On Pseudospectral Concentration for Rank-1 Sampling

TL;DR

The paper develops a quantitative theory for pseudospectral concentration of complex matrices under rank-1 random perturbations by exploiting unitary invariance to reduce to upper-triangular (and diagonal) forms. It establishes two regimes: regular concentration for normal/diagonal matrices with a separation radius $\delta_d$ scaling as $\delta_d\sim 1/\sqrt{d}$, and singular concentration for nilpotent Jordan blocks using complex Hanson–Wright and Carbery–Wright inequalities, also yielding $\delta_d\sim 1/\sqrt{d}$. The results extend to upper-triangular Toeplitz matrices via polynomial symbols, including binomial-symbol and general-symbol cases, with explicit bounds $\delta_d = 2\sqrt{\frac{C\epsilon}{|a_n| d}}$ and related constants. The work provides a rigorous probabilistic framework for pseudospectral analysis under rank-1 perturbations, with practical implications for efficiently probing spectral sensitivity in nonnormal systems. Overall, the study connects random perturbation theory, complex analysis, and structured matrix classes to yield high-probability, dimension-dependent concentration results for pseudospectra.

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 $δ_d \sim 1/\sqrt{d}$. 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.

Figures (5)

• Figure 1: Scatter plots illustrating the spectral behavior of rank-1 sampling perturbations. Parameter: $\epsilon = 2$. Number of samples: $N=1000$.
• Figure 2: Scatter plots illustrating the spectral behavior of the nilpotent Jordan block, as given in \ref{['eqn: nil-jordan']}, under rank-1 sampling perturbations. Parameter: $\epsilon = 2$. Number of samples: $N=1000$.
• Figure 3: Scatter plots illustrating the spectral behavior of a diagonal matrix under rank-1 sampling perturbations. The underlying diagonal matrix contains two distinct eigenvalues, $2$ and $3$, each repeated with multiplicity $d/2$. Parameter: $\epsilon = 2$. Number of samples: $N=1000$.
• Figure 4: Scatter plots illustrating the spectral behavior of the $\mathcal{R}$-Toeplitz matrix defined by a linear symbol $p(z)=3 + 2z$ under rank-1 sampling perturbations. Parameter: $\epsilon = 2$. Number of samples: $N=1000$.
• Figure 5: Scatter plots illustrating the spectral behavior of the $\mathcal{R}$-Toeplitz matrix defined by a quadratic symbol $p(z)=3+2z+z^2$ under rank-1 sampling perturbations. Parameter: $\epsilon = 2$. Number of samples: $N=1000$.

Theorems & Definitions (25)

• Definition 2.1: Pseudospectra
• Theorem 2.2: Rank-1 Characterization of Pseudospectra
• Lemma 2.3: Schur Decomposition, Theorem 7.1.3 in golub2013matrix
• Lemma 2.4: Corollary 7.1.4 in golub2013matrix
• Theorem 2.5
• Lemma 2.6: Matrix Determinant Lemma, Section 2.1.4 in golub2013matrix
• Lemma 2.7: Rouché's Theorem, Theorem 4.3 of Chapter 3 in stein2010complex
• Lemma 2.8: Maximum Modulus Principle, Theorem 4.5 of Chapter 3 in stein2010complex
• Lemma 2.9: Hanson-Wright Concentration Inequality, hanson1971bound
• Lemma 2.10: Carbery–Wright Anti-Concentration Inequality, Theorem 8 in carbery2001distributional