Unstable Eigenvectors & Eigenvalues

• Unstable eigenvectors and eigenvalues are spectral components highly sensitive to even minor perturbations, critical for diagnosing dynamical instability in various systems.
• Perturbation theory and condition number analysis quantify eigenvalue displacements, emphasizing the role of small eigen-gaps and nearly degenerate spectra.
• Diagnostic tools such as pseudospectra, localization metrics, and spectral transformations offer actionable methods to detect and mitigate instability in complex operators.

Unstable eigenvectors and eigenvalues refer to spectral components (eigenvalues and their corresponding eigenvectors) that are sensitive to small perturbations in the underlying operator or matrix. Instability may manifest as large changes in the spectrum under infinitesimal perturbations, high sensitivity of eigenvectors due to near-degenerate spectra, or the emergence of nontrivial dynamical or localization phenomena. Such instability is a central issue across applied mathematics, random matrix theory, numerical analysis, quantum physics, network science, and fluid dynamics, each with distinct mechanisms and diagnostic tools.

1. Fundamental Mechanisms of Instability

Unstable eigenvalues frequently correspond to those of largest real part in physical systems (signaling dynamical instability), or to spectral regions with small eigen-gaps. Instability of eigenvectors is often associated with:

• Near-coalescence or degeneracy of eigenvalues, leading to high eigenvector condition numbers (see (Taylor et al., 2016, Movassagh et al., 2016, Greenbaum et al., 2019));
• Non-normal (especially non-Hermitian) operators for which the pseudospectrum—the region where the resolvent norm is large—can become very broad, so that eigenvalues are extremely sensitive even under norm-small perturbations (Aboud et al., 2016, Yang et al., 2022);
• The presence of Jordan block structure, where eigenvalues and their eigenvectors coalesce (exceptional points; (Yang et al., 2022));
• Heavy-tailed randomness or disorder, leading to both non-Gaussian eigenvector statistics and anomalously large eigenvector overlaps (Aggarwal et al., 2020, Truong et al., 2017).

The mathematical quantification of instability involves derivative estimates (perturbation theory), pseudospectrum analysis, resolvent bounds, and eigenvector condition numbers.

2. Perturbation Theory and Eigenvector Error

Perturbation theory provides precise first-order (and sometimes higher-order) formulas for the displacement of eigenvalues and eigenvectors under small changes in the matrix or operator (Greenbaum et al., 2019). For a simple eigenvalue $\lambda_0$ of a family $A(\tau)$ and right/left eigenvectors $x_0$, $y_0$ (normalized $y_0^* x_0 = 1$), the first-order derivatives are: $$\lambda'(\tau_0) = y_0^* A'(\tau_0) x_0, \qquad x'(\tau_0) = -S A'(\tau_0)x_0, \qquad (y^*)'(\tau_0) = -y_0^* A'(\tau_0) S,$$ where $S$ is the reduced resolvent. The eigenvector condition number $\chi = \|x_0\|\,\|y_0\|$ quantifies sensitivity: large $\chi$ signals high instability.

For multiple or nearly degenerate eigenvalues, these derivatives can exhibit singularities or non-analytic (e.g., Puiseux series) behavior. In high-dimensional empirical matrices, the expected eigenvector error (for population eigenvector $u_i$ and empirical $\tilde{u}_i$) is

$$\mathbb{E}\|u_i - \tilde{u}_i\|^2 \approx \frac{1}{n} h_i,$$

with $$h_i = \sum_{j \neq i} \frac{\lambda_i \lambda_j}{(\lambda_i - \lambda_j)^2}$$. The distribution of $h_i$ has a power-law $1/h^2$ tail, indicating that even if average errors are small, large errors are non-negligible for some eigenvectors, especially as the eigen-gap decreases (Taylor et al., 2016, Cheng et al., 2020).

3. Instability in Operator Classes and Physical Systems

Hamiltonian and Non-Hermitian Operators: For operators with $JL$ structure (Hamiltonian flows), the number of unstable eigenvalues (with $\operatorname{Re}\lambda > 0$) can be bounded using commuting operators $K$:

$\nu(JL) \leq n_{s}(K) + n_{c}(K),$

enabling sharp stability thresholds in spectral problems such as the KP-II equation (Haragus et al., 2016).

Random Matrix Ensembles: In Wigner and Ginibre ensembles, eigenvectors are typically delocalized with eigenvalues stable to perturbation. In contrast, Lévy matrices or certain structured ensembles ($H = W \tilde{H} W + D$) can display transitions from ergodic (delocalized) to multifractal or localized eigenvectors, controlled by scaling parameters in $W$ and $D$. The fluctuation law of eigenvector entries for Lévy matrices is non-Gaussian and energy-dependent, and different eigenvector entries are asymptotically independent but correlated across neighboring eigenvalues, sharply contrasting Wigner-type universality (Aggarwal et al., 2020, Truong et al., 2017).

Nonlinear Eigenvalue Problems: For strongly non-self-adjoint or nonlinear eigenvalue problems, instability is extreme, as small discretization or operator errors produce large deviations in computed eigenvalues and eigenvectors. Condition number analysis and pseudospectrum calculations show very high sensitivity, especially for eigenvalues with large modulus or near the boundary of the spectrum (Aboud et al., 2016).

Non-Hermitian Dynamics and Exceptional Points (EPs): At EPs, multiple eigenvalues and their eigenvectors coalesce (Jordan block formation). The sensitivity of eigenvectors near EPs is path-dependent: "quantum distance" measures show that the eigenvector collapse rate can be different along different parameter directions. Odd-order EPs in systems with sublattice symmetry may lead to anomalous, mixed-type Jordan structures and enhanced eigenvector sensitivity (Yang et al., 2022).

4. Diagnostics: Pseudospectra, Localization, and Condition Number

The analysis of unstable modes leverages several diagnostic tools:

• Pseudospectrum: Defined as the set $\{z: \| (A - zI)^{-1} \| \geq 1/\epsilon\}$, it captures the spectral instability not visible in the spectrum alone. Very large pseudospectra indicate regions where near-eigenvalues and transient amplified responses are possible, even if $z$ is not a true eigenvalue (Aboud et al., 2016).
• Localization Metrics: Localization is quantified via entropy ($H = -\sum_j |v_j|^2 \log_2 |v_j|^2$) and inverse participation ratio (IPR, $=\sum_j |v_j|^4$). Highly localized eigenvectors (large IPR, small entropy) are typical for type II "runaway" eigenpairs in disordered Toeplitz matrices, whereas type I runaways show more delocalized (algebraically decaying) profiles (Movassagh et al., 2016).
• Eigenvector Condition Number: $\kappa$ measures the angle between right and left eigenvectors. Very large $\kappa$ implies that small perturbations (in operator or data) cause large eigenvalue shifts and drastically different eigenvector orientations (Movassagh et al., 2016, Taylor et al., 2016).
• Numerical Experiments and Scaling Laws: Statistical approaches (e.g., in random matrix ensembles) support heavy-tail error laws ($\propto 1/h^2$) and highlight the occurrence of rare, extreme instability events (Taylor et al., 2016, Aggarwal et al., 2020).

5. Computation and Mitigation of Instability

Practical computation of unstable eigenpairs requires specialized algorithms and regularization schemes.

• Spectral Transformation and Möbius Mapping: In large-scale nonsymmetric pencils $(J, L)$ with singular $L$, the Möbius transform

$$C_{(\sigma)} = (J + \overline{\sigma} L)(J - \sigma L)^{-1}$$

maps unstable eigenvalues (positive real part) outside the unit disk, where shift-invert and power iterations converge rapidly. By projecting iterates into the physical subspace (range of $L$), convergence to spurious modes is suppressed. Deflation or inhibition techniques prevent recomputation of converged eigenvalues (Bezerra et al., 2010).

• Constructive and Approximate Eigenvectors: Formulations targeting $\epsilon$-approximate eigenvectors ($\|A v - \lambda v\| \leq \epsilon$) achieve controlled, robust error margins, circumventing ill-posedness due to arbitrarily small singular values or eigenvalue clustering (Osinenko et al., 2016).
• Spectral Approximations for Networks: Eigenvector-dependent observables such as centrality and resistance distance can often be approximated from eigenvalue spectra by using algebraic identities, thus bypassing unstable eigenvector computation in large, dense or nearly degenerate networks (Gutiérrez et al., 2020).
• Taylor and Chebyshev Expansions for Parametric Problems: For parameter-dependent eigenpairs $A(\mu)v(\mu) = \lambda(\mu)v(\mu)$, local Taylor expansions are efficient but break down as the parameter deviates from the expansion point, especially near eigenvalue crossings. Chebyshev expansions provide uniform approximations over intervals but are more computationally intensive due to nonlinear systems introduced by basis mixing (Mach et al., 2023).
• Detection via Operator Determinants: For fluid problems, e.g., linearized 2D Euler dynamics, unstable eigenvalues correspond to zeros of modified Fredholm determinants associated with Birman–Schwinger operators: the analytic function $\mathcal{D}(\lambda, 0)$ vanishes precisely at unstable eigenvalues, translating instability detection to root-finding for a holomorphic function (Latushkin et al., 2018).
• Bifurcation and Normal Form Analysis: For spectral bifurcations, e.g., in Stokes wave stability, Puiseux expansions and Jordan chain analysis reveal that unstable eigenvalues emerge at energy or momentum extrema. The sign of specific coefficients in the normal form equation,

$$\lambda_1^4 \mathcal{B} + \lambda_1^2 \operatorname{sgn}(c - c_0) \mathcal{P}''(c_0) = 0,$$

determines whether unstable real eigenvalues bifurcate—giving a direct spectral criterion for loss of stability (Dyachenko et al., 19 Mar 2025).

6. Applications and Implications

Unstable eigenvectors and eigenvalues have diverse consequences:

• Dynamical Systems: Unstable eigenvalues with $\operatorname{Re}\lambda > 0$ correspond to exponentially growing modes (“instability”) in the linearized phase space or control systems. Effectively computing and tracking these is crucial for transient stability, bifurcation analysis, and control design (Bezerra et al., 2010, Haragus et al., 2016, Dyachenko et al., 19 Mar 2025).
• Statistical Learning and PCA: In high dimensions, eigenvector instability profoundly affects reliability of principal components. Extreme heterogeneity in error is especially problematic without large sample sizes, motivating ensemble-based error estimators (Taylor et al., 2016).
• Random and Disordered Systems: Instability in eigenvectors signals the presence of localizing phenomena, phase transitions in transport, multifractality, or the breakdown of universality, particularly for heavy-tailed ensembles or models near criticality (Truong et al., 2017, Aggarwal et al., 2020).
• Network Science: Eigenvector-based measures (centrality, resistance) can become unstable, especially in networks with nearly degenerate spectra, potentially skewing inference—approximations sidestep such instability (Gutiérrez et al., 2020).
• Non-Hermitian Physics and Sensing: Exceptional points yield extreme sensitivity in observable responses; odd-order EPs in symmetric settings introduce new classes of unstable, path-dependent coalescence phenomena relevant for non-Hermitian quantum systems (Yang et al., 2022).

7. Summary Table: Principal Manifestations and Diagnostics

Phenomenon	Mechanism/Diagnostic	Reference
High eigenvector error, eigen-gap	$\left\|u_i - \tilde{u}_i \right\|^2$ via $h_i$ (power law tail)	(Taylor et al., 2016)
Extreme sensitivity (non-normal, non-Hermitian)	Large pseudospectrum, condition number	(Aboud et al., 2016, Movassagh et al., 2016, Yang et al., 2022)
Localization/delocalization transitions	IPR/entropy/critical scaling	(Movassagh et al., 2016, Truong et al., 2017, Aggarwal et al., 2020)
Spectral bifurcation (fluid dynamics)	Puiseux expansion, normal form	(Dyachenko et al., 19 Mar 2025)
Stability/instability detection	Fredholm determinant zeros	(Latushkin et al., 2018)
Mitigation and analysis	Spectral transformation, constructive methods, Taylor/Chebyshev expansions	(Bezerra et al., 2010, Osinenko et al., 2016, Mach et al., 2023)

Unstable eigenvectors and eigenvalues are a central feature of high-dimensional, non-normal, or critical systems. Their analysis, diagnosis, and mitigation require a combination of spectral theory, perturbation analysis, advanced numerical methods, and, frequently, probabilistic and algebraic tools tailored to the stability properties of each context.