How Close Are the Eigenvectors of the Sample and Actual Covariance Matrices?

How many samples are sufficient to guarantee that the eigenvectors 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 sub-gaussian 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 and the number of samples. Our findings imply \textit{non-asymptotic} concentration bounds for eigenvectors and eigenvalues and carry strong consequences for the non-asymptotic analysis of PCA and its applications. For instance, they provide conditions for separating components estimated from $O(1)$ samples and show that even few samples can be sufficient to perform dimensionality reduction, especially for low-rank covariances.