  Correlation matrices are standardized covariance matrices. They form an
affine space of symmetric matrices defined by setting the diagonal entries to
one. We study the geometry of maximum likelihood estimation for this model and
linear submodels that encode additional symmetries. We also consider the
problem of minimizing two closely related functions of the covariance matrix:
the Stein's loss and the symmetrized Stein's loss. Unlike the Gaussian
log-likelihood these two functions are convex and hence admit a unique positive
definite optimum. Some of our results hold for general affine covariance
models.
