  In this paper, we investigate the asymptotic behaviors of the extreme
eigenvectors in a general spiked covariance matrix, where the dimension and
sample size increase proportionally. We eliminate the restrictive assumption of
the block diagonal structure in the population covariance matrix. Moreover,
there is no requirement for the spiked eigenvalues and the 4th moment to be
bounded. Specifically, we apply random matrix theory to derive the convergence
and limiting distributions of certain projections of the extreme eigenvectors
in a large sample covariance matrix within a generalized spiked population
model. Furthermore, our techniques are robust and effective, even when spiked
eigenvalues differ significantly in magnitude from nonspiked ones. Finally, we
propose a powerful statistic for hypothesis testing for the eigenspaces of
covariance matrices.
