Minimal complete classes of invariant tests are presented for modifications of the problem of testing the independence of $Y$ and $X$, where $(Y, X) \equiv (Y, X_1, \cdots, X_p)$ is a multivariate normal random vector. One modification involves having extra independent observations on $Y$. Others involve extra variates $Z \equiv (Z_1, \cdots, Z_q)$ such that $(Y,X,Z)$ is multivariate normal. Among other results, locally most powerful invariant tests and asymptotically most powerful invariant tests are found; it is shown that for some problems the likelihood ratio test is admissible among invariant tests only for levels less than a specified one; and it is shown that for the problem of testing the independence of $Y$ and $X$ when it is known that $Y$ and $Z$ are independent, the test based on the sample multiple correlation coefficient of $Y$ and $(X, Z)$ is inadmissible.
John I. Marden. "Invariant Tests on Covariance Matrices." Ann. Statist. 9 (6) 1258 - 1266, November, 1981. https://doi.org/10.1214/aos/1176345642