Testing Equality of Principal Eigenvectors of Spiked Covariance Matrices

The problem of detecting differences between the principal eigenvectors of the covariance matrices of two independent multivariate time series is studied. A statistical test is proposed which, under certain separability conditions, consistently converges to a quantity that measures the distance between the two principal subspaces of the true covariance matrices. By assuming a spiked covariance matrix model, a central limit theorem is established that characterizes the asymptotic behavior of the binary hypothesis test in high-dimensional settings. This result enables the specification of the asymptotic false alarm probability of the test and facilitates the analysis of its power under the alternative hypothesis. The study is conducted within a random matrix theory (RMT) framework, where both the sample size and the observation dimension grow proportionally. The findings provide deeper insights into how high-dimensional effects influence the reliability and performance of the test.