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A counterexample to Huang's weak majorization conjecture
Abstract
We give a counterexample to a weak majorization problem, proposed by Z. Huang (Linear Algebra Appl., 434 (2) (2011) 457--462), for singular values of Hadamard products of nonnegative matrices. The problem asks whether, for all nonnegative matrices A and B, {sj2(AHad B)} ≺w {sj(AHad A)sj(BHad B)} holds, where Had denotes the Hadamard product, ≺w means weak majorization, and sj(·) is the jth largest singular value of a matrix. To answer this problem, we construct two 3×3 entrywise positive symmetric matrices. We give rigorous upper and lower bounds for the relevant singular-value sums by Sturm's theorem, and show that the inequality fails for the partial sum with k=2.
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BibTeX
@misc{shi2026counterexample,
title = {{A counterexample to Huang's weak majorization conjecture}},
author = {Shuo Shi and Juan Zhang and Yun Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09749},
eprint = {2609.09749},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09749},
}
RIS
TY - UNPB
TI - A counterexample to Huang's weak majorization conjecture
AU - Shuo Shi
AU - Juan Zhang
AU - Yun Zhang
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09749
UR - https://arxiv.org/abs/2609.09749
AB - We give a counterexample to a weak majorization problem, proposed by Z. Huang (Linear Algebra Appl., 434 (2) (2011) 457--462), for singular values of Hadamard products of nonnegative matrices. The problem asks whether, for all nonnegative matrices A and B, {s_j^2(AHad B)} ≺_w {s_j(AHad A)s_j(BHad B)} holds, where Had denotes the Hadamard product, ≺_w means weak majorization, and s_j(·) is the jth largest singular value of a matrix. To answer this problem, we construct two 3×3 entrywise positive symmetric matrices. We give rigorous upper and lower bounds for the relevant singular-value sums by Sturm's theorem, and show that the inequality fails for the partial sum with k=2.
ER -