Analytical solutions for the nth derivatives of eigenvalues and eigenvectors for a nonlinear eigenvalue problem

I HE nonlinear eigenvalue problem discussed here is defined by the following equations: A(ir,\) x = 0 jttfir(7r,X) x = i (1) (2) where A (TT,X) is an n x n matrix representing a nonlinear operator on the unrepeated eigenvalue X; x is the corresponding 1 x n eigenvector that is /if-normalized by an n x n positive definite hermitian, K\ jc" is the complex conjugate transpose of jc; and TT is a real scalar parameter. The eigenvalue X is obtained by setting the determinant of A to zero and solving the resulting characteristic equation for X(TT). The corresponding eigenvector, jc, is then calculated from Eqs. (1) and (2). It is assumed that X is a differentiable function of TT. Elements of A and K, atj and k^ (1 = 1, ...,«; j = 1, ...,«) must also be differentiable functions of TT and are, in general, nonlinear functions of X and TT. The objective of this paper is to find explicit analytical solutions for the nth derivative of the unrepeated eigenvalue X, X, and the nth derivative of the corresponding eigenvector jc, JC, with respect to TT. Various numerical techniques have been proposed in the past but no analytical solutions of this problem have been available.