The sample canonical correlations between two sets of variates are given a representation as the roots of a determinantal equation involving independent matrix variates having simple standardized distributions. This result is applied to obtain asymptotic formulas for the non-null distributions of three criteria for testing the hypothesis of independence of two sets of variates.
Yoong-Sin Lee. "Distribution of the Canonical Correlations and Asymptotic Expansions for Distributions of Certain Independence Test Statistics." Ann. Math. Statist. 42 (2) 526 - 537, April, 1971. https://doi.org/10.1214/aoms/1177693403