  How many samples are sufficient to guarantee that the eigenvectors and
eigenvalues of the sample covariance matrix are close to those of the actual
covariance matrix? For a wide family of distributions, including distributions
with finite second moment and distributions supported in a centered Euclidean
ball, we prove that the inner product between eigenvectors of the sample and
actual covariance matrices decreases proportionally to the respective
eigenvalue distance. Our findings imply non-asymptotic concentration bounds for
eigenvectors, eigenspaces, and eigenvalues. They also provide conditions for
distinguishing principal components based on a constant number of samples.
