A general density matrix functional theory is formulated in terms of a basis representation of the density matrix in average ((R) over right arrow = ((r) over right arrow (1) + (r) over right arrow (2))/2) and relative ((r) over right arrow = (r) over right arrow (2) - (r) over right arrow (1)) coordinates. This representation involves a parameter set whose dimension by construction grows strictly linearly with system size. Furthermore, the two-electron Coulomb and exchange contributions to the Hartree-Fock and Kohn-Sham energies factorize, and can be computed with reference only to two-index integrals. The problem of,li-representability is addressed and solutions are presented. Kinetic energy transpires to be the hardest term to compute accurately, and three different approaches are discussed. The subtle relationship between N-representability and kinetic energy is investigated.