Before revision:  We  apply our results to the case of empirical correlation matrices, that allow us to estimate reliably the width of the spectrum of the 'true ' underlying correlation matrix, even when the latter is very close to the identity matrix .
After revision:  We find that the overlaps only depend on measurable quantities, and do not require the knowledge of the underlying "true" (noiseless) matrices. We apply our results to the case of empirical correlation matrices, that allow us to estimate reliably the width of the spectrum of the 'true ' underlying correlation matrix, even when the latter is very close to the identity matrix .