  We derive analytic expressions for three flavor neutrino oscillations in the
presence of matter in the plane wave approximation using the Cayley-Hamilton
formalism. Especially, we calculate the time evolution operator in both flavor
and mass bases. Furthermore, we find the transition probabilities, matter mass
squared differences, and matter mixing angles all expressed in terms of the
vacuum mass squared differences, the vacuum mixing angles, and the matter
density. The conditions for resonance in the presence of matter are also
studied in some examples.
