Eigensolutions sensitivity for nonsymmetric matrices with repeated eigenvalues

A perturbation algorithm is developed for evaluating eigenvalue and eigenvector directional derivatives of nonsymmetric defective matrices. Some properties of these matrices are recalled; in particular, chains of generalized right and left eigenvectors and their orthogonal properties are defined. Small perturbations of the matrix are then considered. An asymptotic expansion of the eigensolutions of the perturbed problem is obtained in terms of noninteger powers of the perturbation parameter. Marked sensitivity of the eigensolutions is highlighted. Particular attention is devoted to the eigenvectors of the perturbed system and to the strong coupling that occurs between the chains. An example is developed to illustrate the algorithm and compare perturbative and numerical solutions.