REVIEW 2 major objections 4 minor 12 references
Generalization of Cauchy type theorems for matrix Polynomials
T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read This paper proves that every eigenvalue of a matrix polynomial lies in a disk whose radius is built from ratios of products of the coefficient matrices, generalizing Cauchy's classical zero bound.
desk verdict The paper's headline bounds (Theorems 1 and 4) are unproven because the expansions drop the generic nonzero commutator term at z^m, while Theorems 2 and 3 are correct but unsurprising. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing identity is $P^*(z) = (A_{m-1} - A_m z)P(z) = -A_m^2 z^{m+1} + \sum_{r=1}^m (A_{m-1}A_{m-r} - A_m A_{m-r-1})z^{m-r}$. For a unit eigenvector $u$, the argument shows the leading term $-A_m^2 z^{m+1}u$ dominates the rest whenever $|z|$ exceeds the stated radius; the domination is certified by the subordinate-norm lower bound $\|B u\| \ge \|B^{-1}\|^{-1}$ together with Hölder's inequality applied to the coefficient ratios. Theorem 2 uses the same domination on $P(z)$ directly, splitting off $A_m z^m$.
What would settle it
Choose a concrete matrix polynomial, say degree 2 with $2\times 2$ matrices, compute the eigenvalues of its companion linearization to full precision, and check whether any lie outside the disk of Theorem 1 or Theorem 4; a single eigenvalue outside the claimed radius would disprove the corresponding theorem.
Extended reading notes
Core claim
The central claim is that all eigenvalues of a matrix polynomial $P(z)=\sum_{j=0}^m A_j z^j$ lie in $|z| < \left[\frac{1}{2}\left(1 + (1 + 4\alpha_p^q)^{1/2}\right)\right]^{1/q}$, where $\alpha_p := \left(\sum_{r=0}^m \left(\frac{\|A_{m-1}A_{m-r} - A_m A_{m-r-1}\|}{\|A_m^{-2}\|^{-1}}\right)^p\right)^{1/p}$, with $p>1$, $1/p+1/q=1$, $A_{-1}=0$, and $A_{-2}=(A^2)^{-1}$. The paper proves this as Theorem 1 and gives three companion results: Theorem 2 replaces the product-based quantity by a norm ratio $A_p$ and yields the radius $[1+(A_p)^q]^{1/q}$; Theorem 3 gives a trinomial-equation radius when only coefficients up to a fixed degree are present; and Theorem 4 sharpens the classical Cauchy-type bound $1+M$ to $\frac{1}{2}(1+(1+4M)^{1/2})$ using the same product differences.
Load-bearing premise
The entire argument rests on a subordinate matrix norm and an invertible leading coefficient $A_m$, so that $\|Bu\| \ge \|B^{-1}\|^{-1}$ for unit vectors and products of norms behave multiplicatively; without those, none of the exclusion inequalities hold.
Editorial extensions
If this is right
- Every eigenvalue of a matrix polynomial with invertible leading and constant coefficients lies in a finite disk centered at the origin, so a polynomial eigenvalue problem can be reduced to a bounded search region.
- For $1\times 1$ matrices, Theorem 1 reduces to Mohammad's classical bound for zeros of scalar polynomials, and Theorem 2 reduces to Montel's bound, making the matrix statements genuine generalizations of scalar root-location results.
- Theorem 4's radius $\frac{1}{2}(1+(1+4M)^{1/2})$ is no larger than the older Cauchy-type radius $1+M$ from Theorem C, giving a strictly improved enclosure in the same style.
- Because the Theorem 1 radius depends on products $A_{m-1}A_{m-r} - A_m A_{m-r-1}$, the enclosure can be far smaller than a coefficient-wise Cauchy bound when those products nearly cancel.
- The bounds can be used to shrink the region in which pseudospectra of matrix polynomials are inspected, reducing the cost of numerical algorithms that rely on iterative refinement.
Reading between the lines
- Minimizing the radius over all subordinate norms, or over similarity transformations of the coefficient matrices, is a natural next step that the paper does not address; the sharpness of the bounds may depend strongly on that choice.
- Because the proof only uses the leading term of $P^*(z)$, the same exclusion argument should extend to block-companion linearizations and to rational matrix functions, yielding eigenvalue bounds for those settings as well.
- Testing the radii against random small matrix polynomials with known eigenvalues would show how much tighter Theorem 1 is than Theorem C in practice; the paper gives no numerical comparison.
- The invertibility of $A_m$ is essential to the estimates; a separate treatment for singular leading coefficients, possibly by deflation, is left open.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes four theorems bounding the eigenvalues of a matrix polynomial P(z) = sum_{j=0}^m A_j z^j with nonsingular A_0 and A_m. Theorems 1 and 4 aim to improve on known Cauchy-type bounds by using coefficients of the product (A_{m-1} - A_m z)P(z). Theorems 2 and 3 give bounds via the norms of the coefficient matrices and a maximum norm, respectively, and are proved by direct lower bounds on ||P(z)u||. The paper also states that Theorem 2 contains a result of Montel and Theorem 1 a result of Mohammad as special cases when n = 1.
Significance. If correct, the results would provide new, computable inclusion disks for eigenvalues of matrix polynomials, generalizing classical Cauchy-type bounds. The paper is clearly written and uses only elementary norm inequalities, which makes the intended argument easy to check. However, the main novelty is concentrated in Theorems 1 and 4, and those two theorems rest on an expansion that is currently invalid. Theorems 2 and 3 appear sound and are moderate extensions of known bounds, but they are not the paper's main advertised contribution.
major comments (2)
- [Section 2, Theorem 1 (proof, expansion of P*(z))] The displayed expansion of P*(z) = (A_{m-1} - A_m z)P(z) is incorrect. Direct multiplication gives a z^m term with coefficient A_{m-1}A_m - A_m A_{m-1}, which is generally nonzero because the coefficient matrices need not commute. The proof writes P*(z) = -A_m^2 z^{m+1} + (A_{m-1}^2 - A_m A_{m-2}) z^{m-1} + ... and then applies a Hölder bound to the sum over r = 1 to m. The missing commutator term is of size ||A_{m-1}A_m - A_m A_{m-1}|| |z|^m, which is larger in modulus than any of the retained terms for |z| > 1. Moreover, the definition of α_p in the theorem statement sums r = 0 to m, so the proof's sum over r = 1 to m is inconsistent with the stated α_p. Because of this omitted term, the lower bound for ||P*(z)u|| is not valid, and the inclusion disk (1) is not justified by the argument given.
- [Section 2, Theorem 4 (proof)] The same expansion error occurs in the proof of Theorem 4. The proof states P*(z) = -A_m^2 z^{m+1} + sum_{j=1}^m (A_{m-1}A_{m-j} - A_m A_{m-j-1}) z^{m-j}, omitting the z^m term (A_{m-1}A_m - A_m A_{m-1}) z^m. The defined quantity M also excludes this term. Consequently, the bound (5) does not follow from the displayed argument. As in Theorem 1, the missing term is generally nonzero and cannot be neglected when |z| > 1. The proof also has a typo: it writes ||P(z)u|| where it should write ||P*(z)u||.
minor comments (4)
- [Section 2, Theorem 1 (definition of α_p)] The definition of α_p sums over r = 0 to m, while the proof uses a sum over r = 1 to m. This inconsistency should be resolved; if the intended α_p is the one in the theorem statement, the Hölder step must include the r = 0 term.
- [Notation, Section 2] The notation 'A^{-2} = (A^2)^{-1}' is ambiguous. It should read A_m^{-2} = (A_m^2)^{-1} or (A_m)^{-2}, and the subscript m should be carried consistently through the definitions.
- [Section 2, Theorem 3 (statement)] The statement 'P(z) = A_m z^m + A_p z^p + ... + A_1 z + A_0' is ambiguous about which coefficients are present. Please write P(z) = A_m z^m + sum_{j=0}^p A_j z^j, with A_{p+1} = ... = A_{m-1} = 0, to match the proof.
- [Section 2, Theorem 4 (proof)] The proof contains a notational slip: it uses ||A_m^{-2}||^{-1} but the text defines A_{-2} = (A_2)^{-1}. Please standardize the notation for the inverse of A_m^2.
Circularity Check
No significant circularity: the eigenvalue inclusion theorems are derived self-contained from definitions and standard norm inequalities, with no fitted parameter and no load-bearing self-citation.
full rationale
The paper's derivation chain is self-contained. Theorems 1-4 define explicit inclusion radii in terms of coefficient-norm expressions (e.g., alpha_p, A_p, M) and then prove the bounds directly from the eigenvalue definition P(lambda)u = 0, the subordinate-norm property ||B^{-1}||^{-1} <= ||Bu|| for unit u, and Holder's inequality. No parameter is fitted to data, no result is assumed as an input except the standard scalar Cauchy theorem and prior matrix-polynomial bounds cited for context, and none of the cited results is invoked to justify the new conclusions. The scalar special cases noted in Remarks 1 and 3 are consequences obtained by setting n = 1, not assumptions. The claims are therefore not equivalent to their inputs by construction, and no self-citation is load-bearing. Possible algebraic omissions in the expansion of P*(z) would be a correctness concern, not a circularity concern, and do not affect the circularity score.
Assumptions & free parameters
assumptions (4)
- domain assumption The matrix norm is subordinate to a vector norm
- domain assumption A_m and A_0 are nonsingular
- standard math Holder's inequality holds
- standard math The inequality ||Bv|| >= ||v|| / ||B^{-1}|| for invertible B
Cite this review
Pith. "Pith review of Generalization of Cauchy type theorems for matrix Polynomials." pith.science (2026). https://pith.science/paper/PIQFZ5IX
@misc{pith2026250609013,
author = {Pith},
title = {Pith review of: Generalization of Cauchy type theorems for matrix Polynomials},
year = {2026},
howpublished = {\url{https://pith.science/paper/PIQFZ5IX}},
note = {Machine review of arXiv:2506.09013}
}
read the original abstract
In this paper, we find bounds for the eigenvalues of matrix polynomials. In particular, we find generalizations of Cauchy's classical Theorem for distribution of eigenvalues of matrix polynomial.
Reference graph
Works this paper leans on
-
[1]
E. Artin and A. Milgram, Galois theory, vol. 2, Courier Corporation, (1998)
work page 1998
-
[2]
D.A. Bini, V. Noferini and M. Sharify, Locating the eigenvalues of matrix polynomial, SIAM Journal on matrix Analysis and Applications 34(2013), 1708-1727
work page 2013
-
[3]
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
-
[4]
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
-
[5]
I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials, Academic Press, Inc., New York London, 1982 (republished by SIAM, Philadelphia, 2009)
work page 1982
-
[6]
N.J. Higham and F. Tisseur, 2003, Bounds for eigenvalues of matrix polynomials, Linear Algebra and its Applications, 358(2003), 5-22. 8
work page 2003
-
[7]
M. Marden, The geometry of the zeros of a polynomial in a complex variable, Math- ematical Surveys, Amer. Math. Soc., (1949)
work page 1949
-
[8]
G.V. Milovanovic, D.S. Mitrinovic, Th. M. Rassias, Topics in Polynomials, Extremal Problems, Inequalities, Zeros. World Scientific, Singapore, (1994)
work page 1994
Show all 12 references
-
[9]
Mohammad, On the zeros of Polynomials, Amer
Q.G. Mohammad, On the zeros of Polynomials, Amer. Math. Monthly, 72(1965), 631-633
1965
-
[10]
Montel, Sur la limits superieure du module des racines d’une equation algebrique, C.R
P. Montel, Sur la limits superieure du module des racines d’une equation algebrique, C.R. Soc. Sci. Varsorie, 24(1932), 317-326
1932
-
[11]
Rahman and G
Q.I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Mathe- matical Society Monographs, New Series, 26. The Clarendon Press, Oxford Univer- sity Press, Oxford, (2002)
2002
-
[12]
Simoncini and F
V. Simoncini and F. Perotti, On the numerical solutions of ( λ2A+ λB + C)x = b and application to structural dynamics, SIAM Journal on Scietific Computing, 23(2006), 1875-1897. 9
2006
Reviewed August 7, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.