Eigenvalue and pseudospectrum processes generated by nonnormal Toeplitz matrices with rank 1 perturbations

TL;DR

This work examines how eigenvalues and pseudospectra of two rank-1 perturbed nilpotent Toeplitz matrices evolve over time. By deriving explicit nonzero-eigenvalue equations for Model 1 and its a-perturbed Model 2, and by interpreting the dynamics through Toeplitz symbol curves, the authors reveal a clean separation: outer curves correspond to exact, robust eigenvalues, while inner structures correspond to the pseudospectrum around the degenerate zero eigenvalue. They show that, in the large-n limit, outliers scale with nδ and the remaining eigenvalues align to roots of unity on the unit circle (or rescaled by (1+a) in Model 2), with inner pseudospectral regions contracting as time progresses. These results bridge numerical analysis of nonnormal matrices with pseudospectral theory and offer insights into stability and sensitivity in non-Hermitian systems, with potential relevance to quantum physics applications.

Abstract

We introduce two kinds of matrix-valued dynamical processes generated by nonnormal Toeplitz matrices with the additive rank 1 perturbations $δJ$, where $δ\in {\mathbb{C}}$ and $J$ is the all-ones matrix. For each process, first we report the complicated motion of the numerically obtained eigenvalues. Then we derive the specific equation which determines the motion of non-zero simple eigenvalues and clarifies the time-dependence of degeneracy of the zero-eigenvalue $λ_0=0$. Comparison with the solutions of this equation, it is concluded that the numerically observed non-zero eigenvalues distributing around $λ_0$ are the exact eigenvalues not of the original system, but of the system perturbed by uncontrolled rounding errors of computer. The complex domain in which the eigenvalues of randomly perturbed system are distributed is identified with the pseudospectrum including $λ_0$ of the original system with $δJ$. We characterize the pseudospectrum processes using the symbol curves of the corresponding nonnormal Toeplitz operators without $δJ$. We report new phenomena in our second model such that at each time the outermost closed simple curve cut out from the symbol curve is realized as the exact eigenvalues, but the inner part of symbol curve is reduced in size and embedded in the pseudospectrum including $λ_0$. Such separation of exact simple eigenvalues and a degenerated eigenvalue associated with pseudospectrum will be meaningful for numerical analysis, since the former is stable and robust, but the latter is highly sensitive and unstable with respective to perturbations. The present study will be related to the pseudospectra approaches to non-Hermitian systems developed in quantum physics

Figures (8)

• Figure 1: Numerically obtained eigenvalues are dotted for the non-Hermitian matrix-valued Brownian motion, $(M(t))_{t \geq 0}$, with size $n=300$ starting from the nonnormal matrix $S$. (a) Ring structure ($0<t \ll 1$), (b) Growing annulus ($0 < t < 1$), (c) Fulfilled disk ($t \geq 1$).
• Figure 5: Plots of numerically obtained eigenvalues at time $m=1$ for model 1, $S^{(1)}_{\delta J}(1)$, with $n=200$ and $\delta=0.01$. 199 dots form a unit circle missing one point at $z=1$, and an outlier is observed near $z=3$, which is denoted by $\lambda_1(1)$.
• Figure 6: Numerically obtained eigenvalues are plotted for model 1, $(S^{(1)}_{\delta J}(m))_{m=1}^n$, with $n=200$ and $\delta=0.01$ at $m=2$, 8, 15, and 80, respectively.
• Figure 11: Numerically obtained eigenvalues are plotted for model 2, $(S^{(2)}_{\delta J}(m))_{m=1}^n$, with $n=200$, $\delta=0.01$, and $a=1$ at time $m=1$, 2, 3, and 4, respectively.
• Figure 16: Exact eigenvalues are plotted for model 2, $(S^{(2)}_{\delta J}(m))_{m=1}^n$, with $n=200$, $\delta=0.01$, and $a=1$ at $m=1, 2, 3$, and 4, respectively.

Theorems & Definitions (15)

• Lemma 3.1
• Theorem 3.2
• Proposition 3.3
• Proposition 3.4
• Theorem 3.5
• Lemma 3.6
• Proposition 3.7
• Theorem 4.1
• Proposition 4.2
• Conjecture 4.3