Calculation of eigenvalue and eigenvector derivatives of a defective matrix

Based on Puiseux expansions of perturbation parameter for the solution of the perturbed problem, a modal expansion method for the eigensensitivity analysis of a defective matrix is developed, in which any of eigenvector derivatives is expressed as a linear combination of all the eigenvectors and principal vectors of the matrix. First, an eigenvalue problem related to the differentiable eigenvectors and the first-order eigenvalue derivatives associated with the same order Jordan blocks corresponding to a concerned eigenvalue of the matrix is formulated. Then, under the condition that all of the eigenvalues of the derived eigenvalue problem are simple, the formulas for calculating the differentiable eigenvectors, the first- to third-order eigenvalue derivatives, and the first- to second-order eigenvector derivatives are derived. Two numerical examples show the validity of the method and how to program based on the mathematical derivations.