Suppose that I've used the spectral theorem of linear algebra to completely decompose the covariance matrix. I now know the largest and smallest eigenvalue, which corresponds to the largest and smallest risk.
Can I use the information somehow to denoise the covariance matrix and make it more robust? The smallest eigenvalues appear to have the largest weight in the information matrix, mixing the returns. Therefore I suspect, that if I add an uncorrelated asset, it will shift the whole portfolio. Can you explain, what is usually done in order to exploit the information about the eigenvalues/eigenvectors?