bibkey: zhangli2026eigenvalueperturbation
authors: Lei-Hong Zhang and Ren-Cang Li
year: 2026
title: On a Conjecture Related to Eigenvalue Perturbations
doi: 10.48550/arXiv.2609.25552
url: https://arxiv.org/abs/2609.25552
claim: "Problem 3 asks for a relation between a nonsingular Hadamard norm product minimum and the squared permutation minimum; §5 identifies the rank-one nonsingular case as unresolved."
strata_touched:
  - D5/S3/Quantum/Schatten/ZhangLiRankOneProblemThree
license: citation-only
triage: anchor

On a Conjecture Related to Eigenvalue Perturbations

Lei-Hong Zhang and Ren-Cang Li, arXiv:2609.25552v1, §1 Problem 3 and §5.

Locator

DOI: 10.48550/arXiv.2609.25552

Source: https://arxiv.org/abs/2609.25552

The versioned locator for both quotations below is arXiv:2609.25552v1, §1 Problem 3 and §5.

Problem 3: Relate min_{W nonsingular} ‖W∘G‖_ui ‖W^{-1}∘G^T‖ui to min{P permutation} ‖P∘G‖_ui^2.

In fact, for rank-one G, Romeo and Tilli [23, Theorem 1.2] proved (1.1) with c1=1 and the unitary restriction (2.3) with c2=1. They explicitly left the corresponding nonsingular version of Problem 3 unresolved; see [23, Remark 3.1].

Mathematical scope

Here $G$ is a complex square matrix, $W$ ranges over nonsingular complex matrices, $P$ ranges over permutation matrices, and the same unitarily invariant norm is used in both factors. The second Hadamard factor uses $G^{\mathsf{T}}$, the plain transpose, and $W^{-1}$, the inverse.

For $\mathrm{rank}(G)\le 1$, the delivered assertion states that the infimum over invertible $W$ equals the squared permutation minimum and is attained at a minimizing permutation. The explicit formulation includes the zero matrix and dimension zero. It does not settle the rank-at-least-two question for general unitarily invariant norms in Zhang–Li §5. Zhang–Li Theorem 1.1 proves Conjecture 1.1 with $c=1$ for the Frobenius norm for every $G$, without a rank restriction. Thus higher rank is already solved for the Frobenius norm.

Related literature and boundaries

Zhang–Li cite Romeo–Tilli, Linear Algebra and its Applications 326 (2001), 161–172, Theorem 1.2 and Remark 3.1. Their previously established rank-one unitary case is distinguished from the nonsingular case in the quotation. The wording of Romeo–Tilli Remark 3.1 is known here only through Zhang–Li’s citation; the original remark has not been independently read.

R. Bhatia, F. Kittaneh, R.-C. Li, “Some inequalities for commutators and an application to spectral variation. II”, Linear and Multilinear Algebra 43 (1997), no. 1–3, 207–219, https://doi.org/10.1080/03081089708818526, §4, supplies a related method precedent: a balanced average of a diagonal matrix and its inverse. The exact remark number is ASSUMED-UNVERIFIED; the locator was supplied through a secondary reading. The argument here uses $(tW+t^{-1}{W}^{-\ast })/2$ for an arbitrary invertible complex matrix. This method comparison does not attribute the rank-one nonsingular variational conclusion to that precedent. Classical singular-value and symmetric-gauge background is recorded in the Bhatia note.

The quotations and source metadata are supplied source material. No online verification is included. Citation indexes and subsequent literature have not been exhaustively checked; bibliographic priority is not established by the formal proof.