  We obtain general, exact formulas for the overlaps between the eigenvectors
of large correlated random matrices, with additive or multiplicative noise.
These results have potential applications in many different contexts, from
quantum thermalisation to high dimensional statistics. 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 correlation matrix, even when the latter is very
close to the identity. We illustrate our results on the example of stock
returns correlations, that clearly reveal a non trivial structure for the bulk
eigenvalues. We also apply our results to the problem of matrix denoising in
high dimension.
