  Eigenvector perturbation analysis plays a vital role in various statistical
data science applications. A large body of prior works, however, focused on
establishing $\ell_{2}$ eigenvector perturbation bounds, which are often highly
inadequate in addressing tasks that rely on fine-grained behavior of an
eigenvector. This paper makes progress on this by studying the perturbation of
linear functions of an unknown eigenvector. Focusing on two fundamental
problems -- matrix denoising and principal component analysis -- in the
presence of Gaussian noise, we develop a suite of statistical theory that
characterizes the perturbation of arbitrary linear functions of an unknown
eigenvector. In order to mitigate a non-negligible bias issue inherent to the
natural "plug-in" estimator, we develop de-biased estimators that (1) achieve
minimax lower bounds for a family of scenarios (modulo some logarithmic
factor), and (2) can be computed in a data-driven manner without sample
splitting. Noteworthily, the proposed estimators are nearly minimax optimal
even when the associated eigen-gap is substantially smaller than what is
required in prior theory.
