Here, Λ is the n occ × n occ matrix of singular values λ i , but is the same matrix that appears (as Λ 2 ) in eqs. (3.1) and (3.2). According to Eq. (3.4), the matrices U o and U v contain the left and right singular vectors of x, respectively, but they are identical to the eponymous transformations defined as eigenvectors of ∆P hole and ∆P elec . As compared to the definitions in Section 3.1, the construction in Eq. (3.4) demonstrates more clearly why no more than n occ of the singular values are non-zero, and that eigenvalues of ∆P hole and ∆P elec occur in pairs. From yet another point of view, Eq. (3.4) is a special case of a corresponding orbitals transformation, which selects a subset of virtual orbitals in one-to-one correspondence with the occupied orbitals, diagonalizing the TDM in the process.