Expressions for the second-order energy variations in the density matrix functional theory (DMFT) are derived, resulting in a formalism for time-independent response properties (including absolute electronegativity and hardness) and stability conditions. A quadratically convergent scheme for a direct determination of natural spinorbitals and their occupancy numbers is developed and tested with the Goedecker-Umrigar and the exact two-electron functionals. The derivatives of the electronic energy with respect to the number of electrons are found to be very sensitive to the DMFT description of the exchange-correlation energy, providing a sensitive measure of accuracy that can be readily employed in testing and development of approximate functionals.