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On differentiating eigenvalues and eigenvectors

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  • Magnus, J.R.

    (Tilburg University, School of Economics and Management)

Abstract

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.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • 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.
    3. Antonio Diez de Los Rios, 2015. "A New Linear Estimator for Gaussian Dynamic Term Structure Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 33(2), pages 282-295, April.
    4. 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.
    5. Stéphane Bonhomme & Koen Jochmans & Jean-Marc Robin, 2013. "Nonparametric estimation of finite mixtures," Working Papers hal-00972868, HAL.
    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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