Construction of the Nearest Nonnegative Hankel Matrix for a Prescribed Eigenpair

TL;DR

This paper develops a numerical framework to determine the minimal structured perturbation needed for a nonnegative Hankel matrix to have a prescribed eigenpair, addressing feasibility and sensitivity analysis.

Contribution

It introduces an optimization-based method to compute the nearest nonnegative Hankel matrix with a given eigenpair, handling both feasible and infeasible cases.

Findings

Method effectively computes minimal perturbations for eigenpair realization.

Framework handles both feasible and infeasible eigenpair cases.

Provides numerical examples demonstrating the approach's applicability.

Abstract

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.