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On differentiating eigenvalues and eigenvectors


  • Magnus, J.R.

    (Tilburg University, School of Economics and Management)


Let X0 be a square matrix (complex or otherwise) and u0 a (normalized) eigenvector associated with an eigenvalue λo of X0, so that the triple (X0, u0, λ0) satisfies the equations Xu = λu, . We investigate the conditions under which unique differentiable functions λ(X) and u(X) exist in a neighborhood of X0 satisfying λ(X0) = λO, u(X0) = u0, Xu = λu, and . 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.
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  • Magnus, J.R., 1985. "On differentiating eigenvalues and eigenvectors," Other publications TiSEM f410e3a5-ba9b-4787-b8cc-4, Tilburg University, School of Economics and Management.
  • Handle: RePEc:tiu:tiutis:f410e3a5-ba9b-4787-b8cc-44c6d5d8cdd9

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    Cited by:

    1. Eduardo Abi Jaber & Bruno Bouchard & Camille Illand & Eduardo Jaber, 2018. "Stochastic invariance of closed sets with non-Lipschitz coefficients," Working Papers hal-01349639, HAL.
    2. Abry, Patrice & Didier, Gustavo, 2018. "Wavelet eigenvalue regression for n-variate operator fractional Brownian motion," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 75-104.
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    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. 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.
    8. 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.
    9. David Baqaee & Emmanuel Farhi & Michael J. Mina & James H. Stock, 2020. "Reopening Scenarios," NBER Working Papers 27244, National Bureau of Economic Research, Inc.
    10. Bystrov, Victor & di Salvatore, Antonietta, 2012. "Martingale approximation for common factor representation," MPRA Paper 37669, University Library of Munich, Germany.
    11. 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.
    12. 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.
    13. Maggi, Bernardo, 2017. "A technology-based countries-interaction dynamic model for the study of European growth and stability: Were there the conditions for convergence?," Technological Forecasting and Social Change, Elsevier, vol. 125(C), pages 275-288.
    14. 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.
    15. Gregório, R.M. & Oliveira, P.R. & Alves, C.D.S., 2019. "A two-phase-like proximal point algorithm in domains of positivity," Applied Mathematics and Computation, Elsevier, vol. 343(C), pages 67-89.
    16. Chen, Liang, 2015. "Estimating the common break date in large factor models," Economics Letters, Elsevier, vol. 131(C), pages 70-74.
    17. Eduardo Abi Jaber & Bruno Bouchard & Camille Illand & Eduardo Jaber, 2018. "Stochastic invariance of closed sets with non-Lipschitz coefficients," Post-Print hal-01349639, HAL.
    18. Bystrov, Victor & di Salvatore, Antonietta, 2013. "Martingale approximation of eigenvalues for common factor representation," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 233-237.
    19. Gunnar Nordén, 2004. "The Correspondence Principle and Structural Stability in Non-Maximum," Levine's Bibliography 122247000000000422, UCLA Department of Economics.
    20. 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.
    21. Koebel, Bertrand M. & Falk, Martin & Laisney, François, 2000. "Imposing and testing curvature conditions on a Box-Cox function," ZEW Discussion Papers 00-70, ZEW - Leibniz Centre for European Economic Research.
    22. 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.
    23. Abi Jaber, Eduardo & Bouchard, Bruno & Illand, Camille, 2019. "Stochastic invariance of closed sets with non-Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1726-1748.
    24. 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.

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