Local semicircle law with imprimitive variance matrix

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.