Density matrices for systems with a finite number of states are considered as elements in a vector space. A density matrix that is a convex combination of density matrices that are unitary transformations of some initial density matrix will be closer to, or possibly at the same distance from, the density matrix for the ensemble in which all states are equally occupied, compared with the initial density matrix. This distance is correlated, for few-state systems, with a function simply related to the entropy. This function increases (entropy decreases) with distance from the equal-occupancy density matrix. Other properties of the function are also established.