Eigensensitivity Analysis of a Defective Matrix with Zero First-Order Eigenvalue Derivatives

Abstract

A direct method is developed for calculating the first- to third-order eigenvalue derivatives and first- to second-order eigenvector derivatives of a defective matrix with a zero first-order eigenvalue derivative associated with Jordan blocks of order higher than the lowest. It is found under some conditions that the corresponding eigen-solution of the perturbed problem has a distinct structure that, in the fractional power series of the perturbation parameter, the denominator of the power can not only he the orders of Jordan blocks (as in the normal cases) but also can be the sum of two successive orders of Jordan blocks. The solution method is composed of three parts. One of them is simular to known work and the other two are very different. Numerical example show the correctness of the method.