Eigenvector sensitivity when tracking modes with repeated eigenvalues

While eigenvalue optimization problems have been widely studied during the last three decades, particularly for structural and topology optimization problems, there are very few examples of problems involving mode shapes either in the cost or the constraints. In this paper, we propose a general framework for computing eigenvector sensitivity whenever tracking specific mode shapes selected beforehand. Of course, the approach is valid for either non-repeated or repeated eigenvalues, but here the emphasis is placed on the multiple eigenvalues case. Both mathematical validity of the algorithm and numerical corroboration through several examples are included.