  In this paper, we study the asymptotic behavior of the extreme eigenvalues
and eigenvectors of the high dimensional spiked sample covariance matrices, in
the supercritical case when a reliable detection of spikes is possible.
Especially, we derive the joint distribution of the extreme eigenvalues and the
generalized components of the associated eigenvectors, i.e., the projections of
the eigenvectors onto arbitrary given direction, assuming that the dimension
and sample size are comparably large. In general, the joint distribution is
given in terms of linear combinations of finitely many Gaussian and Chi-square
variables, with parameters depending on the projection direction and the
spikes. Our assumption on the spikes is fully general. First, the strengths of
spikes are only required to be slightly above the critical threshold and no
upper bound on the strengths is needed. Second, multiple spikes, i.e., spikes
with the same strength, are allowed. Third, no structural assumption is imposed
on the spikes. Thanks to the general setting, we can then apply the results to
various high dimensional statistical hypothesis testing problems involving both
the eigenvalues and eigenvectors. Specifically, we propose accurate and
powerful statistics to conduct hypothesis testing on the principal components.
These statistics are data-dependent and adaptive to the underlying true spikes.
Numerical simulations also confirm the accuracy and powerfulness of our
proposed statistics and illustrate significantly better performance compared to
the existing methods in the literature. Especially, our methods are accurate
and powerful even when either the spikes are small or the dimension is large.
