The normalization and phases within the right and left sets are arbitrary, but the biorthogonality condition, Eq. ( ), fixes the relative norms and phases: i.e., if one changes the sign and norm of R K , the sign and norm of L K changes accordingly. In many implementations, the diagonalization is carried out independently for the right and left vectors and the biorthogonality condition, which synchronizes the phases, is applied a posteriori (in the case of degenerate eigenstates, one needs to rotate the degenerate eigenstates such that the overlap matrix between the left and right degenerate manifolds becomes a unit matrix).