Generalizations of Enestrom-Kakeya Type Theorems for matrix Polynomials

Idrees Qasim

REVIEW 3 major objections 8 minor 21 references

Generalizations of Enestrom-Kakeya Type Theorems for matrix Polynomials

T0 review · 3 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read All eigenvalues fit in a disk set by a simple trinomial root.

desk verdict Correct, competent, and modest: four new Eneström–Kakeya-type bounds for matrix polynomials, with checkable proofs and mostly typographical blemishes. read the letter →

arxiv 2506.09112 v1 pith:HW63GTOK submitted 2025-06-10 math.CA math.CV

Idrees Qasim This is my paper

classification math.CAmath.CV MSC 12D1015A1830C15

keywords Eneström-KakeyatheoremmatrixpolynomialeigenvalueboundsproblemPellnumbersnorminequalitieszero-freediskanalyticmatrix-valuedfunctions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves four theorems that locate the eigenvalues of a matrix polynomial $P(z)=\sum_{j=0}^m A_j z^j$ from the sizes of its coefficient matrices, without requiring the coefficients to form an ordered chain of Hermitian matrices. The main result shows that if the norms satisfy $\lVert A_m^{-1}\rVert^{-1} \geq t\lVert A_{m-1}\rVert \geq \cdots \geq t^m\lVert A_0\rVert$ with $A_0>0$ and $A_m$ invertible, then every eigenvalue lies in $|z|\leq k_1/t$, where $k_1$ is the largest positive root of $K^{m+1}-2K^m+1=0$. A second theorem gives a shifted disk under a Frobenius-angle condition, a third gives an annulus with radii built from Pell numbers, and a fourth gives a zero-free disk for analytic matrix-valued functions. A sympathetic reader would care because explicit eigenvalue regions are the practical input to numerical algorithms for polynomial eigenvalue problems.

What carries the argument

The carrying device is the coefficient-norm chain combined with the triangle inequality: for a unit eigenvector $u$, if the norm of the leading term $\lVert A_m z^m u\rVert$ dominates the accumulated norm of the lower terms, then $P(z)u$ cannot vanish. Lemma 1, the Pell-number identity $\sum_{k=0}^m \binom{2m}{m+k} P_k^2 = 2^{3(m-1)}$, creates the cancellation that makes Theorem 3's annulus radii explicit. Lemma 2, an angle-perturbation inequality in the Frobenius inner product, supplies the shifted disk of Theorem 2. Theorem 4 uses Schwarz's lemma through the auxiliary function $u^*(z-t)f(z)u$ to locate a zero-free disk.

What would settle it

Evaluate the eigenvalues of the quadratic pencil $P(z)=z^2 I + z A_1 + A_0$ with $A_1$ a symmetric $2\times2$ matrix of spectral norm $1$ and $A_0=0.1 I$; this satisfies Theorem 1's chain with $t=1$, so the theorem forces every eigenvalue into $|z|\leq 1.618$, and any computed eigenvalue outside that disk refutes the bound. A wider search over positive-definite $A_0$ and norm-chain-satisfying $A_j$ that finds one eigenvalue with $|\lambda|>k_1/t$ would likewise settle whether the theorem is true.

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Extended reading notes

Core claim

The paper's central claim is that Eneström-Kakeya-type monotonicity can be replaced by a norm-chain condition for matrix polynomials. In Theorem 1, for $P(z)=\sum_{j=0}^m A_j z^j$ with $A_m$ invertible and $A_0>0$, the chain $\lVert A_m^{-1}\rVert^{-1} \geq t\lVert A_{m-1}\rVert \geq \cdots \geq t^m\lVert A_0\rVert$ forces all eigenvalues into $|z|\leq k_1/t$, where $k_1$ is the largest positive root of $K^{m+1}-2K^m+1=0$. The proof bounds $\lVert P(z)u\rVert$ from below for unit vectors $u$ and uses the trinomial inequality $(tR)^{m+1}-2(tR)^m+1>0$. Theorem 3 wraps eigenvalues in an annulus $r_1\leq|z|\leq r_2$ whose radii are explicit combinations of binomial coefficients, Pell numbers, and coefficient norms, using the identity $\sum_{k=0}^m \binom{2m}{m+k} P_k^2 = 2^{3(m-1)}$. Theorem 2 and Theorem 4 add a shifted disk and a zero-free disk under angle and analyticity hypotheses.

Load-bearing premise

The load-bearing premise is that $A_0$ and $A_m$ are nonsingular (and in Theorem 1, $A_0>0$); if either fails, 0 becomes an eigenvalue and the inverse norms that anchor all four proofs are unavailable.

Editorial extensions

If this is right

Any matrix polynomial whose coefficient norms decay geometrically with ratio $t$ has an a priori eigenvalue bound $k_1/t$, computable before any iterative solver runs.
The annulus theorem gives both an inner exclusion disk and an outer enclosure in one formula, so it can certify that a polynomial eigenvalue problem has no small eigenvalues.
The disk of Theorem 2 depends on the spread of Frobenius norms and the angle to a fixed matrix, so it remains available when no monotone norm chain exists.
The analytic-function theorem transfers the same technique to infinite series, giving zero-free disks for matrix-valued power series on $|z|<t$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An extension the paper leaves implicit is that the trinomial radius $k_1/t$ is roughly between $1.618/t$ and $2/t$, so the bound is most valuable when the norm chain decays with a large $t$; comparing it against existing matrix-polynomial enclosures on random quadratic pencils would reveal where it wins and where it loses.
The annulus theorem could be turned into a stopping criterion for iterative eigensolvers: once the gap between $r_2$ and $r_1$ is small enough, the eigenvalues are well localized without solving the polynomial eigenvalue problem, a numerical use the paper does not discuss.
A natural testable extension is to relax Theorem 2's single-matrix angle condition to a per-coefficient discrepancy such as the numerical radius of $A_j-C_j$, and to check numerically on random coefficient matrices whether a similar shifted disk survives.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing to know: the mathematics holds up. I read the four theorems and their proofs carefully, and the bounds are valid under the stated hypotheses. The new content is real: Theorem 1 generalizes the Dirr–Wimmer matrix Eneström–Kakeya theorem with a t-parameter norm chain and a clean trinomial root; Theorem 2 adds a Frobenius angle condition to get a disk centered at z = k−1; Theorem 3 uses the Pell-number identity to produce an annulus; Theorem 4 extends the Schwarz-lemma argument to analytic matrix functions with positive semidefinite coefficients. None of these appear in the cited prior work, so the novelty claim is honest, if incremental. The paper is careful in the right way: the standing assumptions (A0 and Am nonsingular) are stated up front, and the derivations do not hide a circular step. The Pell identity and angle inequality come from the literature, but the main results do not reduce to those lemmas.

Soft spots are real but mostly cosmetic. The displayed factorization in Theorem 2 has a typo ('-A_m' should be '-A_m z^m'), Corollary 1's powers of t do not match the substitution P(tz), there is a placeholder citation '[?]' in the introduction, and a few names are misspelled. These are fixable and do not threaten the main bounds. More substantively, Theorem 2's angle condition—every coefficient matrix within α ≤ π/2 of a single matrix C in the Frobenius inner product—is genuinely restrictive; many matrix polynomials will not satisfy it. The paper does not dwell on this limitation, so readers should understand that Theorem 2 is narrower in reach than Theorem 1. The nonsingularity of A0 and Am is also a real restriction, but it is explicitly stated and standard in this area, so I do not count it as a hidden flaw.

Bottom line: this is a modest but correct contribution to an established research program. It does not reshape the field, but the proofs are short and checkable, and the bounds may be useful for numerical linear algebra or control theory. The right audience is researchers working on eigenvalue bounds for matrix polynomials. It deserves a serious referee; after a round of copyediting and a corrected Corollary 1, I would accept it. I would probably not bring it to my own reading group unless the group specifically cares about Eneström–Kakeya-type results.

Referee Report

3 major / 8 minor

Summary. The paper presents four theorems on locating eigenvalues of matrix polynomials, framed as generalizations of the Enestrom-Kakeya theorem. Theorem 1 gives an outer disk bound |z| <= k1/t under a chain of subordinate norms of the coefficient matrices with a parameter t, where k1 is the largest positive root of K^(m+1)-2K^m+1=0. Theorem 2 gives a disk bound |z+k-1| <= R under Frobenius-norm monotonicity and an angular condition on the coefficients relative to a fixed matrix C. Theorem 3 gives an annulus r1 <= |z| <= r2 whose radii are expressed through Pell numbers. Theorem 4 gives a zero-free disk for analytic matrix-valued functions whose coefficient matrices satisfy a Loewner-order chain. A corollary applies Theorem 2 to P(tz). The proofs use geometric sums, the Monga-Shah angle lemma, a Pell-number identity, and Schwarz's lemma.

Significance. If the results are correct, the paper supplies explicit, checkable inclusion regions for polynomial eigenvalue problems, which are useful for pseudospectra and for initializing numerical methods. The bounds depend only on norm chains or Loewner order, so they are directly computable. The Pell-number annulus in Theorem 3 is a genuine new contribution to the matrix-polynomial version of the Enestrom-Kakeya circle of ideas, and the use of the angle lemma in Theorem 2 is a nontrivial extension of earlier scalar work. The paper's contribution is, however, incremental: it combines existing lemmas and does not introduce a new method. Several hypotheses need clarification and at least one stated result (Theorem 4) is false without additional assumptions, so the claims cannot be accepted in their present form.

major comments (3)

[Section 3, Theorem 4] The theorem is false as stated because the hypotheses allow A0 to be singular. For example, take t=1, k=1, A0=diag(1,0), and A_j=0 for j>=1; the chain kA0 >= tA1 >= t^2 A2 >= ... holds, yet f(z)=A0 is singular at every point of the stated disk |z|<1, contradicting the conclusion. The proof also requires u^* A0 u > 0 for every unit vector u and u^*(A_{j-1}-tA_j)u >= 0, which need A0 positive definite and all A_j Hermitian; these assumptions are not stated. In addition, the proof uses analyticity on |z| <= t and lim_{j->infty} A_j z^j = 0, while the theorem only assumes analyticity in |z| < t. The theorem and its proof need to be repaired, for example by assuming A0 > 0, Hermitian coefficients, and analyticity on a neighborhood of the closed disk, or by a limiting argument.
[Section 3, Corollary 1] The powers of t in the corollary do not match the substitution P(tz). If Q(z)=P(tz)=sum_{j=0}^m (t^j A_j) z^j, then Theorem 2 applied to Q gives the chain k t^m ||A_m||_F >= t^{m-1} ||A_{m-1}||_F >= ... >= ||A_0||_F, not the displayed chain k t^n ||A_m||_F >= t||A_{m-1}||_F >= ... >= ||A_0||_F. The resulting radius also contains inconsistent powers of t. The corollary must be rederived, and the parameter n should be clarified or removed.
[Section 3, Theorem 2 proof] The displayed factorization in the proof of Theorem 2 is algebraically incorrect. The paper writes F(z)=(1-z)P(z) and then claims ||F(z)u|| >= ||(A_m z^{m+1}+k A_m z^m - A_m)u|| - ||((k A_m - A_{m-1})z^m + sum_{j=0}^{m-1}(A_j-A_{j-1})z^j)||, but F(z) is not equal to the difference of those two expressions. The correct identity is F(z) = -A_m z^m(z+k-1) + ((k A_m - A_{m-1})z^m + sum_{j=0}^{m-1}(A_j-A_{j-1})z^j). With that correction the subsequent use of the triangle inequality and Lemma 2 is valid, but as printed the key estimate does not follow.

minor comments (8)

[Section 1] In the paragraph defining the Loewner order, the phrase "A > Bmeans" is missing a space and is grammatically incomplete.
[Section 2] The sentence "any matrix-valued function F(z) analytic in |z| <= t can be expressed as a power series ... (for ref. see [ ?].)" contains a placeholder reference that should be filled or removed.
[Section 2, Lemma 4] Lemma 4 is stated but never used in the paper; either use it or remove it.
[Section 3, Theorem 2] The statement of Theorem 2 should explicitly assume A_m is invertible, since the bound involves ||A_m^{-1}||_F^{-1}; without this, the displayed formula is not well defined.
[Section 3, Theorem 4] The proof of Theorem 4 contains a typo: "|(k-1)z + t - k|z|}" should be "|(k-1)z+t| - k|z|". The Loewner chain also requires the matrices A_j to be Hermitian, which the statement should say explicitly.
[Section 3, Theorem 1 proof] The proof divides by ||A_{m-1}||; if A_{m-1}=0, then the norm chain forces all lower coefficients to vanish and the conclusion is trivial, but this case should be mentioned for completeness.
[Section 3, Theorem 2 proof] After invoking Lemma 3, the factor |z|^m is dropped without comment. The argument is still a valid sufficient-condition argument, but as a displayed inequality it is not literally correct; the authors should clarify that they are proving positivity of the bracket.
[References] Reference [20] lists "A.E. Taylor and C. David"; the author names appear garbled and should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bounds are derived from stated hypotheses and external lemmas, with no fitted parameters or self-citation chains.

full rationale

The derivation chain is self-contained against external benchmarks. Theorem 1 derives a root bound from the norm chain and a geometric series estimate; the constant k1 is the root of an independent trinomial, not fitted to the eigenvalues. Theorem 2 uses the cited angle/Frobenius-norm lemma (Monga-Shah) and numerical radius inequalities as external tools; no parameter in the bound is calibrated to the eigenvalues being bounded. Theorem 3 defines r1 and r2 by explicit coefficient/Pell-number formulas and then proves the inequalities (4) and G(r2) >= 0; this is a direct estimate, not a definitional identity with the conclusion. Theorem 4 uses Schwarz's lemma on an auxiliary function; the disk center and radius are explicit and no prior result is imported wholesale. The manuscript explicitly states the standing nonsingularity assumption on A0 and Am in the introduction ('we always assume that A0 and Am are non-singular'), so that restriction is disclosed rather than hidden. No self-citation is load-bearing; citations to Dirr-Wimmer, Monga-Shah, Popescu-Diaz-Barrero, and Higham-Tisseur supply lemmas and context, but the main eigenvalue bounds are proven in the paper from the stated hypotheses. There are presentation-type typos (e.g., the omitted z^m in Theorem 2's factorization and the t-powers in Corollary 1), but these do not reduce any result to its inputs. Therefore no circularity is present.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities or fitted constants. The bounds depend on user-chosen parameters t and k, and on several cited background lemmas. The non-singularity of A0 and Am is a standing, load-bearing assumption.

free parameters (2)

t (Theorem 1) = user-specified, t > 0
A positive parameter calibrating the norm chain; the bound becomes |z| ≤ k1/t, so smaller t gives looser but more widely applicable bounds. Not fitted to data.
k (Theorems 2 and 4) = user-specified, k ≥ 1
A scaling parameter in the norm/PSD chains and the disk center. Not fitted to data; it generalizes the classical k=1 case.

assumptions (5)

standard math Lemma 1: Pell number identity sum_{k=0}^m binom(2m, m+k) P_k^2 = 2^{3(m-1)}
Used in Theorem 3 to show the ring radii are exact. Cited from Popescu and Díaz-Barrero [16].
standard math Lemma 2: Frobenius norm angle inequality ||A-B||_F ≤ (||A||_F - ||B||_F)cos α + (||A||_F + ||B||_F)sin α when ||A||_F ≥ ||B||_F and angle(A,B) ≤ 2α
Used in Theorem 2 to bound differences of coefficient matrices. Cited from Monga and Shah [15].
standard math Lemma 3: r(A) ≤ ||A|| ≤ ||A||_F where r(A) is the numerical radius
Used to switch between operator norms and Frobenius norms in Theorem 2. Cited from Horn and Johnson [10].
standard math Lemma 4: Rayleigh quotient bounds λ_min(A) ≤ u*Au ≤ λ_max(A) for Hermitian A
Used implicitly to justify the scalar inequalities in Theorem 4. Cited from Horn and Johnson [10].
domain assumption Standing assumption: A0 and Am are non-singular
Stated in the introduction and used in Theorem 3 (A0^{-1}) and Theorem 1 (A_m^{-1}). If either is singular, 0 is an eigenvalue and the derived bounds are not proven.

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Cite this review

Pith. "Pith review of Generalizations of Enestrom-Kakeya Type Theorems for matrix Polynomials." pith.science (2026). https://pith.science/paper/HW63GTOK

@misc{pith2026250609112,
  author       = {Pith},
  title        = {Pith review of: Generalizations of Enestrom-Kakeya Type Theorems for matrix Polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HW63GTOK}},
  note         = {Machine review of arXiv:2506.09112}
}

read the original abstract

In this paper, we establish bounds for the eigenvalues of matrix polynomials. Specifically, we find different generalizations of the Enestrom-Kakeya Theorem for matrix polynomials.

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Reference graph

Works this paper leans on

21 extracted references · 19 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Artin and A

E. Artin and A. Milgram, Galois theory, vol. 2, Courier Corporation, 1998

work page 1998

[2] [2]

Aziz and Q

A. Aziz and Q. G. Mohammad, On the zeros of a certain class of polynomials and related analytic functions, J. Math. Anal. and Appl., 75 (1980), 495-502

work page 1980

[3] [3]

D.A. Bini, V. Noferini and M. Sharify, Locating the eigenvalues of matrix polynomial, SIAM Journal on matrix Analysis and Applications 34: 1708-1727, (2013)

work page 2013

[4] [4]

A.L. Cauchy, Sur la r´ esolution des ´ equations num´ eriqes et sur la th´ eorie de l’´ elimination, Exercices de Math´ ematiques, Quatri´ eme Ann´ ee, Paris, Also in: Oeu- vres Compl´ etes, S´ erie 2, Tome 9(1829), 86-161

[5] [5]

Cˆ ong-T´ rinh Lˆ e, Thi-Hoa-Binh Du and Tran-Duc Nguyen, On the location of eigen- values of matrix polynomials, Operators and Matrices, 13(4)(2019) 937-954

work page 2019

[6] [6]

Dirr and H.K

G. Dirr and H.K. Wimmer, An Enestr¨ om-Kakeya theorem for hermitian polynomial matrices, IEEE Trans. Automat. Control 52:2151-2153(2007)

work page 2007

[7] [7]

Enestr¨ om, Remarque sur un th´ eor´ em´ e relatif aux racines de I’equationanzn + an−1zn−1 + · · ·+ a1z + a0 = 0 o´ u tous les coefficients sont re´ els et positifs, Tˆ ohoku Math

G. Enestr¨ om, Remarque sur un th´ eor´ em´ e relatif aux racines de I’equationanzn + an−1zn−1 + · · ·+ a1z + a0 = 0 o´ u tous les coefficients sont re´ els et positifs, Tˆ ohoku Math. J., 18 (1920), 34-36

work page 1920

[8] [8]

Gohberg, P

I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials, Academic Press, Inc., New York London, 1982 (republished by SIAM, Philadelphia, 2009)

1982