  Estimating eigenvectors and low-dimensional subspaces is of central
importance for numerous problems in statistics, computer science, and applied
mathematics. This paper characterizes the behavior of perturbed eigenvectors
for a range of signal-plus-noise matrix models encountered in both statistical
and random matrix theoretic settings. We prove both first-order approximation
results (i.e. sharp deviations) as well as second-order distributional limit
theory (i.e. fluctuations). The concise methodology considered in this paper
synthesizes tools rooted in two core concepts, namely (i) deterministic
decompositions of matrix perturbations and (ii) probabilistic matrix
concentration phenomena. We illustrate our theoretical results via simulation
examples involving stochastic block model random graphs.
