DERIVATIVES OF EIGENVALUES AND EIGENVECTORS OF MATRIX FUNCTIONS

For an n x n matrix-valued function L(rho, lambda), where rho is a vec tor of independent parameters and lambda is an eigenparameter, the eig envalue-eigenvector problem has the form L(rho, lambda)(rho))x(rho) = 0. Real or complex values for rho and lambda are admitted, and L is as sumed to depend analytically on these variables. In particular, nonlin ear dependence on lambda is the main concern. On the assumption that t he eigenvalue-eigenvector problem itself can be satisfactorily solved, a study is made of the derivatives (sensitivities) of lambda and x wi th respect to rho. Analysis is made of the existence of derivatives, t he effect of normalization strategies, and the solvability and conditi on of the bordered matrix equations arising naturally in this context. Implications for various classical eigenvalue problems L(rho, lambda) = A(rho) - lambdaI are clarified.