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arxiv: 2606.24620 · v1 · pith:V7EACKOAnew · submitted 2026-06-23 · 🧮 math.LO

Linear Systems and Eigenvectors in Constructive Mathematics

Bob Driessen , C.J.F. van de Ven This is my paper

Pith reviewed 2026-06-25 22:21 UTC · model grok-4.3

classification 🧮 math.LO
keywords constructive mathematicslinear systemseigenvectorsGauss-Jordan eliminationsingular matricesapproximate rankBishop-style mathematics
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The pith

Constructive mathematics yields explicit methods to solve linear systems and approximate eigenvectors via a new characterization of singular matrices.

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

The paper develops constructive versions of two standard linear algebra tasks: solving linear systems and computing eigenvectors. It presents new constructive results for Gauss-Jordan elimination and for approximating the rank of a matrix. The central advance is a procedure that builds approximate eigenvectors from an original characterization of singular matrices. A reader would care because the approach keeps all steps explicit and accounts for numerical indeterminacy without non-constructive existence arguments. This keeps the computations verifiable within the same framework used for the rest of the development.

Core claim

Within Bishop-style constructive mathematics the authors establish new results on Gauss-Jordan elimination and on constructive approximation of matrix rank; they further give a method that constructs approximate eigenvectors by means of a previously unexplored characterization of singular matrices.

What carries the argument

A previously unexplored characterization of singular matrices that directly supplies an explicit construction for approximate eigenvectors.

If this is right

  • Gauss-Jordan elimination acquires fully constructive versions that track accuracy at each step.
  • Matrix rank can be approximated to any desired precision inside the same framework.
  • Eigenvector approximations become available without classical non-constructive detours.
  • Indeterminacy in numerical data is handled uniformly across linear systems and spectral problems.

Where Pith is reading between the lines

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

  • The same characterization might extend to other matrix decompositions that currently lack constructive treatments.
  • Explicit error bounds produced by the method could be fed directly into downstream constructive proofs.
  • The approach suggests a route for making other classical numerical algorithms fully explicit in constructive settings.

Load-bearing premise

The characterization of singular matrices supplies enough information to build an explicit constructive procedure for approximating eigenvectors.

What would settle it

A concrete matrix for which the proposed construction, when followed step by step, fails to produce a vector that satisfies the approximate eigenvector relation would show the method does not work.

read the original abstract

In this work we study two classical problems of (numerical) linear algebra: (i) solving linear systems and (ii) computing eigenvectors, within a constructive framework. Numerical accuracy and indeterminacy are naturally incorporated through Bishop-style constructive mathematics. Our contributions include new results on Gauss-Jordan elimination and on approximating the rank of a matrix. Additionally, we introduce a novel method for constructing approximate eigenvectors, based on a previously unexplored characterization of singular matrices.

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 / 2 minor

Summary. The paper develops constructive approaches, in the sense of Bishop-style mathematics, to two classical problems in numerical linear algebra: solving linear systems and computing approximate eigenvectors. It reports new results on Gauss-Jordan elimination and on approximating matrix rank, together with a novel method for approximate eigenvectors that rests on a claimed previously unexplored characterization of singular matrices.

Significance. If the claimed characterization of singular matrices supplies an explicit, computable procedure that produces approximate eigenvectors together with explicit error bounds, the work would strengthen the constructive treatment of linear algebra by furnishing methods that respect numerical indeterminacy. The Gauss-Jordan and rank results would likewise be of interest if they are fully constructive and avoid non-computable existence statements.

major comments (1)
  1. [§4] §4 (Characterization of singular matrices): the abstract asserts that a previously unexplored characterization of singular matrices directly yields a constructive approximation procedure for eigenvectors. The manuscript must exhibit the precise definition (presumably an equation or theorem in this section) and the subsequent derivation showing how the characterization produces an explicit algorithm together with computable error bounds; if the characterization merely restates the classical determinant-zero condition without a constructive extraction step, the central claim for the eigenvector method does not follow.
minor comments (2)
  1. Notation for matrices and vectors should be introduced once and used consistently; several passages switch between boldface and non-boldface without explanation.
  2. The statement of the main theorem on approximate eigenvectors should include an explicit reference to the error bound that is claimed to be computable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for major revision. We address the major comment on §4 below.

read point-by-point responses

  1. Referee: [§4] §4 (Characterization of singular matrices): the abstract asserts that a previously unexplored characterization of singular matrices directly yields a constructive approximation procedure for eigenvectors. The manuscript must exhibit the precise definition (presumably an equation or theorem in this section) and the subsequent derivation showing how the characterization produces an explicit algorithm together with computable error bounds; if the characterization merely restates the classical determinant-zero condition without a constructive extraction step, the central claim for the eigenvector method does not follow.

    Authors: We agree that the connection between the characterization and the explicit constructive procedure should be stated more directly. In the revised manuscript we will add an explicit formulation of the characterization as a numbered theorem in §4 together with a dedicated derivation subsection that extracts the algorithm and the computable error bounds from the constructive proof. revision: yes

Circularity Check

0 steps flagged

No circularity; claims rest on novel characterization without visible self-referential reduction.

full rationale

The abstract and context present a novel method based on a previously unexplored characterization of singular matrices, with no equations, self-citations, or fitted parameters shown that reduce any prediction or result to its own inputs by construction. No load-bearing steps match the enumerated circularity patterns. The derivation appears self-contained pending full text verification, consistent with the default expectation that most papers are not circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the background framework of Bishop-style constructive mathematics; no free parameters, invented entities, or additional ad-hoc axioms are mentioned in the abstract.

axioms (1)
  • domain assumption Principles of Bishop-style constructive mathematics
    The entire study is conducted inside this framework, as stated in the abstract.

pith-pipeline@v0.9.1-grok · 5589 in / 1047 out tokens · 20766 ms · 2026-06-25T22:21:24.904503+00:00 · methodology

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Reference graph

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