  In the high-energy physics literature one finds statements such as ``matrix
algebras converge to the sphere''. Earlier I provided a general precise setting
for understanding such statements, in which the matrix algebras are viewed as
quantum metric spaces, and convergence is with respect to a quantum
Gromov-Hausdorff-type distance.
  But physicists want even more to treat structures on spheres (and other
spaces), such as vector bundles, Yang-Mills functionals, Dirac operators, etc.,
and they want to approximate these by corresponding structures on matrix
algebras. In the present paper we provide a somewhat unified construction of
Dirac operators on coadjoint orbits and on the matrix algebras that converge to
them. This enables us to prove our main theorem, whose content is that, for the
quantum metric-space structures determined by the Dirac operators that we
construct, the matrix algebras do indeed converge to the coadjoint orbits, for
a quite strong version of quantum Gromov-Hausdorff distance.
