  Spectral methods include a family of algorithms related to the eigenvectors
of certain data-generated matrices. In this work, we are interested in studying
the geometric landscape of the eigendecomposition problem in various spectral
methods. In particular, we first extend known results regarding the landscape
at critical points to larger regions near the critical points in a special case
of finding the leading eigenvector of a symmetric matrix. For a more general
eigendecomposition problem, inspired by recent findings on the connection
between the landscapes of empirical risk and population risk, we then build a
novel connection between the landscape of an eigendecomposition problem that
uses random measurements and the one that uses the true data matrix. We also
apply our theory to a variety of low-rank matrix optimization problems and
conduct a series of simulations to illustrate our theoretical findings.
