Eigenvector derivatives with repeated eigenvalues

In this paper an algorithm is derived for computing the derivatives of eigenvalues and eigenvectors for real symmetric matrices in the case of repeated eigenvalues, where the matrices are functions of real parameters such as mass density or moment of inertia. The algorithm is an extension of recent work by I. U. Ojalvo; the key step in this extended derivation is to differentiate the eigenvalue equation twice. The algorithm preserves the symmetry and band structure of the matrices, allowing efficient computer storage and solution techniques. Applications include sensitivity analysis and optimization of the normal modes of finite-element modeled structures, such as large space structures. A cantilever beam finite-element example is included.