$).I would have thought that in order to have a real Lagrangian describing the dynamics that this matrix would have to be complex and Hermitian in general. Is it in fact real and symmetric?