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Eigenvector dynamics: general theory and some applications

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  • Romain Allez
  • Jean-Philippe Bouchaud
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    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
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    Bibliographic Info

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

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    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

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    1. 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.
    2. Gilles Zumbach, 2009. "The empirical properties of large covariance matrices," Papers 0903.1525, arXiv.org.
    3. Sébastien Laurent & Luc Bauwens & Jeroen V. K. Rombouts, 2006. "Multivariate GARCH models: a survey," Journal of Applied Econometrics, John Wiley & Sons, Ltd., John Wiley & Sons, Ltd., vol. 21(1), pages 79-109.
    4. Marc Potters & Jean-Philippe Bouchaud & Laurent Laloux, 2005. "Financial Applications of Random Matrix Theory: Old Laces and New Pieces," Science & Finance (CFM) working paper archive 500058, Science & Finance, Capital Fund Management.
    5. 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, Springer, vol. 55(2), pages 201-207, 01.
    6. Baik, Jinho & Silverstein, Jack W., 2006. "Eigenvalues of large sample covariance matrices of spiked population models," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 97(6), pages 1382-1408, July.
    7. Emeric Balogh & Ingve Simonsen & Balint Zs. Nagy & Zoltan Neda, 2010. "Persistent collective trend in stock markets," Papers 1005.0378, arXiv.org.
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