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Zhang–Li Problem 3 for matrices of rank at most one

Abstract

Zhang–Li Problem 3 for matrices of rank at most one.

Lei-Hong Zhang and Ren-Cang Li, On a Conjecture Related to Eigenvalue Perturbations, 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].

Zhang–Li Theorem 1.1 proves Conjecture 1.1 with c = 1 for the Frobenius norm for every G. The result here settles the rank-at-most-one case for every unitarily invariant norm. Higher rank remains open for general unitarily invariant norms; the Frobenius case is already solved by Zhang–Li.

The averaging method has a related precedent in 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, §4, https://doi.org/10.1080/03081089708818526. The exact remark number is ASSUMED-UNVERIFIED.

Definition 1.1 (The complete rank-one variational assertion).

Lean statement: D5/S3/Quantum/Schatten/ZhangLiRankOneProblemThree.claim

Formalization. D5/S3/Quantum/Schatten/ZhangLiRankOneProblemThree.claim (✓ std3).

Citation. Lei-Hong Zhang and Ren-Cang Li (2026). On a Conjecture Related to Eigenvalue Perturbations. DOI: 10.48550/arXiv.2609.25552. URL: https://arxiv.org/abs/2609.25552.

Commentary.

For every natural n, every complex n by n matrix G of rank at most one, and every unitarily invariant norm N, there exists a permutation σ minimizing N(Pσ∘G) over all permutations, with N(Pσ∘G)²≤N(W∘G)N(W⁻¹∘Gᵀ) for every invertible W and equality in that product when W=Pσ. This includes zero G and dimension zero.

Theorem 1.2 (The rank-one infimum equals the permutation minimum).

Lean statement: D5/S3/Quantum/Schatten/ZhangLiRankOneProblemThree.result

Proof. Machine-checked in Lean as D5/S3/Quantum/Schatten/ZhangLiRankOneProblemThree.result (✓ std3). ∎

Resolves. Problems/zhang-li-2026-rank-one-hadamard-product-variational (proved) by D5/S3/Quantum/Schatten/ZhangLiRankOneProblemThree.result.

Source. Repository-derived.

Commentary.

For every natural n, every complex n by n G of rank at most one and every unitarily invariant norm N, the infimum over invertible W of N(W∘G)N(W⁻¹∘Gᵀ) equals the minimum over permutation matrices P of N(P∘G)², attained at a minimizing permutation. The statement supplies that permutation together with the lower bound for every invertible W and the equality at the permutation. Factor G and remove its phases to obtain common nonnegative diagonal weights. The expanding diagonal minimum bounds every balanced inverse-adjoint average, and optimization over its positive scale gives the product bound.

References