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# Eigenvector dynamics: general theory and some applications

## Author Info

• Romain Allez
• Jean-Philippe Bouchaud
Registered author(s):

## Abstract

We propose a general framework to study the stability of the subspace spanned by $P$ consecutive eigenvectors of a generic symmetric matrix ${\bf H}_0$, when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (${\bf H}_0$ is then the Hamiltonian) and financial risk control (in which case ${\bf H}_0$ is the assets return covariance matrix). We argue that the problem can be formulated in terms of the singular values of an overlap matrix, that allows one to define a "fidelity" distance. We specialize our results for the case of a Gaussian Orthogonal ${\bf H}_0$, for which the full spectrum of singular values can be explicitly computed. We also consider the case when ${\bf H}_0$ is a covariance matrix and illustrate the usefulness of our results using financial data. The special case where the top eigenvalue is much larger than all the other ones can be investigated in full detail. In particular, the dynamics of the angle made by the top eigenvector and its true direction defines an interesting new class of random processes.

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File URL: http://arxiv.org/pdf/1203.6228

## Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1203.6228.

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 Length: Date of creation: Mar 2012 Date of revision: Jul 2012 Publication status: Published in Phys. Rev. E 86, 046202 (2012) Handle: RePEc:arx:papers:1203.6228 Contact details of provider: Web page: http://arxiv.org/

## References

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1. Emeric Balogh & Ingve Simonsen & Balint Zs. Nagy & Zoltan Neda, 2010. "Persistent collective trend in stock markets," Papers 1005.0378, arXiv.org.
2. BAUWENS, Luc & LAURENT, Sébastien & ROMBOUTS, Jeroen, 2003. "Multivariate GARCH models: a survey," CORE Discussion Papers 2003031, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
3. Baik, Jinho & Silverstein, Jack W., 2006. "Eigenvalues of large sample covariance matrices of spiked population models," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1382-1408, July.
4. J.-P. Bouchaud & L. Laloux & M. A. Miceli & M. Potters, 2007. "Large dimension forecasting models and random singular value spectra," The European Physical Journal B - Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 55(2), pages 201-207, 01.
5. M. Potters & J. P. Bouchaud & L. Laloux, 2005. "Financial Applications of Random Matrix Theory: Old Laces and New Pieces," Papers physics/0507111, arXiv.org.
6. Jean-Philippe Bouchaud & Laurent Laloux & M. Augusta Miceli & Marc Potters, 2005. "Large dimension forecasting models and random singular value spectra," Science & Finance (CFM) working paper archive 500066, Science & Finance, Capital Fund Management.
7. Gilles Zumbach, 2009. "The empirical properties of large covariance matrices," Papers 0903.1525, arXiv.org.
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