Eigensolution sensitivity revisited

An explicit formulation is developed for the sensitivity of eigensolutions associated with matrix-valued operator pencils depending nonlinearly on a scalar parameter. The eigenproblem is expressed as a nonlinear algebraic system for the eigenvalue and eigenvector, including a normalization constraint. Differentiation of this system yields the sensitivity equations. Regularization of the augmented system followed by symbolic inversion and limiting yields compact, closed-form expressions for the derivatives, applicable also in the presence of algebraic multiplicity. The interpretation of the sensitivity problem relies on an inversion formula for the sum of singular complex matrices, extended through decomposition-based constructions to ensure numerical robustness. Applications to structural dynamics, rigid-body degeneracies, time-delay systems, and continuum eigenproblems illustrate the method's capability to handle ill-conditioning near eigenvalue coalescence and to provide stable sensitivity estimates for both self-adjoint and non-self-adjoint operators.