Three new test criteria are proposed for overall tests of hypotheses in multivariate analysis. They are based on the characteristic roots of certain matrices obtained from the product moment matrices of samples drawn from multivariate normal populations. The approximate distributions of the statistics involved in the tests are found as Type I or Type II Beta distributions.
K. C. S. Pillai. "Some New Test Criteria in Multivariate Analysis." Ann. Math. Statist. 26 (1) 117 - 121, March, 1955. https://doi.org/10.1214/aoms/1177728599