  This paper is concerned with the interplay between statistical asymmetry and
spectral methods. Suppose we are interested in estimating a rank-1 and
symmetric matrix $\mathbf{M}^{\star}\in \mathbb{R}^{n\times n}$, yet only a
randomly perturbed version $\mathbf{M}$ is observed. The noise matrix
$\mathbf{M}-\mathbf{M}^{\star}$ is composed of zero-mean independent (but not
necessarily homoscedastic) entries and is, therefore, not symmetric in general.
This might arise, for example, when we have two independent samples for each
entry of $\mathbf{M}^{\star}$ and arrange them into an {\em asymmetric} data
matrix $\mathbf{M}$. The aim is to estimate the leading eigenvalue and
eigenvector of $\mathbf{M}^{\star}$. We demonstrate that the leading eigenvalue
of the data matrix $\mathbf{M}$ can be $O(\sqrt{n})$ times more accurate --- up
to some log factor --- than its (unadjusted) leading singular value in
eigenvalue estimation. Further, the perturbation of any linear form of the
leading eigenvector of $\mathbf{M}$ --- say, entrywise eigenvector perturbation
--- is provably well-controlled. This eigen-decomposition approach is fully
adaptive to heteroscedasticity of noise without the need of careful bias
correction or any prior knowledge about the noise variance. We also provide
partial theory for the more general rank-$r$ case. The takeaway message is
this: arranging the data samples in an asymmetric manner and performing
eigen-decomposition could sometimes be beneficial.
