  This paper aims to address two fundamental challenges arising in eigenvector
estimation and inference for a low-rank matrix from noisy observations: (1) how
to estimate an unknown eigenvector when the eigen-gap (i.e. the spacing between
the associated eigenvalue and the rest of the spectrum) is particularly small;
(2) how to perform estimation and inference on linear functionals of an
eigenvector -- a sort of "fine-grained" statistical reasoning that goes far
beyond the usual $\ell_2$ analysis. We investigate how to address these
challenges in a setting where the unknown $n\times n$ matrix is symmetric and
the additive noise matrix contains independent (and non-symmetric) entries.
Based on eigen-decomposition of the asymmetric data matrix, we propose
estimation and uncertainty quantification procedures for an unknown
eigenvector, which further allow us to reason about linear functionals of an
unknown eigenvector. The proposed procedures and the accompanying theory enjoy
several important features: (1) distribution-free (i.e. prior knowledge about
the noise distributions is not needed); (2) adaptive to heteroscedastic noise;
(3) minimax optimal under Gaussian noise. Along the way, we establish optimal
procedures to construct confidence intervals for the unknown eigenvalues. All
this is guaranteed even in the presence of a small eigen-gap (up to
$O(\sqrt{n/\mathrm{poly}\log (n)})$ times smaller than the requirement in prior
theory), which goes significantly beyond what generic matrix perturbation
theory has to offer.
