---
title: Nonlinear Matrix Eigenvalue Problem
url: https://www.emergentmind.com/topics/nonlinear-matrix-eigenvalue-problem
type: topic
---

# Nonlinear Matrix Eigenvalue Problem

A nonlinear matrix eigenvalue problem (NMEP) is the task of computing nontrivial vectors $x \in \mathbb{C}^n$ and associated scalars $\lambda \in \mathbb{C}$ such that $T(\lambda)x = 0$, where $T: \Omega \to \mathbb{C}^{n\times n}$ is a holomorphic (often analytic, rational, or meromorphic) matrix-valued function defined on some domain $\Omega \subset \mathbb{C}$. NMEPs generalize the linear matrix eigenproblem by permitting arbitrary (not necessarily linear or polynomial) dependence on the spectral parameter $\lambda$, and arise in a wide range of applications including quantum mechanics, photonics, delay-differential equations, structural vibrations, and control theory. Their mathematical and computational analysis combines complex operator theory, nonlinear approximation, numerical linear algebra, and high-performance algorithms.

## 1. Mathematical Foundations

The canonical NMEP is
\[
T(\lambda)x = 0, \quad x \neq 0,
\]
where typically $T(\lambda)$ is analytic on an open subset $\Omega \subset \mathbb{C}$. The spectrum $\Lambda(T) = \{ \lambda \in \Omega \mid \det T(\lambda) = 0 \}$ consists of isolated points (possibly accumulating at the boundary or at infinity), each counted with algebraic multiplicity determined by the order of vanishing of $\det T$ at $\lambda$. Keldysh's theorem extends the structure theory of the spectrum: the matrix function $T(\lambda)^{-1}$, viewed as a meromorphic function in the resolvent set, admits a partial fraction expansion whose singular terms encode the right and left eigenvectors and Jordan structure of $T$ [1003.1580].

In the context of infinite-dimensional spaces (e.g., operator-valued NMEPs representing PDEs or delay equations), $T(\lambda)$ becomes a family of closed, Fredholm operators acting on a Hilbert space, and the discrete eigenvalues correspond to isolated singularities of $T(\lambda)^{-1}$. The pseudospectrum, and related localization theorems, generalize to the nonlinear setting, and play a crucial role in spectral verification and error analysis [1303.4668, 2305.01691].

## 2. Classification and Model Problems

NMEPs naturally stratify according to the analytic structure of $T(\lambda)$ and the source of nonlinearity:
- **Polynomial eigenvalue problems (PEPs)**: $T(\lambda) = \sum_{k=0}^d \lambda^k A_k$, which admit companion linearizations and are a classical subject.
- **Rational eigenvalue problems (REPs)**: $T(\lambda) = P(\lambda) + \sum_{i=1}^m (C_i - \lambda D_i) r_i(\lambda)$, with $r_i$ rational [1801.08622, 1901.01188, 2006.14189].
- **Transcendental-type NMEPs**: $T(\lambda)$ contains terms such as exponentials or roots, e.g., $e^{-\sqrt{\lambda}}$ [1801.08622].
- **Eigenvector-nonlinear NMEPs (NEPv)**: The matrix depends explicitly on the eigenvector, $T(x, \lambda)x = 0$ [1212.0417, 2002.12805, 2510.02900].
- **Infinite-dimensional/Operator NMEPs**: $T(\lambda): D(T)\rightarrow H$, $H$ Hilbert, often arising from spatially discretized or semi-discretized PDEs [2305.01691].

Many physically important cases reduce to block or Schur complements of more general NMEPs, and some nonlinearities enter via domain decomposition or parametric model-reduction schemes [1907.00913].

## 3. Analytical and Perturbation Theory

The spectral and perturbation theory for NMEPs is more intricate than in the linear case. The nonlinear generalization of the Bauer–Fike theorem provides a sharp bound for eigenvalue perturbations:
\[
| \tilde{\lambda} - \lambda_0 | \leq \kappa(T, \lambda_0) \cdot \| T(\lambda_0) - \tilde{T}(\tilde{\lambda}) \| + O(\|T-\tilde{T}\|^2),
\]
where $\kappa(T, \lambda_0) = \|T'(\lambda_0)^{-1}\|$ is a nonlinear condition number [2409.11098, 1303.4668]. Keldysh’s theorem justifies representing $T(z)^{-1}$ as a Laurent expansion near simple or semi-simple eigenvalues, enabling the definition of spectral projectors and invariance subspaces via contour integration.

For eigenvector-nonlinear NMEPs, the appropriate notion of conditioning and convergence is subtler. Local convergence rates for iterative methods can often be derived from the linearization of the map $x\mapsto T(x, \lambda)x$ at a solution, and closely mirror those of the associated Jacobian [1212.0417, 2002.12805].

Localization theorems generalizing Gershgorin’s disks and the pseudospectral inclusion for analytically parameterized matrix functions enable a priori demarcation of regions containing eigenvalues and rigorous eigenvalue counting [1303.4668].

## 4. Numerical Methods

A variety of algorithms have been developed for NMEPs, each targeting different structures and scales. The primary classes are:

**Contour-integral and moment-based methods**:  
- **Beyn's integral method** computes all eigenvalues inside a contour $\Gamma$ by contour integrating resolvent-moment matrices, reducing the problem to a small projected linear eigenproblem whose spectrum matches the targeted eigenvalues [1003.1580, 2312.13117, 2012.14979].
- **Riesz-projection-based methods** use scalar functionals of the resolvent to form moment equations for the eigenvalues within a contour, allowing prioritization of physically relevant (coupled) modes without linearization [1811.11624].

**Krylov–Arnoldi–Lanczos extensions**:  
- **Infinite bi-Lanczos** and variants construct Krylov subspaces for infinite-dimensional linearizations of $T(\lambda)$ to simultaneously approximate right and left eigenvectors [1607.03454].
- **Rational/truncated Krylov methods** (e.g., CORK, TS-CORK) exploit rational approximations and compact linearizations, achieving efficient shift-and-invert strategies for large-scale REPs [1801.08622, 2006.14189].

**Quasi-Newton, Newton, and Broyden-type methods**:  
- **Newton-type iteration** for $(\lambda, x)$ seeks stationary points of the system $F(\lambda, x) = (T(\lambda)x, c^H x - 1)$, using Block-Newton or quasi-Newton linearizations in each iteration [1702.08492]. Several classical methods can be interpreted as quasi-Newton schemes, such as residual inverse iteration and the method of successive linear problems, each with quantifiable convergence rates based on Keldysh-theoretic resolvent expansions.
- **Broyden's method** adapts the secant-based approach to the structured augmented NMEP system, achieving local superlinear convergence for simple eigenvalues and supporting robust deflation strategies [1802.07322].

**Rational approximation and linearization**:  
- **AAA rational approximations** and set-valued adaptations construct rational surrogates matching the nonlinear terms in $T(\lambda)$, enabling efficient CORK-style companion linearizations and resulting in pencils of minimal dimension that embed the true spectrum [1801.08622, 1901.01188].

**Multigrid and multilevel schemes**:  
- **Multilevel correction methods** decompose the nonlinear eigenproblem into a sequence of linear boundary-value problems on multigrid hierarchies and small-dimensional nonlinear corrections, reducing overall computational work to that of a linear problem of the same scale [1502.04657, 1405.7158].

**NEPv-focused algorithms**:  
- For eigenvector-dependent problems, implicit Newton schemes and structure-exploiting inverse iterations form the prevailing methodology. Quadratic convergence is achievable for Newton-type updates (proper Jacobian handling) while self-consistent field (SCF) iterations offer linear, robust progress [1212.0417, 2002.12805].

A summary table of selected method classes, key ideas, and cost structure:

| Methodology      | Core Principle              | Complexity per Iteration                  |
|------------------|----------------------------|-------------------------------------------|
| Contour methods  | Resolvent moment integrals  | $O(N \cdot l)$ solves, $N \sim 30$–$50$   |
| Rational Krylov  | Linearization via AAA/CORK  | $O(n (\text{block size})^2)$ per step     |
| Quasi-Newton     | Fixed-shift Jacobian solve  | 1 LU + 1 mat-vec or eigenproblem          |
| Multigrid FMG    | Nested mesh linear solves   | $O(\mathrm{total \; dof})$ per full solve |
| Eigenvector NEP  | Eigendecomposition of Jacobian at $v$ | $O(n^3)$ |  

## 5. Implementation Issues and Spectral Verification

Critical challenges arise in avoiding artifacts due to discretization of operator-valued NMEPs:
- **Spectral pollution**: Discretized spectra may exhibit spurious modes, poor convergence, or missing eigenvalues, especially when dealing with unbounded operators or non-compact resolvents [2305.01691]. The infBeyn method circumvents this by constructing the contour-integral reduced pencil directly in the operator setting and only discretizing after projection.
- **Pseudospectra and stability**: Localizing eigenvalues robustly is achievable via nonlinear analogues of the pseudospectrum, featured in both localization theorems and the explicit error control in infinite-dimensional settings [1303.4668, 2305.01691].
- **Deflation and multiple eigenvalues**: Augmented or block-deflated systems permit systematic computation of several (or all) modes, given suitable orthogonality conditions [1802.07322, 1003.1580].

## 6. Practical Applications and Example Problems

NMEPs are central in several domains:
- **Quantum mechanics**: Nonselfadjoint boundary value problems (e.g., resonances in open quantum wells or nanophotonic structures) frequently reduce to rational or transcendental NMEPs [1811.11624].
- **Photonic and nanophotonic systems**: Maxwell or Helmholtz equations with dispersive, frequency-dependent permittivities yield large-scale rational NMEPs after finite element discretization [1801.08622, 1811.11624].
- **Delay-differential and control systems**: Time-delay in feedback leads to matrix-valued functions involving exponentials or other transcendental functions [1607.03454, 2312.13117].
- **Gross–Pitaevskii and nonlinear Schrödinger equations**: NEPv with quadratic or rational dependence, especially in the presence of normalization and symmetry properties, appear in Bose–Einstein condensate modeling [1212.0417, 2002.12805, 2510.02900].

Designed numerical experiments confirm that the most advanced methods recover all eigenvalues inside a given region to machine precision, up to sizes $n \sim 10^5$–$10^6$, and efficiently separate physical from spurious modes in physics-informed settings [1003.1580, 1811.11624, 1801.08622, 2305.01691].

## 7. Advanced Topics and Recent Developments

Several frontiers are being actively pursued:
- **Subspace and derivative-interpolatory frameworks** combine model reduction concepts with Hermite interpolation to achieve quadratic (or higher) convergence for select eigenvalues, especially in large-scale rational NMEPs [2006.14189].
- **Multiparameter linearizations** for NEPv with rational eigenvector dependencies realize spectral inclusion via structured operator determinants and exploit tailored Krylov–Arnoldi filtering to mitigate the swamping of spurious solutions inherent in generic linearizations [2510.02900].
- **Adaptive and parallel contour-integration**: Domain decomposition and parallelized quadrature permit scaling to complex eigenvalue landscapes without a priori enumeration of eigenvalue count or distribution [2312.13117].
- **Sensitivity, conditioning, and stability**: Recent advances extend the Bauer–Fike theorem explicitly to nonlinear and operator-valued settings, providing rigorous frameworks for step-size control in iterative solvers, verification of computed eigenvalues, and parametric tracking in bifurcation scenarios [2409.11098].
- **Automatic rational approximation**: Black-box frameworks (e.g., AAA-CORK) automate pole/zero assignment for linearization while achieving optimal linearization sizes and negligible user overhead [1801.08622].

Open research challenges include effective iterative solvers for massive-scale operator-NMEPs, universal spectral localization tools for highly defective spectra, structure-preserving generalizations for non-Hermitian or indefinite problems, and integration with parameter continuation and bifurcation tracking mechanisms.

---
**References (arXiv IDs):**

- [1003.1580], [1607.03454], [1811.11624], [1502.04657], [1303.4668], [1801.08622], [2305.01691], [1212.0417], [2312.13117], [1907.00913], [1802.07322], [1405.7158], [1702.08492], [2012.14979], [2409.11098], [2006.14189], [2002.12805], [2510.02900]

Source: https://www.emergentmind.com/topics/nonlinear-matrix-eigenvalue-problem