  This is the second part of a study of the limiting distributions of the top
eigenvalues of a Hermitian matrix model with spiked external source under a
general external potential. The case when the external source is of rank one
was analyzed in an earlier paper. In the present paper we extend the analysis
to the higher rank case. If all the eigenvalues of the external source are less
than a critical value, the largest eigenvalue converges to the right end-point
of the support of the equilibrium measure as in the case when there is no
external source. On the other hand, if an external source eigenvalue is larger
than the critical value, then an eigenvalue is pulled off from the support of
the equilibrium measure. This transition is continuous, and is universal,
including the fluctuation laws, for convex potentials. For non-convex
potentials, two types of discontinuous transitions are possible to occur
generically. We evaluate the limiting distributions in each case for general
potentials including those whose equilibrium measure have multiple intervals
for their support.
