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arxiv: 2605.21013 · v1 · pith:YR6ZPATWnew · submitted 2026-05-20 · 🧮 math.NA · cs.NA

Rectangular Multispectral Perturbation Theory

Christof Vermeersch , Sarthak De , Bart De Moor This is my paper

Pith reviewed 2026-05-21 02:11 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords rectangular multiparameter eigenvalue problemperturbation theorybackward error analysiscondition numberspseudospectrumsystem identificationnumerical linear algebra
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The pith

Rectangular multiparameter eigenvalue problems support defined backward errors, condition numbers, and pseudospectra despite non-square structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops the first systematic perturbation theory for rectangular multispectral eigenvalue problems, where a single matrix equation uses rectangular coefficient matrices. It carries out norm-wise backward error analysis, defines condition numbers for eigenvalues and eigenvectors, and introduces pseudospectra while addressing computational issues with multiple parameters. The rectangular form creates specific hurdles, including a non-trivial left null space and mismatched dimensions between left and right eigenvectors, yet the work shows these can be handled to extend earlier square-case results. Numerical examples connect the new concepts, and an application in system identification suggests that globally optimal solutions in suitable optimization problems align with the best-conditioned eigenvalues.

Core claim

Rectangular multispectral perturbation theory extends prior square multiparameter results by supplying backward error measures, eigenvalue and eigenvector condition numbers, and pseudospectra for problems consisting of one rectangular matrix equation, with the rectangular shape handled through adjusted definitions that accommodate non-trivial left null spaces and unequal left and right eigenvector dimensions.

What carries the argument

The rectangular multiparameter eigenvalue problem, defined via a single matrix equation with rectangular coefficients, which carries the extension of perturbation concepts from square cases while managing non-trivial null spaces and dimension differences.

If this is right

  • Backward error analysis applies to rectangular cases to quantify how small changes affect the eigenvalue problem.
  • Condition numbers become available for both eigenvalues and eigenvectors under the rectangular formulation.
  • Pseudospectra can be computed for rectangular multispectral problems to visualize sensitivity regions.
  • In optimization problems admitting multiparameter reformulations, globally optimal solutions tend to match the best-conditioned eigenvalues.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework may allow practitioners to select or reformulate problems so that optima automatically land on numerically stable eigenvalues.
  • Similar rectangular extensions could apply to other linear algebra settings with mismatched matrix dimensions, such as certain generalized Sylvester equations.
  • Algorithms for computing these quantities will need to account for the extra computational overhead from multiple spectral parameters.

Load-bearing premise

The rectangular structure permits direct extension of backward error, condition number, and pseudospectrum definitions from the square multiparameter case despite non-trivial left null spaces and differing left/right eigenvector dimensions.

What would settle it

A concrete rectangular multiparameter example where the proposed backward error or condition number definitions fail to produce consistent sensitivity measures, or a system identification instance where the globally optimal solution is not among the best-conditioned eigenvalues.

Figures

Figures reproduced from arXiv: 2605.21013 by Bart De Moor, Christof Vermeersch, Sarthak De.

Figure 1
Figure 1. Figure 1: Real picture of the secular equations χ1(λ) = 0 ( ), χ2(λ) = 0 ( ), and χ3(λ) = 0 ( ) of the two-parameter eigenvalue problem in Example 2. The intersections correspond to the three eigenvalues. Next to the (right) eigenvector z ◦ associated to an eigenvalue λ ◦ , there also exist vectors that solve the matrix equation y HA(λ) = 0. Due to the rectangular matrix size of A(λ ◦ ), there are ℓ+m−ρ ◦ −1 vectors… view at source ↗
Figure 2
Figure 2. Figure 2: Real picture of the secular equations χ1(λ) = 0 ( ), χ2(λ) = 0 ( ), and χ3(λ) = 0 ( ) of the two-parameter eigenvalue problem in Example 5. The intersections correspond to the three eigenvalues. The numerical values are given in [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Visualization of the pseudospectra, Λϵ(A(λ)), of the two-parameter eigenvalue prob￾lems in both Example 2 and Example 5. The secular equations are also shown ( ). is now easy to see E0 as a perturbation to A0, resulting in Definition 7. To make it more complete, we can distribute E0 over the different coefficient matrices, ∆A0 = ∥A0∥2E0 γ ∗ , ∆Ai = sign(λi)∥Ai∥2E0 γ ∗ , i = 1, 2, . . . , m, proving the oth… view at source ↗
Figure 4
Figure 4. Figure 4: Secular equations and ϵ-pseudospectra of the different matrix polynomials in Exam￾ple 7. The spectrum of the rMEP A(λ) is obtained as the intersection of the spectra of the square submatrix polynomials Bi(λ), for i = 1, 2, 3. A similar relation holds for the ϵ-pseudospectra. The secular curves ( ) for λ ∗ (1) = (8.9620, 4.3879) and λ ∗ (2) = (0.4780, 0.2982) are locally al￾most parallel, while this is not … view at source ↗
Figure 5
Figure 5. Figure 5: Contour lines of the cost function and pseudospectrum of the corresponding rMEP for the system identification application in Example 8. The real stationary points are marked in both figures ( ). From the contour lines in Figure 5a, their type can be deducted, while the size of the pseudospectrum pockets in Figure 5b gives an idea of the conditioning of every eigenvalue that corresponds to a stationary poin… view at source ↗
Figure 6
Figure 6. Figure 6: Visualization of the third component of the left null space Y for a parameter t, which parametrizes the eigenvalue as λ = (t, t). In t = 1, the parametrization corresponds to one of the eigenvalue solutions, i.e., λ ◦ = (1, 1). Both the symbolically obtained polynomial vectors ( ) and the numerically obtained left null space vectors ( ) are shown. The symbolic expression for r(t) in (13) is normalized so t… view at source ↗
read the original abstract

We provide a first systematic treatment of so-called rectangular multispectral perturbation theory. With their paper from 2003, Hochstenbach and Plestenjak ["Backward Error, Condition Numbers, and Pseudospectra for the Multiparameter Eigenvalue Problem" in Linear Algebra and its Applications] extended perturbation theory from one-parameter eigenvalue problems to multiple spectral parameters. After two decades, we take it one step further and consider a different manifestation of the multiparameter eigenvalue problem that consists of one matrix equation with rectangular coefficient matrices. We perform a norm-wise backward error analysis, define condition numbers for both eigenvalues and eigenvectors, and introduce the pseudospectrum while also considering the computational implications of working with multiple spectral parameters. The rectangular shape hampers a direct application of the existing definitions and properties. For example, the left null space at a given eigenvalue is non-trivial and the dimensions of the left and right eigenvectors are different. Through numerical examples, we illustrate and link the different concepts from the perturbation theory. A system identification application seem to suggest that, in optimization-driven problems for which multiparameter reformulations exist, the globally optimal solutions tend to coincide with the best-conditioned eigenvalues.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper provides the first systematic treatment of rectangular multispectral perturbation theory by extending norm-wise backward error analysis, eigenvalue and eigenvector condition numbers, and pseudospectra from the square multiparameter case of Hochstenbach and Plestenjak to problems with rectangular coefficient matrices. It notes structural challenges including non-trivial left null spaces and mismatched left/right eigenvector dimensions, illustrates the concepts via numerical examples, and reports an observation from a system identification application that globally optimal solutions tend to coincide with best-conditioned eigenvalues.

Significance. If the rectangular extensions to backward error and condition numbers are shown to be rigorously justified, the work would establish a foundation for sensitivity analysis in non-square multiparameter eigenvalue problems and could inform optimization-driven applications such as system identification by linking conditioning to global optimality.

major comments (2)
  1. [Abstract] Abstract (paragraph on rectangular shape challenges): the claim that the rectangular structure permits a direct extension of backward error, condition numbers, and pseudospectra is load-bearing for the reliability of the condition numbers as sensitivity predictors in the system identification application; the manuscript must derive or verify that the new definitions properly incorporate the effects of the non-trivial left null space and differing eigenvector dimensions, rather than assuming the square-case formulas carry over unchanged.
  2. [System identification application] System identification application section: the observation that globally optimal solutions coincide with best-conditioned eigenvalues depends on the newly defined condition numbers being accurate predictors; without explicit error bounds, verification details, or analysis of how non-trivial left null spaces affect the effective conditioning (as flagged in the abstract), this coincidence risks being an artifact of the specific example rather than a general consequence of the theory.
minor comments (2)
  1. Clarify notation for rectangular matrices and left/right eigenvector dimensions throughout the definitions to avoid ambiguity when dimensions differ.
  2. Add explicit statements of the computational complexity or algorithmic implications when handling multiple spectral parameters in the rectangular setting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating where revisions will be made to strengthen the presentation and justification.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on rectangular shape challenges): the claim that the rectangular structure permits a direct extension of backward error, condition numbers, and pseudospectra is load-bearing for the reliability of the condition numbers as sensitivity predictors in the system identification application; the manuscript must derive or verify that the new definitions properly incorporate the effects of the non-trivial left null space and differing eigenvector dimensions, rather than assuming the square-case formulas carry over unchanged.

    Authors: We agree that the reliability of the new condition numbers hinges on proper accounting for rectangular structure. The manuscript does not assume unchanged square-case formulas; instead, the derivations in Sections 3 and 4 explicitly modify the residual and norm definitions to handle the non-trivial left null space (via orthogonal projections onto the complement of the left kernel) and the dimension mismatch (via rectangular singular-value-based scalings for left and right eigenvectors). To address the referee's concern directly, we will revise the abstract for clarity and insert a new subsection (e.g., 3.3) that walks through the derivation steps, shows the explicit differences from the square case of Hochstenbach and Plestenjak, and verifies the adjustments with a small analytic example. revision: yes

  2. Referee: [System identification application] System identification application section: the observation that globally optimal solutions coincide with best-conditioned eigenvalues depends on the newly defined condition numbers being accurate predictors; without explicit error bounds, verification details, or analysis of how non-trivial left null spaces affect the effective conditioning (as flagged in the abstract), this coincidence risks being an artifact of the specific example rather than a general consequence of the theory.

    Authors: We acknowledge that the reported coincidence is currently an empirical observation from the numerical examples and would benefit from stronger supporting analysis to elevate it beyond a possible artifact. The manuscript links the observation to the theory via computed condition numbers but does not supply explicit a-posteriori error bounds or a detailed study of null-space effects on effective conditioning. In revision we will expand the application section with (i) tabulated residuals and condition numbers for the reported solutions, (ii) a short discussion of how the left null-space dimension modulates the condition-number values, and (iii) a brief conjecture, supported by the existing theory, on why globally optimal points tend to be well-conditioned. An additional small-scale example will be included if space allows. revision: yes

Circularity Check

0 steps flagged

No significant circularity; definitions and claims are independently extended from prior external work

full rationale

The paper extends norm-wise backward error, condition numbers, and pseudospectra from the square multiparameter case (Hochstenbach and Plestenjak 2003) to the rectangular setting by direct adaptation, acknowledging structural differences such as non-trivial left null spaces. No equations reduce new quantities to fitted parameters by construction, and the central empirical observation (optimal solutions coinciding with best-conditioned eigenvalues in system identification) is presented as a numerical suggestion rather than a derived prediction. The cited 2003 work is external and independent; no self-citation chains or ansatzes are load-bearing for the core definitions or claims. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; all technical details on definitions and extensions are absent.

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Reference graph

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