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.

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