Calculating Derivatives of Repeated and Nonrepeated Eigenvalues Without Explicit Use of Eigenvectors

Abstract

The analysis of inverse problems in parametric model updating often require the sensitivities of eigenvalues. The calculation of these sensitivities is mathematically related to the derivatives of the eigenvalues with respect to the model parameters. A common method to calculate these derivatives is the Nelson method, which requires the eigenvectors. The method introduced in this paper is derived from the characteristic equation of the underlying general eigenvalue problem and allows the derivatives of eigenvalues with respect to the model parameters to be calculated without explicit use of the eigenvectors. The method is extended for the case of repeated eigenvalues, which leads to restrictions on the parameterization. For repeated eigenvalues of multiplicity two, these restrictions are formulated expicitly. Applications and limitations of the method are demonstrated by examples.