Construction of the Nearest Nonnegative Hankel Matrix for a Prescribed Eigenpair

We study the problem of determining whether a prescribed eigenpair $(\lambda,x)$ can be made an exact eigenpair of a nonnegative Hankel matrix through the smallest possible structured perturbation. The task reduces to check 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 $\|(H+\Delta H)x-\lambda x\|_{2}$ 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 nonnegativity-preserving Hankel perturbations. Numerical examples illustrate both feasible and infeasible cases.