OP-00073 Open

Chollet's permanent inequality for Hadamard products

Let $\operatorname{per}$ be the permanent and $A \circ B$ the Hadamard (entrywise) product. The inequality below was proved for $n = 2, 3$ by Gregorac and Hentzel, and for $n = 4$ by Rodtes.

Problem. Is $$\operatorname{per}(A \circ B) \le \operatorname{per}(A) \operatorname{per}(B)$$ for every $n \ge 1$ and all Hermitian positive semidefinite $A, B \in \mathbb{C}^{n \times n}$?

Source: Chollet (1982).

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Imported from github.com/MColbrook/AIM.

Classification

Primary: FA — Functional Analysis
Secondary: CO
Keywords: Hadamard product, OpenProblemsInNLA, permanents
Literature checked: Oct. 2, 2026 by clemens-admin

References

@article{chollet1982,
  author  = {Chollet, John},
  title   = {Is there a permanental analogue to {O}ppenheim's inequality?},
  journal = {Amer. Math. Monthly},
  volume  = {89},
  number  = {1},
  year    = {1982},
  pages   = {57--58},
  doi     = {10.1080/00029890.1982.11995380},
}
@article{gregorac1987,
  author  = {Gregorac, Robert J. and Hentzel, Irvin Roy},
  title   = {A note on the analogue of {O}ppenheim's inequality for permanents},
  journal = {Linear Algebra Appl.},
  volume  = {94},
  year    = {1987},
  pages   = {109--112},
  doi     = {10.1016/0024-3795(87)90082-6},
}
@article{rodtes2024,
  author        = {Rodtes, Kijti},
  title         = {Chollet's permanent conjecture for $4\times 4$ matrices},
  journal       = {Linear Multilinear Algebra},
  volume        = {72},
  number        = {16},
  year          = {2024},
  pages         = {2633--2638},
  doi           = {10.1080/03081087.2023.2279150},
  eprint        = {2211.02839},
  archivePrefix = {arXiv},
}
@misc{pant2026,
  author        = {Pant, Priyanshu and Singh, Ranveer},
  title         = {Structural Classes for {C}hollet's Permanent Conjecture},
  year          = {2026},
  eprint        = {2604.24192},
  archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mi10,
  author       = {Townsend, Alex},
  title        = {{MI-10}: Chollet's permanent inequality for Hadamard products},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-10/README.md},
  note         = {Read at commit e375a6f, 2026-09-30},
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00073, "Chollet's permanent inequality for Hadamard products", clopen.solutions.