Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Construction of the Nearest Nonnegative Hankel Matrix for a Prescribed Eigenpair

Prince Kanhya, Udit Raj

TL;DR

The paper addresses whether a prescribed eigenpair can be realized by a nonnegative Hankel matrix through the smallest structured perturbation and, when impossible, computes the closest residual-minimizing Hankel matrix. It introduces a convex two-stage framework that encodes Hankel structure and nonnegativity via a Hankel-structure matrix $S$ and linear constraints, yielding Stage A (minimum-norm exact correction) and Stage B (best residual approximation). A feasibility equivalence is established between exact realizability and a linear system, with convex optimization guaranteeing global solutions. Numerical experiments demonstrate both exact realizability and residual-minimizing approximations across matrix sizes, and reveal clear performance patterns tied to the feasibility of the eigenpair constraints.

Abstract

We study the problem of determining whether a prescribed eigenpair $(λ,x)$ 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 $ΔH$ such that $(H+ΔH)x=λx$. When no such perturbation exists, we compute the nearest nonnegative Hankel matrix in a residual sense by minimizing $\|(H+ΔH)x-λx\|_{2}$ 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.

Construction of the Nearest Nonnegative Hankel Matrix for a Prescribed Eigenpair

TL;DR

The paper addresses whether a prescribed eigenpair can be realized by a nonnegative Hankel matrix through the smallest structured perturbation and, when impossible, computes the closest residual-minimizing Hankel matrix. It introduces a convex two-stage framework that encodes Hankel structure and nonnegativity via a Hankel-structure matrix and linear constraints, yielding Stage A (minimum-norm exact correction) and Stage B (best residual approximation). A feasibility equivalence is established between exact realizability and a linear system, with convex optimization guaranteeing global solutions. Numerical experiments demonstrate both exact realizability and residual-minimizing approximations across matrix sizes, and reveal clear performance patterns tied to the feasibility of the eigenpair constraints.

Abstract

We study the problem of determining whether a prescribed eigenpair 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 such that . When no such perturbation exists, we compute the nearest nonnegative Hankel matrix in a residual sense by minimizing 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.

Paper Structure

This paper contains 8 sections, 1 theorem, 49 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Preliminaries and Notations
  4. Hankel Structure Matrix
  5. Main theorem & its proof
  6. Numerical Experiments and Performance Analysis
  7. Numerical Experiments and Performance Analysis
  8. Performance Analysis of Algorithm

Key Result

Theorem 1

Let $H\in\mathbb{R}^{n\times n}$ be a Hankel matrix, let $\lambda\in\mathbb{C}$ and $x\in\mathbb{C}^n$ be a nonzero vector. Let $S\in\{0,1\}^{n^2\times(2n-1)}$ (defined previously) be the Hankel structure matrix, and let $z\in\mathbb{R}^{2n-1}$ denote the vector of Hankel parameters so that $\operat and Consider the two optimization problems used by the algorithm: Then the following statements h

Figures (3)

  • Figure 1: Stage A vs Stage B usage across matrix sizes. Blue denotes exact feasible cases; orange denotes infeasible cases requiring Stage B optimization.
  • Figure 2: Residual distribution across all trials. Two distinct clusters appear: (i) very small residuals ($10^{-16}$–$10^{-8}$) corresponding to feasible eigenpairs (Stage A), (ii) large residuals ($\approx 1$–$10$) corresponding to infeasible eigenpairs requiring approximation (Stage B).
  • Figure 3: CPU time as a function of matrix size $n$. The overall growth reflects increased SDP complexity. Occasional spikes correspond to infeasible eigenpairs where Stage B dominates, confirming that infeasibility significantly increases the computational burden.

Theorems & Definitions (4)

  • Remark
  • Theorem 1
  • proof
  • Remark