Spectral Perturbation Theory of Hermitian Matrices
Marcus Carlsson · IntechOpen eBooks · 2025
While Hermitian Perturbation theory is significantly simpler in many aspects than the non-Hermitian one, explicit formulas for first- and second-order perturbations are often not present in standard reference works on the topic and hard to find elsewhere, especially in the presence of degenerate eigenvalues (i.e., of multiplicity higher than one). This chapter aims to fill in these gaps and also presents some new contributions, pertaining to degenerate eigenvalues. We focus on the local behavior of eigenvalues and eigenvectors as functions of all the matrix entries in the perturbation, giving concrete formulas for the gradient and Hessian of the corresponding multivariable Taylor series.