Spectral properties for a type of heptadiagonal symmetric matrices

Jo\~ao Lita da Silva

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.

Desk Editor's Note private letter to a colleague

This paper gives explicit rational expressions for eigenvalues, eigenvectors, det, and inverse of heptadiagonal symmetric matrices with constant entries on each diagonal, derived via a third-order recurrence.

read the letter

The core result is explicit rational functions whose roots are the eigenvalues of this narrow matrix class, plus bounds on those roots, eigenvector formulas from the nullspace, and parameter-free expressions for the determinant and inverse. The approach relies on a linear recurrence of order three for the principal minors, solved to produce the characteristic polynomial in rational form, then applied to the other quantities. Within the stated class the derivations are consistent and avoid hidden parameters. The work is new for the heptadiagonal constant-diagonal case; the recurrence technique itself is standard for banded Toeplitz-like matrices, but the explicit rational forms and the full set of accompanying formulas for this bandwidth appear to be the contribution. It does the job cleanly for the restricted pattern. The main limitation is the narrow scope: everything requires constant values along each of the seven diagonals. If the entries vary within a diagonal the recurrence and closed forms do not apply, so this supplies no general method for arbitrary heptadiagonal symmetric matrices or for wider or narrower bands beyond what is already known. There is also no comparison showing whether the rational functions simplify computation relative to direct characteristic-polynomial evaluation or overlap with earlier results for tridiagonal or pentadiagonal cases. The evidence in the derivations supports the claims for the defined class, with no visible algebraic inconsistencies. This is for readers who need concrete formulas for this exact matrix family, perhaps for verification of numerical methods or applications with uniform banded structure. A serious referee should see it because the results are reproducible from the recurrence and the scope is clearly delimited.

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)

[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.
[§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.
[§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.
[§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.

pith-pipeline@v0.9.0 · 5571 in / 1228 out tokens · 16335 ms · 2026-05-24T20:47:41.294605+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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Usmani, T.H

R.A. Usmani, T.H. Andres, D.J. Walton, Error estimation in the inte gration of ordinary diﬀerential equations, Int. J. Comput. Math. 5 (1976) 241–256. 18

work page 1976

[1] [1]

Anderson, A secular equation for the eigenvalues of a diagona l matrix perturbation, Linear Algebra Appl

J. Anderson, A secular equation for the eigenvalues of a diagona l matrix perturbation, Linear Algebra Appl. 246 (1996) 49–70

work page 1996

[2] [2]

Asplund, Finite boundary value problems solved by Green’s mat rix, Math

S.O. Asplund, Finite boundary value problems solved by Green’s mat rix, Math. Scand. 7 (1959) 49–56

work page 1959

[3] [3]

Bini and M

D. Bini and M. Capovani, Spectral and computational propertie s of band symmetric Toeplitz matrices, Linear Algebra Appl. 52/53 (1983) 99–126

work page 1983

[4] [4]

Bunch, C.P

J.R. Bunch, C.P. Nielsen, D.C. Sorensen, Rank-one modiﬁcation of the symmetric eigen- problem, Numer. Math. 31 (1978) 31–48

work page 1978

[5] [5]

Demko, Inverses of band matrices and local convergence of spline projections, SIAM J

S. Demko, Inverses of band matrices and local convergence of spline projections, SIAM J. Numer. Anal. 14(4) (1977) 616–619

work page 1977

[6] [6]

Fasino, Spectral and structural properties of some penta diagonal symmetric matrices, Calcolo 25(4) (1988) 301–310

D. Fasino, Spectral and structural properties of some penta diagonal symmetric matrices, Calcolo 25(4) (1988) 301–310

work page 1988

[7] [7]

Fischer and R.A

C.F. Fischer and R.A. Usmani, Properties of some tridiagonal matr ices and their application to boundary value problems, SIAM J. Numer. Anal. 6(1) (1969) 127 –142

work page 1969

[8] [8]

Haley, Solution of band matrix equations by projection-recur rence, Linear Algebra Appl

S. Haley, Solution of band matrix equations by projection-recur rence, Linear Algebra Appl. 32 (1980) 33–48

work page 1980

[9] [9]

Harville, Matrix Algebra From a Statistician’s Perspective, Spr inger-Verlag, 1997

D.A. Harville, Matrix Algebra From a Statistician’s Perspective, Spr inger-Verlag, 1997

work page 1997

[10] [10]

Horn, C.R

R.A. Horn, C.R. Johnson, Matrix Analysis (second edition), Camb ridge University Press, 2013. 17

work page 2013

[11] [11]

Keeping, Band matrices arising from ﬁnite diﬀerence approx imations to a third order partial diﬀerential, SIAM J

A.J. Keeping, Band matrices arising from ﬁnite diﬀerence approx imations to a third order partial diﬀerential, SIAM J. Numer. Anal. 7(1) (1970) 142–156

work page 1970

[12] [12]

Miller, On the inverse of the sum of matrices, Math

K.S. Miller, On the inverse of the sum of matrices, Math. Mag. 54( 2) (1981) 67–72

work page 1981

[13] [13]

Pissanetsky, Sparse Matrix Technology, Academic Press, 1 984

S. Pissanetsky, Sparse Matrix Technology, Academic Press, 1 984

work page

[14] [14]

Usmani, T.H

R.A. Usmani, T.H. Andres, D.J. Walton, Error estimation in the inte gration of ordinary diﬀerential equations, Int. J. Comput. Math. 5 (1976) 241–256. 18

work page 1976