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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?

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There is a rich body of knowledge about spectral decomposition of the covariance matrix. Gatheral and Cucuringu have very readable lectures on this topic. I also found a larger and more formal review of "Cleaning large Correlation Matrices: tools from Random Matrix Theory" by Bun, Bouchaud, Potters.

Among basic results is the Marčenko-Pastur bound - eigenvalues that fall within it are often considered noise and cleansed. Spiked covariance model also serves as a basic model for how PCA behaves in the presence of estimation noise; a paper "Minimum Variance Portfolio Optimization in the Spiked Covariance Model" by Yang, Couillet, McKay should be of interest.

My favourite technique is Ledoit and Wolf's nonlinear shrinkage, which expands on their earlier linear approach from 2003 by working in the eigenvalue domain. The simplest exposition can be found here, particularly in section 4.7, but there were several earlier papers developing the theory.

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