On an eigenvector-dependent nonlinear eigenvalue problem from the perspective of relative perturbation theory

Abstract

We are concerned with the eigenvector-dependent nonlinear eigenvalue problem (NEPv) H(V)V=VΛ, where H(V)∈ℂn×n is a Hermitian matrix-valued function of V∈ℂn×k with orthonormal columns, i.e., VHV=Ik, k≤n (usually k≪n). Sufficient conditions on the solvability and solution uniqueness of NEPv are obtained, based on the well-known results from the relative perturbation theory. These results are complementary to recent ones in Cai et al. (2018), where, among others, one can find conditions for the solvability and solution uniqueness of NEPv, based on the well-known results from the absolute perturbation theory. Although the absolute perturbation theory is more versatile in applications, there are cases where the relative perturbation theory produces better results.