  Eigenvector perturbation analysis plays a vital role in various 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 {\em substantially smaller} than what is required in prior
statistical theory.
