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On Differentiating Eigenvalues and Eigenvectors

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  • Magnus, Jan R.

Abstract

Let X 0 be a square matrix (complex or otherwise) and u 0 a (normalized) eigenvector associated with an eigenvalue λ o of X 0 , so that the triple ( X 0 , u 0 , λ 0 ) satisfies the equations Xu = λ u , null. We investigate the conditions under which unique differentiable functions λ( X ) and u ( X ) exist in a neighborhood of X 0 satisfying λ( X 0 ) = λ O , u ( X 0 ) = u 0 , X = λ u , and null. We obtain the first and second derivatives of λ( X ) and the first derivative of u ( X ). Two alternative expressions for the first derivative of λ( X ) are also presented.

Suggested Citation

  • Magnus, Jan R., 1985. "On Differentiating Eigenvalues and Eigenvectors," Econometric Theory, Cambridge University Press, vol. 1(02), pages 179-191, August.
  • Handle: RePEc:cup:etheor:v:1:y:1985:i:02:p:179-191_01
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    Cited by:

    1. Eduardo Abi Jaber & Bruno Bouchard & Camille Illand, 2016. "Stochastic invariance of closed sets with non-Lipschitz coefficients," Working Papers hal-01349639, HAL.
    2. Broda, S. & Paolella, M.S., 2009. "Evaluating the density of ratios of noncentral quadratic forms in normal variables," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1264-1270, February.
    3. Stéphane Bonhomme & Koen Jochmans & Jean-Marc Robin, 2013. "Nonparametric estimation of finite mixtures," Working Papers hal-00972868, HAL.
    4. Boneva, Lena & Linton, Oliver & Vogt, Michael, 2015. "A semiparametric model for heterogeneous panel data with fixed effects," Journal of Econometrics, Elsevier, vol. 188(2), pages 327-345.
    5. Baringhaus, Ludwig & Gaigall, Daniel, 2017. "Hotelling’s T2 tests in paired and independent survey samples: An efficiency comparison," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 177-198.
    6. Stéphane Bonhomme & Koen Jochmans & Jean-Marc Robin, 2014. "Nonparametric spectral-based estimation of latent structures," CeMMAP working papers CWP18/14, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. repec:eee:econom:v:202:y:2018:i:1:p:45-56 is not listed on IDEAS
    8. Cho, Jin Seo & Phillips, Peter C.B., 2018. "Pythagorean generalization of testing the equality of two symmetric positive definite matrices," Journal of Econometrics, Elsevier, vol. 202(1), pages 45-56.
    9. Chen, Liang, 2015. "Estimating the common break date in large factor models," Economics Letters, Elsevier, vol. 131(C), pages 70-74.
    10. Bystrov, Victor & di Salvatore, Antonietta, 2013. "Martingale approximation of eigenvalues for common factor representation," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 233-237.
    11. Eleonora Cavallaro & Bernardo Maggi, 2016. "State of confidence, overborrowing and the macroeconomic stabilization puzzle: a system dynamic approach," Working Papers 174, University of Rome La Sapienza, Department of Public Economics.
    12. Gunnar Nordén, 2004. "The Correspondence Principle and Structural Stability in Non-Maximum," Levine's Bibliography 122247000000000422, UCLA Department of Economics.
    13. Bystrov, Victor & di Salvatore, Antonietta, 2012. "Martingale approximation for common factor representation," MPRA Paper 37669, University Library of Munich, Germany.
    14. Stéphane Bonhomme & Koen Jochmans & Jean-Marc Robin, 2014. "Nonparametric estimation of finite measures," CeMMAP working papers CWP11/14, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    15. Koebel, Bertrand M. & Falk, Martin & Laisney, François, 2000. "Imposing and testing curvature conditions on a Box-Cox function," ZEW Discussion Papers 00-70, ZEW - Zentrum für Europäische Wirtschaftsforschung / Center for European Economic Research.
    16. Guohua Feng & Apostolos Serletis, 2009. "Efficiency and productivity of the US banking industry, 1998-2005: evidence from the Fourier cost function satisfying global regularity conditions," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 24(1), pages 105-138.
    17. Cavallaro, Eleonora & Maggi, Bernardo, 2016. "State of confidence, overborrowing and macroeconomic stabilization in out-of-equilibrium dynamics," Economic Modelling, Elsevier, vol. 59(C), pages 210-223.
    18. Nieuwenhuis, Herman J. & Schoonbeek, Lambert, 1997. "Stability and the structure of continuous-time economic models," Economic Modelling, Elsevier, vol. 14(3), pages 311-340, July.
    19. repec:eee:tefoso:v:125:y:2017:i:c:p:275-288 is not listed on IDEAS

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