  The correlated Wishart model provides a standard tool for the analysis of
correlations in a rich variety of systems. Although much is known for complex
correlation matrices, the empirically much more important real case still poses
substantial challenges. We put forward a new approach, which maps arbitrary
statistical quantities, depending on invariants only, to invariant Hermitian
matrix models. For completeness we also include the quaternion case and deal
with all three cases in a unified way. As an important application, we study
the statistics of the largest eigenvalue and its limiting distributions in the
correlated Wishart model, because they help to estimate the behavior of large
complex systems. We show that even for fully correlated Wishart ensembles, the
Tracy-Widom distribution can be the limiting distribution of the largest as
well as the smallest eigenvalue, provided that a certain scaling of the
empirical eigenvalues holds.
