  Correlation matrices are an essential tool for investigating the dependency
structures of random vectors or comparing them. We introduce an approach for
testing a variety of null hypotheses that can be formulated based upon the
correlation matrix. Examples cover MANOVA-type hypothesis of equal correlation
matrices as well as testing for special correlation structures such as, e.g.,
sphericity. Apart from existing fourth moments, our approach requires no other
assumptions, allowing applications in various settings. To improve the small
sample performance, a bootstrap technique is proposed and theoretically
justified. Based on this, we also present a procedure to simultaneously test
the hypotheses of equal correlation and equal covariance matrices. The
performance of all new test statistics is compared with existing procedures
through extensive simulations.
