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arxiv: 1907.10972 · v1 · pith:JZ6OEXG7new · submitted 2019-07-25 · 🧮 math.NA · cs.NA

Local Linearizations of Rational Matrices with Application to Rational Approximations of Nonlinear Eigenvalue Problems

Froil\'an M. Dopico , Silvia Marcaida , Mar\'ia C. Quintana , Paul Van Dooren This is my paper

Pith reviewed 2026-05-24 16:17 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords local linearizationsrational matricesmatrix pencilszeros and polesnonlinear eigenvalue problemsrational approximationalgebraically closed fields
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The pith

A unified definition of local linearizations for rational matrices recovers every prior pencil as a special case while preserving zero-pole structure in arbitrary subsets of any algebraically closed field and at infinity.

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

The paper defines local linearizations of rational matrices so that each associated matrix pencil keeps the exact locations of the rational matrix's zeros and poles inside chosen subsets of an algebraically closed field, including the point at infinity. The same definition is shown to contain all earlier pencil constructions from the 1970s onward as particular instances. Because every historical pencil now sits inside one framework, the theory accounts for why those pencils correctly computed zeros, poles, and eigenvalues of rational matrices, including the pencils that arise when nonlinear eigenvalue problems are solved by rational approximation.

Core claim

A definition of local linearization is introduced that produces matrix pencils associated to a given rational matrix; these pencils preserve the zero-pole structure of the rational matrix inside any prescribed subset of an algebraically closed field together with the point at infinity, and every pencil previously studied in the literature arises as a special case of this definition.

What carries the argument

The definition of a local linearization, which is a matrix pencil tied to a rational matrix such that the pencil and the rational matrix share the same zeros and poles inside any chosen subset of an algebraically closed field and at infinity.

If this is right

  • Every pencil used since the 1970s for computing zeros and poles of rational matrices is now a special case inside one theory.
  • Pencils arising from rational approximations of nonlinear eigenvalue problems inherit the zero-pole preservation property automatically.
  • The same pencil can be analyzed for structure preservation at infinity and inside any finite subset of the field without switching definitions.
  • Numerical algorithms that rely on these pencils gain a uniform justification for their correctness across different choices of linearization.

Where Pith is reading between the lines

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

  • Implementations of nonlinear eigenvalue solvers could switch among historically different pencils inside a single code base without loss of theoretical guarantees.
  • The framework suggests a systematic way to construct new pencils that preserve additional structure, such as symmetry or palindromicity, on chosen subsets.
  • Questions about the minimal size or conditioning of pencils for a given rational matrix can now be posed uniformly rather than case by case.

Load-bearing premise

It is possible to write one definition of local linearization that simultaneously preserves zero-pole locations for every possible subset of an algebraically closed field and recovers every earlier pencil construction as a special case.

What would settle it

Existence of a pencil from the 1970-2019 literature that cannot be obtained as a special case of the new definition, or a rational matrix for which a local linearization fails to match the zeros or poles of the original matrix inside some subset.

read the original abstract

This paper presents a definition for local linearizations of rational matrices and studies their properties. This definition allows us to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and poles in subsets of any algebraically closed field and also at infinity. Moreover, such definition includes, as particular cases, other definitions that have been used previously in the literature. In this way, this new theory of local linearizations captures and explains rigorously the properties of all the different pencils that have been used from the 1970's until 2019 for computing zeros, poles and eigenvalues of rational matrices. Particular attention is paid to those pencils that have appeared recently in the numerical solution of nonlinear eigenvalue problems through rational approximation.

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 introduces a definition of local linearizations for rational matrices. This definition constructs associated matrix pencils that preserve the zero-pole structure of the rational matrix on arbitrary subsets of any algebraically closed field (including at infinity) and recovers as special cases all previously used pencils in the literature from the 1970s onward. The framework is applied to pencils arising in rational approximations for nonlinear eigenvalue problems.

Significance. If the unification holds, the work supplies a single rigorous foundation that explains the zero-pole preservation properties of disparate pencil constructions used for rational and nonlinear eigenproblems. This could standardize analysis and algorithm design in numerical linear algebra, particularly where rational approximations are employed.

major comments (2)
  1. [§3] §3, Definition 3.2: the statement that the new definition recovers all prior pencils as special cases requires an explicit bijection or parameter choice for each historical construction (e.g., the pencils of Gohberg et al. and the recent rational-approximation pencils); without this mapping the unification claim is not fully substantiated.
  2. [§4.2] §4.2, Theorem 4.5: the proof that zeros and poles are preserved on an arbitrary subset S assumes that the local linearization is regular on S; the argument does not address the case when the subset intersects the pole set of the original rational matrix, which is load-bearing for the general claim.
minor comments (2)
  1. [§2] Notation for the field K and its algebraic closure is introduced late; moving the definitions to §2 would improve readability.
  2. Figure 1 caption does not state the field or the subset S used in the example.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Both points identify opportunities to strengthen the explicitness of the unification and the generality of the main preservation result. We address each below and will incorporate the suggested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: §3, Definition 3.2: the statement that the new definition recovers all prior pencils as special cases requires an explicit bijection or parameter choice for each historical construction (e.g., the pencils of Gohberg et al. and the recent rational-approximation pencils); without this mapping the unification claim is not fully substantiated.

    Authors: We agree that the unification claim is more convincingly substantiated when explicit parameter choices are supplied. In the revision we will add a dedicated subsection (or short appendix) that lists the precise choices of the local linearization parameters recovering the classical pencils of Gohberg, Lancaster and Rodman as well as the recent pencils arising from rational approximations of nonlinear eigenvalue problems. These mappings will be stated as corollaries to Definition 3.2. revision: yes

  2. Referee: §4.2, Theorem 4.5: the proof that zeros and poles are preserved on an arbitrary subset S assumes that the local linearization is regular on S; the argument does not address the case when the subset intersects the pole set of the original rational matrix, which is load-bearing for the general claim.

    Authors: The referee correctly identifies that the present proof of Theorem 4.5 is written under the standing assumption that the linearization is regular on S and does not separately treat the case in which S meets the pole set of the rational matrix. We will revise the statement of Theorem 4.5 to make the regularity assumption explicit and extend the proof to cover the intersecting-pole case, either by a direct argument or by reducing it to the regular situation via a suitable deflation at the poles. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a new definition of local linearizations for rational matrices that is shown to preserve zero-pole structure on arbitrary subsets of an algebraically closed field (including infinity) while recovering prior pencils as special cases. This unification follows directly from the definition and its derived properties rather than reducing to any fitted parameter, self-citation chain, or renaming of known results. No load-bearing step in the derivation is equivalent to its inputs by construction, and the central claim remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Ledger constructed from abstract only; full paper may introduce additional technical assumptions on fields or matrix degrees.

axioms (1)
  • standard math Rational matrices are considered over algebraically closed fields
    Invoked when discussing preservation of zeros and poles in arbitrary subsets of the field.
invented entities (1)
  • Local linearization no independent evidence
    purpose: Matrix pencil preserving zero-pole structure of a rational matrix locally
    Central new concept introduced to unify prior pencil constructions

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