Ten invariant multivariate testing problems involving the real, complex, or quaternion structure of covariance matrices are considered. In each problem the maximal invariant statistic and its distribution are described, as well as the maximum likelihood estimators and likelihood ratio test statistics. These results are obtained by means of a new, unified method based on invariance arguments.
"Distribution of Eigenvalues in Multivariate Statistical Analysis." Ann. Statist. 11 (2) 392 - 415, June, 1983. https://doi.org/10.1214/aos/1176346149