Re-examination of eigenvector derivatives

Analytical expressions for eigenvector derivatives for general non-self-adjoint systems using a modal expansion approach have not been correctly derived in several papers and books that address this problem. A common mistake in several developments on deriving eigenvector derivatives (or its perturbation forms) has been to ignore the eigenvector in question, in the expansion of its derivative, although the remaining eigenvectors forming the basis are generally non-orthogonal. This assumption is based upon explicit or implicit heuristic arguments that only directional changes contribute to eigenvector sensitivity. It is shown in this paper that the above assumption and the resulting equations are incorrect, except for the classical and most commonly encountered case of a self-adjoint problem having orthogonal eigenvectors. For the non-self-adjoint case, certain basis coefficients in the eigenvector derivative expansion have not been resolved correctly in the literature. A careful re-examination of the eigenvalue problem reveals that the two independent sets of normalizations (required for uniquely specifying the right and left eigenvectors) can be used to uniquely determine the basis coefficients. The solution derived herein for the eigenvector derivatives is shown to have generally nonzero projections onto all eigenvectors. It is also shown that basis coefficients for left and right eigenvector derivatives are related by a simple expression. A numerical example is included to demonstrate the present formulation for eigenvector derivatives with respect to a scalar parameter. We also extend Nelson's algebraic approach (for self-adjoint eigenvalue problems) to the general non-self-adjoint problem and the modal truncation approach to approximate eigenvector derivatives for the non-self-adjoint case.