  The evolution and propagation of a partially coherent matter wave (PCMW) is
investigated theoretically by the correlation function method. The ABCD matrix
formalism previously used for a fully coherent matter wave is extended to the
PCMW domain. A new ABCD law is derived, using a tensor method to describe the
evolution of a PCMW. An analytical solution of the first-order correlation
function is obtained that makes the propagation and evolution of a PCMW very
simple and clear. As an example, the propagation of a PCMW in a gravitational
field is calculated numerically.
