In this work, we study the problem of determining whether a prescribed pair \((\lambda ,x)\) can be made an exact eigenpair of a nonnegative Hankel matrix obtained from a given Hankel matrix \(H\in \mathbb {R}^{n\times n}\) through the smallest possible structured perturbation. The task reduces to checking the feasibility of a set of linear constraints that encode both the Hankel structure and entrywise nonnegativity. When the feasibility set is nonempty, we compute the minimum-norm perturbation \(\Delta H\) such that \((H+\Delta H)x=\lambda x\) . When no such perturbation exists, we compute the nearest nonnegative Hankel matrix in a residual sense by minimizing \(\Vert (H+\Delta H)x-\lambda x\Vert _{2}\) subject to the imposed constraints. Because closed-form formulas for the structured backward error are generally unavailable, our method provides a fully numerical and optimization-based framework for evaluating eigenpair sensitivity under Hankel perturbations that keep the perturbed matrix nonnegative. Numerical examples illustrate feasible, infeasible, and complex eigenpair cases, together with validation experiments.
