The transformations in Eqs. (3.1) and (3.2) fully defines the NTOs in TTDFT, but an equivalent and illustrative definition is possible. Keeping to the TDA case for simplicity, we considder x to be a rectangular matrix of dimension n occ × n virt . Hole and particle NTOs are defined by separate unitary transformations of the occupied and virtual MOs (U o and U v , respectively), and an equivalent definition of these two transformations involves a singular value decomposition (SVD) of x: