Advertisement

Numerische Mathematik

, Volume 70, Issue 1, pp 45–72 | Cite as

Sensitivity analysis of all eigenvalues of a symmetric matrix

  • J.-B. Hiriart-Urruty
  • D. Ye

Summary.

Given \(A(x)= [a_{ij} (x)]\) a \(n\)-by-\(n\) symmetric matrix depending (smoothly) on a parameter $x$, we study the first order sensitivity of all the eigenvalues\(\lambda_m (x)\) of \(A(x)\), \(1\le m\le n\). Under some smoothness assumption like the \(a_{ij}\) be \(C^1\), we prove that the directional derivatives\( d\mapsto \lambda^\prime_m (x,d) = \lim_{t \to o^+} [\lambda_m (x + td) - \lambda_m (x)] / t \) do exist and give an explicit expression of them in terms of the data of the parametrized matrix. The key idea to circumvent the difficulties inherent to the study of each\(\lambda_m\) taken separately, is to consider the functions\(f_m (x)\) , \(1\le m\le n\), defined as the sums of the\(m\) largest eigenvalues of \(A(x)\). Based on Ky Fan's variational formulation of \(f_m\) and some chain rule from nonsmooth analysis, we derive an explicit formula for the generalized gradient of \(f_m\) and a computationally useful formula for the directional derivative of \(f_m\). Using these formulas and the relation \(\lambda_m= f_m - f_{m-1}\), we then derive the directional derivative of \(\lambda_m\). Some properties of this directional derivative as well as an illustrative example are presented.

Mathematics Subject Classification (1991): 65F15, 58C40, 15A18, 90C31, 49M45 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • J.-B. Hiriart-Urruty
    • 1
  • D. Ye
    • 1
  1. 1.Laboratoire d'Analyse Num\'erique, U.F.R. Math\'ematiques, Informatique, Gestion, Universit\'e Paul Sabatier, 118, route de Narbonne, F-31062 Toulouse Cedex, France Fax: (+33) 61.55.61.83; e.mail: jbhu@cict.fr FR

Personalised recommendations