Generalized Eigenspaces and Pseudospectra of Nonnormal and Defective Matrix-Valued Dynamical Systems

TL;DR

This work analyzes discrete-time matrix-valued dynamical systems built from nilpotent shift operators, focusing on a defective zero eigenvalue whose geometric multiplicity grows with time. By constructing explicit generalized eigenspaces and performing a Jordan decomposition of the resolvent, it derives a time-dependent description of the spectrum and, in particular, the shrinking of the $\varepsilon$-pseudospectrum around $0$ as time advances, while nonzero eigenvalues form a polynomial-root structure. The authors provide rigorous bounds linking resolvent growth to pseudospectral dynamics and validate the theory with numerical experiments using Gaussian perturbations, revealing a relaxation from far-from-normal to near-normal behavior and connections to Toeplitz-symbol geometry. They also outline future directions, including general perturbation schemes, computation of generalized condition numbers, and links to representation-theoretic structures such as Young diagrams, aiming to broaden the understanding of defectivity in time-evolving nonnormal systems.

Abstract

We consider nonnormal matrix-valued dynamical systems with discrete time. For an eigenvalue of matrix, the number of times it appears as a root of the characteristic polynomial is called the algebraic multiplicity. On the other hand, the geometric multiplicity is the dimension of the linear space of eigenvectors associated with that eigenvalue. If the former exceeds the latter, then the eigenvalue is said to be defective and the matrix becomes nondiagonalizable by any similarity transformation. The discrete-time of our dynamics is identified with the geometric multiplicity of the zero eigenvalue $λ_0=0$. Its algebraic multiplicity takes about half of the matrix size at $t=1$ and increases stepwise in time, which keeps excess to the geometric multiplicity until their coincidence at the final time. Our model exhibits relaxation processes from far-from-normal to near-normal matrices, in which the defectivity of $λ_0$ is recovering in time. We show that such processes are realized as size reductions of pseudospectrum including $λ_0$. Here the pseudospectra are the domains on the complex plane which are not necessarily exact spectra but in which the resolvent of matrix takes extremely large values. The defective eigenvalue $λ_0$ is sensitive to perturbation and the eigenvalues of the perturbed systems are distributed densely in the pseudospectrum including $λ_0$. By constructing generalized eigenspace for $λ_0$, we give the Jordan block decomposition for the resolvent of matrix and characterize the pseudospectrum dynamics. Numerical study of the systems perturbed by Gaussian random matrices supports the validity of the present analysis.

Figures (6)

• Figure 1: (a) One sampling result of numerically obtained eigenvalues of the system with Gaussian perturbation matrix $Z$; $S^{(b)}(m, \delta Z)=S^m+bS^{m+1}+ (1/\sqrt{2n}) Z$, where $n=5000$, $m=3$, and $b=1$. (b) Superpositions of 50 samples. (c) Symbol curve of $\widehat{S}^m+ b \widehat{S}^{m+1}$ with $m=3$ and $b=1$.
• Figure 5: Time-evolution of the geometric multiplicity $g_0$, the algebraic multiplicity $a_0$, and the degree $k_0$ of $\lambda_0$ are shown by black, red, and blue dots, respectively, for $t=1, 2, \dots, T:=n-2$ with $n=100$.
• Figure 6: Time evolution of Young diagram representing the dynamics of the Jordan canonical form associated with $\lambda_0$. The upper-left diagram is for the initial time $t=1$ with $\lfloor n/2 \rfloor$ boxes and the right diagram is for the final time $t=T$ with $n-2$ boxes. Here we have drawn the case with $n=17$. The lower-left diagram shows the intermediate state at time $t=5$. In this case, $\xi(t,n):=n\bmod{(t+1)}=5$. Then $k_0=\lfloor n/(t+1) \rfloor +1=3$ and $n-(t+1) \lfloor n/(t+1) \rfloor -1=\xi(t,n)-1=4$.
• Figure 7: Numerically obtained eigenvalues are superposed for 200 i.i.d Gaussian random perturbations $Z$ added to $S^{(b)}(t, \delta J)$ as \ref{['eq:Gaussian1']} with each value of $\widetilde{\delta}$, where $n=500$, $t=2$, $b=1$, and $\delta=10^{-2}$. The outlier eigenvalue $\lambda_1(t)$ is not drawn in each figure, since it is located outside the frame.
• Figure 11: Numerically obtained eigenvalues are superposed for 200 i.i.d Gaussian random perturbations $Z$ added to $S^{(b)}(t, \delta J)$ as \ref{['eq:Gaussian1']} with $b=1$, $\delta=10^{-2}$, and $\widetilde{\delta}=10^{-10}$. Dependence on $n$ and $t$ is shown. The pseudospectrum including $\lambda_0$ is observed as an inner domain fulfilled by dots. It shrinks with increment of $t$ showing the relaxation process of the defectivity of $\lambda_0$ for each $n$. On the other hand, as $n$ increases the inner part shows expansion. The curves lined up by the eigenvalues of perturbed systems seem to draw the inner parts of symbol curves of $\widehat{S}^{t+1}+b \widehat{S}^{t+2}$.

Theorems & Definitions (17)

• Theorem 2.1
• Remark 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 3.1
• Proposition 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Proposition 3.6