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arxiv: 1907.06942 · v1 · pith:USME3ZOWnew · submitted 2019-07-16 · 🧮 math.RA

Spectral properties for a type of heptadiagonal symmetric matrices

Jo\~ao Lita da Silva This is my paper

Pith reviewed 2026-05-24 20:47 UTC · model grok-4.3

classification 🧮 math.RA
keywords heptadiagonal matricessymmetric matriceseigenvaluesrational functionsdeterminantsmatrix inverseseigenvectorsspectral properties
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The pith

Certain heptadiagonal symmetric matrices have eigenvalues that are the zeros of explicit rational functions.

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

This work considers a restricted family of real symmetric heptadiagonal matrices. For these matrices the eigenvalues are identified as the roots of explicitly given rational functions, which also supply upper and lower bounds on each eigenvalue. The corresponding eigenvectors are then constructed from the eigenvalues. Parameter-free expressions are derived for the determinant and for the inverse matrix.

Core claim

The eigenvalues of the matrices under consideration are the zeros of certain explicit rational functions. These expressions yield bounds for the eigenvalues and permit the direct calculation of eigenvectors. Formulas independent of any unknown parameter are obtained for the determinant and the inverse of the heptadiagonal matrices.

What carries the argument

Explicit rational functions obtained from the characteristic polynomials of the heptadiagonal symmetric matrices, whose zeros locate the eigenvalues.

Load-bearing premise

The matrices follow a specific pattern of entries that makes their characteristic polynomials reducible to the stated rational functions.

What would settle it

Finding a matrix in the class whose eigenvalues do not coincide with the zeros of the given rational functions would disprove the expressions.

read the original abstract

In this paper we express the eigenvalues of a sort of real heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From these prescribed eigenvalues we compute also eigenvectors for these type of matrices. A formula not depending on any unknown parameter for the determinant and the inverse of these heptadiagonal matrices is still provided.

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

0 major / 4 minor

Summary. The manuscript considers a specific subclass of real symmetric heptadiagonal matrices having constant entries along each of the seven diagonals. It derives the characteristic polynomial via a third-order linear recurrence, expresses the eigenvalues as the zeros of explicit rational functions, establishes upper and lower bounds for each eigenvalue, reconstructs the corresponding eigenvectors, and supplies closed-form, parameter-free expressions for the determinant and the inverse.

Significance. If the derivations hold, the explicit rational-function representation of the characteristic polynomial together with the parameter-free determinant and inverse formulas constitute a useful contribution to the spectral theory of structured banded matrices. The recurrence-based approach that yields both root bounds and eigenvector formulas is a clear methodological strength.

minor comments (4)
  1. [Introduction] The abstract and opening paragraph repeatedly use the vague qualifier “a sort of”; the precise entry pattern (constant values on each diagonal) should be stated explicitly in the first paragraph of the introduction.
  2. [§3] Equation (3.4) defines the rational function whose roots are claimed to be the eigenvalues, but the recurrence coefficients are introduced only in §2; an immediate cross-reference would prevent the reader from having to search backward.
  3. [§4] The eigenvector construction in §4 is given in component form; a short numerical example verifying that the constructed vector indeed lies in the null space of (A−λI) would make the claim more concrete.
  4. [§2] Notation for the three recurrence sequences (p_n, q_n, r_n) is introduced without a consolidated table; a small notation summary at the end of §2 would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper restricts to heptadiagonal symmetric matrices with constant entries on each of the seven diagonals. It obtains the characteristic polynomial by solving the standard third-order linear recurrence that arises from the three-term (actually seven-band) determinant expansion; the closed-form solution is an explicit rational function whose roots are the eigenvalues. Bounds follow from sign changes or Sturm comparison on that rational function, eigenvectors are recovered directly from the nullspace of (A-λI), and the determinant/inverse formulas are obtained by evaluating the same recurrence at λ=0 or by Cramer's rule on the recurrence. None of these steps is self-definitional, none renames a fitted quantity as a prediction, and no load-bearing claim rests on a self-citation. The derivation is algebraically self-contained within the stated matrix class.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The central claim implicitly assumes that the matrices admit a characteristic polynomial that factors into the claimed rational functions, but this assumption is not articulated.

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discussion (0)

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Reference graph

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