Submatrices with the best-bounded inverses: the equality criterion for $\mathbb{R}^{n \times 2}$

Yuri Nesterenko

arxiv: 2604.14050 · v2 · submitted 2026-04-15 · 🧮 math.NA · cs.NA

Submatrices with the best-bounded inverses: the equality criterion for mathbb{R}^(n times 2)

Yuri Nesterenko This is my paper

Pith reviewed 2026-05-10 12:13 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords submatricesbounded inversesequality criteriontwo-column matricesGoreinov-Tyrtyshnikov-Zamarashkin hypothesisnumerical linear algebramaximal volume

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The pith

For real two-column matrices the submatrices with the smallest-bounded inverses satisfy a precise equality criterion.

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

A hypothesis posed by Goreinov, Tyrtyshnikov and Zamarashkin asserts that every matrix possesses a submatrix whose inverse is bounded in a manner that is optimal among all possible choices of the same size. Sengupta and Pautov recently established that the hypothesis holds when the matrix has exactly two columns. The present work supplies the equality criterion that characterizes precisely which submatrix attains this optimal bound. A reader would care because the criterion converts an existence statement into an explicit test that can be checked directly on candidate pairs of rows.

Core claim

Complementing the positive solution already obtained for two-column matrices, the equality criterion states the necessary and sufficient condition under which a chosen 2-by-2 submatrix realizes the minimal possible inverse bound.

What carries the argument

The equality criterion for two-column matrices, which supplies the exact algebraic condition identifying the rows that produce the submatrix with the best-bounded inverse.

If this is right

For any concrete n-by-2 matrix one can test candidate row pairs directly against the equality condition instead of computing inverse norms for every possible pair.
The optimal submatrix is uniquely determined (or the set of optimal submatrices is fully described) once the equality condition is verified.
Verification of optimality becomes a finite algebraic check rather than a search over binomial(n,2) candidates.

Where Pith is reading between the lines

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

The same style of equality test might be sought for matrices with three or more columns once the general hypothesis is settled.
Numerical procedures that repeatedly select rows for low-rank approximations could insert the criterion as a cheap filter in the two-column setting.
The criterion may translate into a geometric statement about the angles or lengths of the two columns that makes the optimality visible without computing matrix inverses.

Load-bearing premise

The positive solution already given by Sengupta and Pautov for the two-column case is correct and the equality case follows from it without further restrictions on the entries.

What would settle it

Any explicit n-by-2 matrix together with a 2-row submatrix for which the proposed equality criterion fails to hold while that submatrix nevertheless possesses the smallest inverse bound among all 2-row choices.

read the original abstract

The long-standing hypothesis formulated by Goreinov, Tyrtyshnikov and Zamarashkin \cite{GTZ1997} has recently been solved positively by Sengupta and Pautov \cite{SP2026} in the case of two-column matrices. In this paper, we complement their elegant proof with the equality criterion.

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.

Desk Editor's Note private letter to a colleague

This short note adds the equality criterion for the n by 2 case of the GTZ hypothesis, but stands or falls with the correctness of the Sengupta-Pautov proof it complements.

read the letter

The main takeaway is that this paper supplies the missing equality case for submatrices with best-bounded inverses when the matrix has two columns over the reals. It positions itself explicitly as a complement to Sengupta and Pautov's recent positive resolution of the 1997 Goreinov-Tyrtyshnikov-Zamarashkin hypothesis, without re-deriving the inequality part. That keeps the scope narrow and the contribution focused on finishing the characterization for this specific setting. The authors do a clean job of crediting the prior work and stating what is new: the equality condition itself. For anyone who has already read the Sengupta-Pautov paper, this addition rounds out the result without unnecessary expansion or new machinery. The citation pattern is straightforward and appropriate for a short complement. The soft spots are real but limited in scope. The entire argument inherits whatever assumptions or potential gaps exist in the base proof, including any handling of singular submatrices, the choice of matrix norm, or conditions on the entries. The abstract does not spell out the derivation steps, so it is difficult to confirm whether the equality holds unconditionally or requires generic-position restrictions that the stress-test note flags. If those conditions turn out to be needed, the criterion would not apply to the full class of n by 2 matrices as claimed. That dependence is the main limitation, though it is typical for this kind of follow-up note rather than a fatal flaw. This is for readers already working in numerical linear algebra on matrix selection and approximation problems. Someone tracking the GTZ line of results will find it useful as a completion, but it is unlikely to interest a broader audience. The paper shows clear, honest engagement with the literature and deserves a serious referee as a short note to check the details of the equality derivation against the prior proof.

Referee Report

1 major / 0 minor

Summary. The manuscript complements the positive solution of the Goreinov-Tyrtyshnikov-Zamarashkin (GTZ1997) hypothesis for real n×2 matrices, recently obtained by Sengupta and Pautov, by supplying the corresponding equality criterion that identifies when the bound is attained.

Significance. If the equality case is correctly characterized without hidden restrictions, the work completes the extremal description for the two-column case of this long-standing problem in numerical linear algebra. It thereby supplies a concrete, usable criterion for identifying submatrices achieving the optimal inverse bound, which is a modest but tangible advance for the special case n×2.

major comments (1)

The equality criterion is presented solely as a complement to Sengupta-Pautov (2026) without re-deriving or verifying the underlying inequality. The manuscript must explicitly address whether the transition to equality imposes any additional conditions (e.g., non-singularity of all 2×2 submatrices, strict convexity of the chosen norm, or generic position of entries) that are not required for the strict inequality. If such conditions are tacit, the claim that the criterion holds for the full class of n×2 matrices is not yet substantiated. This point is load-bearing for the central contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive overall assessment and for the detailed major comment. We address the concern directly below and will incorporate a clarifying revision in the next version of the manuscript.

read point-by-point responses

Referee: The equality criterion is presented solely as a complement to Sengupta-Pautov (2026) without re-deriving or verifying the underlying inequality. The manuscript must explicitly address whether the transition to equality imposes any additional conditions (e.g., non-singularity of all 2×2 submatrices, strict convexity of the chosen norm, or generic position of entries) that are not required for the strict inequality. If such conditions are tacit, the claim that the criterion holds for the full class of n×2 matrices is not yet substantiated. This point is load-bearing for the central contribution.

Authors: We appreciate the referee drawing attention to this load-bearing point. The manuscript is intentionally a short complement that takes the inequality from Sengupta-Pautov (2026) as given and derives the corresponding equality case by direct inspection of the equality conditions in their argument. No additional restrictions are introduced: the criterion applies to every real n×2 matrix under exactly the same hypotheses used for the inequality (in particular, without requiring non-singularity of every 2×2 submatrix, without invoking strict convexity of the norm, and without any genericity assumption on the entries). The underlying norm is the standard induced operator norm, which is convex but not strictly convex; our equality analysis does not rely on strict convexity. To make this explicit and remove any possibility of tacit conditions, we will revise the manuscript by inserting a short clarifying paragraph in the introduction together with a dedicated remark immediately following the statement of the main theorem. The revised text will state that the equality criterion holds for the full class of n×2 matrices under the hypotheses of Sengupta-Pautov (2026) and contains no further restrictions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; equality criterion complements independent external result

full rationale

The paper's derivation chain begins from the positive solution of the GTZ1997 hypothesis established in the independent Sengupta-Pautov 2026 work for n×2 matrices, then adds the equality criterion as a complement. No equations or steps in the provided abstract or description reduce by construction to self-defined quantities, fitted inputs renamed as predictions, or load-bearing self-citations. The central claim does not invoke uniqueness theorems from the same authors or smuggle ansatzes via prior self-work; it treats SP2026 as an external benchmark. This satisfies the condition for a self-contained addition against external results, yielding no circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the work relies on the external proof and standard linear-algebra background.

pith-pipeline@v0.9.0 · 5344 in / 1021 out tokens · 26673 ms · 2026-05-10T12:13:59.355157+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · 1 internal anchor

[1]

Berman and R

A. Berman and R. J. Plemmons.Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics 9. SIAM, 1994

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[2]

J. Blinn. Consider the lowly 2 x 2 matrix.IEEE Computer Graphics and Applications, 16(2):82–88, 1996

work page 1996
[3]

Goreinov, E.E

S.A. Goreinov, E.E. Tyrtyshnikov, and N.L. Zamarashkin. A theory of pseudo-skeleton approximations.Linear Algebra Appl., 261:1–21, 1997

work page 1997
[4]

R. A. Horn and C. R. Johnson.Matrix Analysis. Cambridge University Press, 2nd edition, 2012

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Nesterenko

Yu. Nesterenko. Submatrices with the best-bounded inverses: StudyingR n×2 andC n×2. arXiv, 2408.16631, 2024

work page arXiv 2024
[6]

On the submatrices with the best-bounded inverses

R. Sengupta and M. Pautov. On the submatrices with the best-bounded inverses.arXiv, 2604.05944, 2026. Email address:yuri.r.nesterenko@gmail.com

work page internal anchor Pith review Pith/arXiv arXiv 2026

[1] [1]

Berman and R

A. Berman and R. J. Plemmons.Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics 9. SIAM, 1994

work page 1994

[2] [2]

J. Blinn. Consider the lowly 2 x 2 matrix.IEEE Computer Graphics and Applications, 16(2):82–88, 1996

work page 1996

[3] [3]

Goreinov, E.E

S.A. Goreinov, E.E. Tyrtyshnikov, and N.L. Zamarashkin. A theory of pseudo-skeleton approximations.Linear Algebra Appl., 261:1–21, 1997

work page 1997

[4] [4]

R. A. Horn and C. R. Johnson.Matrix Analysis. Cambridge University Press, 2nd edition, 2012

work page 2012

[5] [5]

Nesterenko

Yu. Nesterenko. Submatrices with the best-bounded inverses: StudyingR n×2 andC n×2. arXiv, 2408.16631, 2024

work page arXiv 2024

[6] [6]

On the submatrices with the best-bounded inverses

R. Sengupta and M. Pautov. On the submatrices with the best-bounded inverses.arXiv, 2604.05944, 2026. Email address:yuri.r.nesterenko@gmail.com

work page internal anchor Pith review Pith/arXiv arXiv 2026