Eigensensitivity Analysis of a Defective Matrix
Zhen-Yu Zhang, Huisheng Zhang · AIAA Journal · 2001
The formulas for calculating the first- to third-order perturbation coefficients of the eigenvalues and the first-order perturbation coefficients of eigenvectors of a defective matrix are derived by use of the direct perturbation method for the case where the eigenvalue problem for the first-order perturbation coefficients of the eigenvalues of the defective matrix has repeated eigenvalues. Associated with νth-order Jordan blocks of the matrix, the perturbation equations are derived by expanding the perturbed eigenvalues and eigenvectors in power series of η= ∈ 1/ν , where ∈ is the small parameter of the problem, substituting the series into the perturbed eigenvalue problem and comparing the powers of η. The kth-order perturbation equations for k = 1,., ν - 1 can be solved formally. The solutions contain some undetermined quantities. By use of the solvability conditions of the kth-order equations for k = v, ν + 1,..., the undetermined quantities in the solutions can be calculated in turns. When we run all orders of the Jordan blocks, the problem is solved. The numerical example shows the validity of the method.