Eigenvalue Reparametrization in Spectral Analysis

Updated 28 November 2025

Eigenvalue reparametrization is a systematic process of altering eigenvalue problems to enable closed-form characterizations and simplify complex, parameter-dependent structures.
It facilitates the reduction of multi-parameter and nonlinear eigenproblems through analytic transforms such as Taylor and Chebyshev series, ensuring robust numerical and backward error analysis.
The approach is pivotal in diverse applications including algorithmic filtering, domain-based spectral reordering, and quantum eigenvalue transformations for efficient computation.

Eigenvalue reparametrization refers to the systematic process of altering the parameterization, structure, or functional dependence of eigenvalue problems in order to facilitate analysis, computation, or geometric or algorithmic interpretation. The concept permeates spectral theory, applied linear algebra, differential equations, optimal control, numerical analysis, quantum algorithms, and geometric analysis, and takes many highly specific forms according to context, including analytic reparameterizations, polynomial/nonlinearization reductions, domain reparametrization in PDE discretizations, or parameter tracking in parametric problems.

1. Foundational Principles and Definitions

Eigenvalue reparametrization typically involves mapping an original parameter, variable, geometric domain, or matrix/operator structure to a new parameter space or functional dependence, thereby transforming the associated spectrum or eigendata. Motivating objectives include:

Enabling closed-form or recursive characterization of eigenvalues, especially under structural or parametric constraints.
Facilitating efficient or stable computation by reducing multi-parameter or nonlinear eigenproblems to tractable forms.
Tracking or “filtering” specific eigenvalues through adaptive parameter tuning or functional transforms.
Revealing geometric, topological, or structural features of eigen-spectra under perturbation, deformation, or domain mappings.

No single definition suffices, but archetypal instantiations include the replacement of a two-parameter eigenproblem with a single-parameter (possibly nonlinear) problem (Ringh et al., 2019), constructing continuous eigenvalue paths under parameterized families of matrices (Jankowski et al., 2019), reindexing eigenmodes under domain or geometric transformations (Noureddine et al., 14 Oct 2025), or recasting the spectral parameter to expose monotonic or analytic structure for numerical root-finding (Peters et al., 2018).

2. Reparametrization in Parameter-Dependent and Nonlinear Eigenvalue Problems

A major area of application is the reduction or transformation of parameter-dependent eigenvalue problems. For instance, two-parameter eigenvalue problems (2PEP) of the form

$A_1 x + \lambda A_2 x + \mu A_3 x = 0, \quad B_1 y + \lambda B_2 y + \mu B_3 y = 0$

can be “nonlinearized” by eliminating $\mu$ blockwise to yield a family of nonlinear eigenvalue problems (NEPs) in $\lambda$ of the type

$M_i(\lambda)x = (A_1 + \lambda A_2 + g_i(\lambda)A_3)x=0, \quad \mu=g_i(\lambda)$

where each branch $g_i(\lambda)$ arises as a root of an associated generalized eigenproblem dependent on $\lambda$ (Ringh et al., 2019). This reparametrization is locally invertible under simplicity conditions on the spectrum and unifies several well-known linearization concepts (including the companion linearization for polynomial eigenproblems), while supporting efficient numerical solution, algorithmic structure exploitation, and robust backward error analysis.

In the context of parametric dependence, eigenvalue functions $\lambda(\mu)$ with $A(\mu)$ analytic or smooth in a parameter $\mu$ may be expanded via Taylor or Chebyshev series (Mach et al., 2023). Taylor expansion provides high-fidelity local reparametrization, with the eigenvalue and eigenvector derivatives computed recursively via bordered system solves, while Chebyshev expansion yields uniformly accurate global reparametrizations on prescribed intervals, exploiting spectral convergence in analytic regions and supporting efficient Monte Carlo sampling or uncertainty quantification. Both frameworks depend on the analytic or smooth dependence of $A(\mu)$ and break down at points of non-analyticity or spectral coalescence.

3. Geometric and Domain-Based Eigenvalue Reparametrization

Reparametrization also arises naturally in geometric analysis and operator theory, particularly in the paper of Sturm–Liouville and Hill-type equations on smooth normed planes. In this setting, eigenfunctions of the cycloid equation

$u''(t) + \lambda \cdot \omega(\varphi(t), \varphi'(t)) u(t) = 0$

are used to reparametrize the Minkowski unit circle, with special conditions under which the reparametrization can be induced by nontrivial eigenvalues $\lambda \ne 1$ (Balestro et al., 2018). The critical insight is that such reparametrizations are possible for $\lambda \ne 1$ only if the geometry is Euclidean (i.e., if the associated weight $f(t) = \omega(\varphi, \varphi')$ is constant), reflecting a deep correspondence between analytic structure of the spectrum, reparametrization class, and underlying geometric symmetry.

In numerical PDEs and spectral approximation, domain reparametrization (e.g., transforming $x = \varphi(t)$ on $[0,1]$ to $[a,b]$ ) fundamentally alters the spectral distribution of discretized operators (Noureddine et al., 14 Oct 2025). Using generalized locally Toeplitz (GLT) theory, the spectral (eigenvalue) symbol $\, \omega_\varphi^p(x,\theta) = e_p(\theta)/[\varphi'(x)]^2\,$ for isogeometric Galerkin Laplacians encodes the reparametrization’s effect, controlling the limiting distribution and ordering of eigenfrequencies. This enables the derivation of asymptotic Weyl-type laws, two-scale eigenfrequency estimates, and spectrum “ordering” results that directly reflect the analytic properties and convexity/concavity of $\Psi_\varphi^p(y)$ , the cumulative distribution function of the symbol.

4. Algorithmic and Filtering-Based Reparametrization

Eigenvalue reparametrization can be exploited in iterative or filtering-based algorithms to target specific eigenvalues. The “filtered power” method (Sudiarta et al., 2022) replaces the classic power operator $A$ by a parametric filter $F(A,\mu)=A\exp(-A/\mu)$ , so that the function $f(\lambda;\mu) = \lambda e^{-\lambda/\mu}$ peaks selectively at $\lambda\approx\mu$ . By scanning or sweeping the parameter $\mu$ across the spectrum, the method can recover all eigenpairs without explicit deflation. The convergence rate and stability are dictated by the separation of filter maxima and the proximity of $\mu$ to target eigenvalues, with spectral proximity supporting rapid isolation and extraction of individual modes.

In quantum linear algebra and quantum simulation, polynomial and functional eigenvalue reparametrization via Chebyshev or Faber polynomial expansions enables the implementation of the Quantum EigenValue Transformation (QEVT) framework (Low et al., 11 Jan 2024). Arbitrary polynomial (or analytic function) transforms can be applied to all eigenvalues of block-encoded non-normal matrices, facilitating quantum linear system solvers, ground state preparation, and the simulation of non-Hermitian dynamics. Faber polynomials extend Chebyshev polynomial reparametrizations to general compact regions in the complex plane, with efficient quantum circuit generation of their coefficients. Eigenvalue estimation (QEVE) leveraging these reparametrizations achieves Heisenberg-limited scaling in accuracy for diagonalizable operators.

5. Structural and Analytical Reparametrizations in Special Matrix Families

In optimal control contexts, matrix families admitting displacement structure allow highly explicit reparametrization of eigenvalues in terms of monotone transcendental equations (Peters et al., 2018). For the class $A(\theta)=U(\theta)L(\theta)$ , with structural parameters $\theta_i$ , one establishes that all eigenvalues are roots of

$f(\lambda;\theta) = \arctan\left(\frac{1}{\sqrt{4\lambda-1}}\right) + 2\sum_{i=1}^n \arctan\left(\frac{\theta_i}{\sqrt{4\lambda-1}}\right) = (2k-1)\frac{\pi}{2}$

for $k=1,\ldots,n$ . This reparametrization transforms spectral analysis into efficient, robust root-finding with monotonicity and explicit parametric sensitivity, facilitating parameter design, bounds, and model-order reduction. Closed-form solutions are available for symmetric or degenerate parameter settings.

6. Continuity, Path-Following, and Perturbation Theory

A rigorous understanding of eigenvalue reparametrization under continuous or analytic dependence of a matrix on one or more parameters is furnished by path theory and spectral perturbation results (Jankowski et al., 2019). Given a continuous family $C_\alpha$ of matrices, one may construct continuous or piecewise-analytic eigenpaths $\gamma_j(\alpha)$ satisfying

$\{\gamma_j(\alpha)\}_{j=1}^n = \mathrm{Spec}\,C_\alpha$

globally for $\alpha\in[0,1]$ , with explicit control over pairings, branch crossings, and singular ambiguities (points where eigenvalues coalesce). Stability theorems guarantee that small perturbations of the matrix path induce small perturbations in eigenvalue paths, and techniques exist to remove ambiguities by perturbed path construction. These results extend directly to monic polynomial families via the companion matrix construction.

7. Parametric and Geometric Effects on Eigenfrequency Distributions

In discretized operator theory, domain reparametrization induces nontrivial and quantifiable changes in eigenfrequency distributions, as formalized by the spectral symbol within the GLT framework (Noureddine et al., 14 Oct 2025). The limiting eigenvalue distribution is given by push-forward of the symbol, and the cumulative function $\Psi_\varphi^p(y)$ controls the ordering and spacing of eigenfrequencies. Convexity or concavity of $\Psi_\varphi^p$ (reflecting stretching/compressing the domain) shifts the packing of eigenvalues, with implications for observability and control in wave propagation and spectral method design. Asymptotic linearity near zero yields two-scale Weyl-type estimates for low-frequency modes, directly connecting eigenvalue growth to reparametrization properties.

The spectrum of eigenvalue reparametrization approaches encompasses analytic, structural, geometric, algorithmic, and quantum frameworks, unified by the systematic alteration of spectral parameterization to achieve tractable analysis, computational efficiency, structural understanding, or geometric insight. Contemporary research continues to extend these paradigms across operator theory, applied mathematics, quantum algorithms, and computational physics, leveraging reparametrization as a foundational mechanism for extracting, controlling, and interpreting the spectral properties of complex systems (Noureddine et al., 14 Oct 2025, Ringh et al., 2019, Mach et al., 2023, Low et al., 11 Jan 2024, Sudiarta et al., 2022, Jankowski et al., 2019, Peters et al., 2018, Balestro et al., 2018).