On Pseudospectral Concentration for Rank-1 Sampling

TL;DR

This paper develops a probabilistic framework for pseudospectral concentration of complex matrices under rank-1 random perturbations, revealing dimension-dependent scaling laws for normal, nilpotent, and Toeplitz matrices.

Contribution

It introduces a new concentration theory for pseudospectra under rank-1 sampling, extending to various matrix classes using probabilistic inequalities and polynomial representations.

Findings

Separation radius scales as 1/√dimension for normal matrices

Singular concentration bounds hold for nilpotent Jordan blocks

Extension of concentration results to upper triangular Toeplitz matrices

Abstract

Pseudospectral analysis serves as a powerful tool in matrix computation and the study of both linear and nonlinear dynamical systems. Among various numerical strategies, random sampling, especially in the form of rank-$1$ perturbations, offers a practical and computationally efficient approach. Moreover, due to invariance under unitary similarity, any complex matrix can be reduced to its upper triangular form, thereby simplifying the analysis. In this study. we develop a quantitative concentration theory for the pseudospectra of complex matrices under rank-$1$ random sampling perturbations, establishing a rigorous probabilistic framework for spectral characterization. First, for normal matrices, we derive a regular concentration inequality and demonstrate that the separation radius scales with the dimension as $\delta_d \sim 1/\sqrt{d}$. Next, for the equivalence class of nilpotent Jordan blocks, we exploit classical probabilistic tools, specifically, the Hanson-Wright concentration inequality and the Carbery-Wright anti-concentration inequality, to obtain singular concentration bounds, and demonstrate that the separation radius exhibits the same dimension-dependent scaling. This yields a singular pseudospectral concentration framework. Finally, observing that upper triangular Toeplitz matrices can be represented via the symbolic polynomials of nilpotent Jordan blocks, we employ partial fraction decomposition of rational functions to extend the singular framework to the equivalence class of upper triangular Toeplitz matrices.