  We extend the proof of the local semicircle law for generalized Wigner
matrices given in [4] to the case when the matrix of variances has an
eigenvalue $ -1 $. In particular, this result provides a short proof of the
optimal local Marchenko-Pastur law at the hard edge (i.e. around zero) for
sample covariance matrices $ \boldsymbol{\mathrm{X}}^\ast
\boldsymbol{\mathrm{X}} $, where the variances of the entries of $
\boldsymbol{\mathrm{X}} $ may vary.
