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arxiv: 1907.08616 · v1 · pith:IGC34MUKnew · submitted 2019-07-19 · 🧮 math.CO

On special matrices related to Cauchy and Toeplitz matrices

Sajad Salami This is my paper

Pith reviewed 2026-05-24 19:34 UTC · model grok-4.3

classification 🧮 math.CO
keywords determinantsCauchy matricesToeplitz matricesmatrix rankspecial matricessquare matricesnon-square matrices
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The pith

Determinants of certain square matrices blending Cauchy and Toeplitz structures can be found explicitly and then used to compute ranks of related non-square matrices.

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

The paper sets out to derive explicit determinant formulas for a family of square matrices that share structural features with both Cauchy matrices and Toeplitz matrices. Once those determinants are obtained, they serve as the direct tool for establishing the ranks of certain non-square matrices built from the same entries. A reader would value the work if the formulas replace case-by-case computation with closed expressions that hold for arbitrary size.

Core claim

We calculate the determinant of a certain type of square matrices, which are related to the well-known Cauchy and Toeplitz matrices. Then, we will use the results to determine the rank of special non-square matrices.

What carries the argument

The certain type of square matrices whose entries combine Cauchy-type and Toeplitz-type patterns, allowing determinant evaluation from known properties of each class.

If this is right

  • Explicit determinant formulas become available for the described family of hybrid matrices.
  • Ranks of the associated non-square matrices follow immediately from the determinant values.
  • The method supplies a uniform way to handle determinant and rank questions for matrices of this form without invoking general-purpose algorithms.

Where Pith is reading between the lines

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

  • The same structural pattern might allow analogous formulas for other matrix invariants such as permanents or characteristic polynomials.
  • The rank results could be used to decide linear dependence among rows or columns in the non-square cases without row reduction.

Load-bearing premise

These square matrices possess a structure that permits explicit determinant formulas derived from properties of Cauchy and Toeplitz matrices.

What would settle it

A concrete numerical example of one such square matrix whose determinant differs from the claimed closed-form expression would refute the result.

read the original abstract

In this paper, we are going to calculate the determinant of a certain type of square matrices, which are related to the well-known Cauchy and Toeplitz matrices. Then, we will use the results to determine the rank of special non-square 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 / 0 minor

Summary. The paper claims to calculate the determinant of a certain type of square matrices related to the well-known Cauchy and Toeplitz matrices, and then use the results to determine the rank of special non-square matrices.

Significance. Explicit determinant formulas for structured matrices related to Cauchy and Toeplitz classes would be of interest in linear algebra if derived and verified, but the manuscript supplies no specific matrix definitions, derivations, examples, or verification, rendering any potential contribution impossible to evaluate.

major comments (1)
  1. The manuscript consists solely of the abstract with no definitions of the matrices under consideration, no explicit determinant formulas, no proofs, and no examples or numerical checks; this absence makes the central claims unevaluable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing the manuscript and for the detailed comments. We acknowledge the concerns raised regarding the completeness of the submission and will address them directly in a revised version.

read point-by-point responses

  1. Referee: The manuscript consists solely of the abstract with no definitions of the matrices under consideration, no explicit determinant formulas, no proofs, and no examples or numerical checks; this absence makes the central claims unevaluable.

    Authors: We agree that the submitted version contained only the abstract and lacked the required definitions, formulas, proofs, and examples. This was an inadvertent submission error. The revised manuscript will include explicit definitions of the special square matrices related to Cauchy and Toeplitz types, the derived determinant formulas, full proofs of the results, and numerical examples or verifications to support the claims about ranks of non-square matrices. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper abstract states an intent to derive explicit determinant formulas for a class of square matrices from known properties of Cauchy and Toeplitz matrices, then apply those to rank computations for related non-square matrices. No equations, fitted parameters, self-citations, or ansatzes are visible in the provided text. The claimed steps rely on external, standard matrix identities rather than reducing to the paper's own inputs by construction. This is the normal case for explicit determinant calculations in linear algebra.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5544 in / 993 out tokens · 30601 ms · 2026-05-24T19:34:18.437704+00:00 · methodology

discussion (0)

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Reference graph

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