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arxiv: 1907.03213 · v1 · pith:DYEFZ5BOnew · submitted 2019-07-07 · 🧮 math.RT

A constructive proof of Pokrzywa's theorem about perturbations of matrix pencils

Vyacheslav Futorny , Tetiana Klymchuk , Vladimir V. Sergeichuk , Nadya Shvai This is my paper

Pith reviewed 2026-05-25 01:42 UTC · model grok-4.3

classification 🧮 math.RT
keywords matrix pencilsKronecker formperturbationsPokrzywa theoremminiversal deformationstrict equivalenceorbit closure
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The pith

A direct constructive proof establishes Pokrzywa's description of nearby Kronecker forms for any matrix pencil.

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

The authors provide a new proof of Pokrzywa's theorem that describes all Kronecker canonical pencils appearing in every neighborhood of a given pencil. Their approach reduces the general case to the similarity problem for single matrices and to the special case of pencils that are direct sums of exactly two indecomposable Kronecker blocks. For those special pencils they compute explicitly which Kronecker forms occur nearby by working within a miniversal deformation. This calculation is sufficient for the general case because every nearby pencil can be transformed into the deformation by smooth strict equivalence.

Core claim

Pokrzywa's theorem characterizes the closure of the orbit of a Kronecker canonical pencil A−λB as the set of all pencils K−λL whose invariants satisfy a system of inequalities coming from the invariants of A−λB. The proof is obtained by handling the matrix similarity case separately, then treating pencils that decompose as direct sums of two indecomposables, and finally listing the Kronecker forms of all pencils inside the miniversal deformation of such a sum.

What carries the argument

The miniversal deformation of a pencil that is the direct sum of two indecomposable Kronecker canonical pencils; it serves as a local parameter space in which all nearby pencils up to smooth strict equivalence are represented and their canonical forms can be read off directly.

If this is right

  • The orbit closure consists exactly of the pencils whose invariants obey the stated inequalities.
  • All pencils sufficiently close to a given pencil reduce via smooth strict equivalence to the miniversal deformation.
  • The Kronecker forms in any neighborhood are completely determined by the forms found inside the miniversal deformation of the two-block sum.

Where Pith is reading between the lines

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

  • The reduction to two-block sums suggests that the perturbation behavior of more complicated pencils is built from interactions between pairs of indecomposable components.
  • Similar direct calculations might be feasible for other canonical form problems under equivalence or similarity.

Load-bearing premise

Every pencil in a neighborhood of P−λQ can be reduced to one inside its miniversal deformation by smooth strict equivalence transformations.

What would settle it

Discovery of a pencil arbitrarily close to P−λQ whose Kronecker canonical form does not appear among the forms computed inside the miniversal deformation of P−λQ.

read the original abstract

Our purpose is to give new proofs of several known results about perturbations of matrix pencils. Andrzej Pokrzywa (1986) described the closure of orbit of a Kronecker canonical pencil $A-\lambda B$ in terms of inequalities with pencil invariants. In more detail, Pokrzywa described all Kronecker canonical pencils $K-\lambda L$ such that each neighborhood of $A-\lambda B$ contains a pencil whose Kronecker canonical form is $K-\lambda L$. Another solution of this problem was given by Klaus Bongartz (1996) by methods of representation theory. We give a direct and constructive proof of Pokrzywa's theorem. We reduce its proof to the cases of matrices under similarity and of matrix pencils $P-\lambda Q$ that are direct sums of two indecomposable Kronecker canonical pencils. We calculate the Kronecker forms of all pencils in a neighborhood of such a pencil $P-\lambda Q$. In fact, we calculate the Kronecker forms of only those pencils that belong to a miniversal deformation of $P-\lambda Q$, which is sufficient since all pencils in a neighborhood of $P-\lambda Q$ are reduced to them by smooth strict equivalence transformations.

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

1 major / 1 minor

Summary. The manuscript presents a constructive proof of Pokrzywa's theorem describing the closure of the orbit of a Kronecker canonical pencil A−λB under small perturbations. It reduces the general case to the already-solved matrix similarity problem and to pencils P−λQ that are direct sums of exactly two indecomposable Kronecker blocks, then computes the Kronecker forms of all pencils inside a miniversal deformation of such a P−λQ, asserting that this determines the full set of admissible degenerations because every nearby pencil reduces to the deformation via smooth strict equivalence transformations.

Significance. If the reduction step holds and the explicit calculations are complete and gap-free, the paper supplies a direct, elementary proof that complements Bongartz's representation-theoretic approach and makes the admissible degenerations computable by hand for the two-block case. The constructive character and the reduction strategy are genuine strengths.

major comments (1)
  1. [Abstract] Abstract (final sentence): the claim that 'all pencils in a neighborhood of P−λQ are reduced to them by smooth strict equivalence transformations' is load-bearing for the completeness of the orbit-closure list. The manuscript must supply an explicit argument (or a precise reference) showing that the miniversal slice intersects every nearby orbit and that the required equivalences remain smooth; without this, forms lying outside the computed family could still appear arbitrarily close to P−λQ.
minor comments (1)
  1. Clarify the precise statement of the reduction to the matrix-similarity case (which section or theorem is invoked) so that the logical dependence is fully transparent.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point where the manuscript's reliance on standard deformation theory could be made more explicit. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final sentence): the claim that 'all pencils in a neighborhood of P−λQ are reduced to them by smooth strict equivalence transformations' is load-bearing for the completeness of the orbit-closure list. The manuscript must supply an explicit argument (or a precise reference) showing that the miniversal slice intersects every nearby orbit and that the required equivalences remain smooth; without this, forms lying outside the computed family could still appear arbitrarily close to P−λQ.

    Authors: We agree that the completeness of the orbit-closure description rests on this property of the miniversal deformation. The claim follows from the general theory of versal and miniversal deformations of matrix pencils under strict equivalence (the slice is transverse to the orbit and the equivalence transformations can be chosen smoothly in a neighborhood). However, the manuscript states the fact without an explicit reference or short derivation. We will therefore insert a concise paragraph (or footnote) in the revised version, citing the relevant background on miniversal deformations of Kronecker pencils, to make the reduction fully self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation reduces to external cases with independent justification

full rationale

The paper's proof reduces Pokrzywa's theorem to the matrix similarity case and to direct sums of two indecomposable Kronecker blocks, then explicitly computes Kronecker forms inside a miniversal deformation of such a pencil. The sufficiency of this computation is justified by the general property that all nearby pencils reduce to the miniversal deformation via smooth strict equivalence transformations, presented as a standard fact rather than derived from the target result. No quoted step shows a self-definitional reduction, a fitted parameter renamed as prediction, or a load-bearing self-citation whose content depends on the present theorem. The similarity case and two-block calculations are treated as independently solvable, making the overall chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are visible. The reduction steps implicitly rely on standard facts about Kronecker forms and miniversal deformations that are treated as background.

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