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Numerical Radius of Non-Hermitian Random Matrices

Zhigang Bao, Giorgio Cipolloni

TL;DR

This work studies the fine-scale behavior of the numerical radius and inner numerical radius for complex non-Hermitian random matrices and their elliptic variants. By linking the radii to extrema of a stationary Airy-like process via the Hermitian parts $H(\theta)$ and establishing a correlation-decorrelation transition at scale $N^{-1/6}$, the authors derive the first two terms in the asymptotic expansions for $r_+(A)$ and $r_-(A)$ and, for the elliptic model, show that fluctuations converge to the maximum or minimum of two independent $ ext{TW}_2$ variables in natural parameter regimes. The elliptic ensemble exhibits localization of extremal fluctuations near specific angles, reflecting nonstationarity, while a combination of Dyson Brownian motion and Green function comparison extends Gaussian results to general ensembles. The work also outlines further directions, including higher-order terms, real matrices, and transitional regimes, with potential implications for numerical linear algebra and random matrix theory.

Abstract

For a square matrix, the range of its Rayleigh quotients is known as the numerical range, which is a compact and convex set by the Toeplitz-Hausdorff theorem. The largest value and the smallest boundary value (in magnitude) of this convex set are known as the numerical radius and inner numerical radius respectively. The numerical radius is often used to study the convergence rate of iterative methods for solving linear systems. In this work, we investigate these radii for complex non-Hermitian random matrix and its elliptic variants. For the former, remarkably, these radii can be represented as extrema of a stationary Airy-like process, which undergoes a correlation-decorrelation transition from small to large time scale. Based on this transition, we obtain the precise first and second order terms of the numerical radii. In the elliptic case, we prove that the fluctuation of the numerical radii boils down to the maximum or minimum of two independent Tracy-Widom variables.

Numerical Radius of Non-Hermitian Random Matrices

TL;DR

This work studies the fine-scale behavior of the numerical radius and inner numerical radius for complex non-Hermitian random matrices and their elliptic variants. By linking the radii to extrema of a stationary Airy-like process via the Hermitian parts and establishing a correlation-decorrelation transition at scale , the authors derive the first two terms in the asymptotic expansions for and and, for the elliptic model, show that fluctuations converge to the maximum or minimum of two independent variables in natural parameter regimes. The elliptic ensemble exhibits localization of extremal fluctuations near specific angles, reflecting nonstationarity, while a combination of Dyson Brownian motion and Green function comparison extends Gaussian results to general ensembles. The work also outlines further directions, including higher-order terms, real matrices, and transitional regimes, with potential implications for numerical linear algebra and random matrix theory.

Abstract

For a square matrix, the range of its Rayleigh quotients is known as the numerical range, which is a compact and convex set by the Toeplitz-Hausdorff theorem. The largest value and the smallest boundary value (in magnitude) of this convex set are known as the numerical radius and inner numerical radius respectively. The numerical radius is often used to study the convergence rate of iterative methods for solving linear systems. In this work, we investigate these radii for complex non-Hermitian random matrix and its elliptic variants. For the former, remarkably, these radii can be represented as extrema of a stationary Airy-like process, which undergoes a correlation-decorrelation transition from small to large time scale. Based on this transition, we obtain the precise first and second order terms of the numerical radii. In the elliptic case, we prove that the fluctuation of the numerical radii boils down to the maximum or minimum of two independent Tracy-Widom variables.

Paper Structure

This paper contains 14 sections, 6 theorems, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proof Strategy
  4. Notations
  5. Outline
  6. Proof of Theorem \ref{['maintheorem']}
  7. Correlation Regime: Proof of Proposition \ref{['lem.correlation']}
  8. Decorrelation Regime: Proof of Proposition \ref{['lem.decorrelation']}
  9. Elliptic ensemble
  10. Further discussion
  11. Further expansion of numerical radii
  12. Real non-Hermitian random matrix
  13. General matrix-valued process
  14. Transition regime for Elliptic model

Key Result

Theorem 1.1

Let $A$ be the non-Hermitian random matrix defined in (082701). We have the following expansion in probability,

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 2.1: Correlation Regime
  • Remark 2.2
  • Proposition 2.3: Decorrelation Regime
  • Proposition 2.4
  • Remark 2.5
  • ...and 7 more