  We give a short and completely elementary method to find the full spectrum of
the exclusion process and a nicely limited superset of the spectrum of the
interchange process (a.k.a.\ random transpositions) on the complete graph. In
the case of the exclusion process, this gives a simple closed form expression
for all the eigenvalues and their multiplicities. This result is then used to
give an exact expression for the distance in $ L^2 $ from stationarity at any
time and upper and lower bounds on the convergence rate for the exclusion
process. In the case of the interchange process, upper and lower bounds are
similarly found. Our results strengthen or reprove all known results of the
mixing time for the two processes in a very simple way.
