OP-00263 Partially resolved

The maximum permanent on a positive-semidefinite unitary orbit

Let $\mathcal U_n$ be the group of complex unitary $n \times n$ matrices. For $n \ge 1$ and $\lambda = (\lambda_1, \dots, \lambda_n)$ with every $\lambda_i \ge 0$ let $$M_n(\lambda) = \max_{U \in \mathcal U_n} \operatorname{per}\bigl(U^* \operatorname{diag}(\lambda_1, \dots, \lambda_n)\, U\bigr),$$ the largest permanent of a Hermitian positive semidefinite matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. These permanents are real, and the maximum is attained by compactness. It is not always attained at a matrix whose diagonal entries are all equal: Drew and Johnson gave a counterexample to Mehta's conjecture that it is. They also determined the largest permanent of a real symmetric positive semidefinite $3 \times 3$ matrix with prescribed eigenvalues, and Hu showed that the same value is the largest over Hermitian matrices, which gives $M_3(\lambda)$.

Problem. For every $n \ge 1$ and every $\lambda \in [0, \infty)^n$, determine the exact value of $M_n(\lambda)$, expressed in terms of $\lambda_1, \dots, \lambda_n$.

Source: Marcus and Minc (1965), as recorded by Zhang (2016), Section 7 (the "Marcus–Minc max-per-U problem").

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Classification

Primary: FA — Functional Analysis
Secondary: CO
Keywords: OpenProblemsInNLA, permanents, positive semidefinite matrices, unitary orbits
Literature checked: Oct. 8, 2026 by clemens-admin

References

@article{marcus1965,
  author  = {Marcus, Marvin and Minc, Henryk},
  title   = {Permanents},
  journal = {Amer. Math. Monthly},
  volume  = {72},
  number  = {6},
  year    = {1965},
  pages   = {577--591},
  doi     = {10.1080/00029890.1965.11970575},
}
@article{zhang2016,
  author        = {Zhang, Fuzhen},
  title         = {An update on a few permanent conjectures},
  journal       = {Spec. Matrices},
  volume        = {4},
  number        = {1},
  year          = {2016},
  pages         = {305--316},
  doi           = {10.1515/spma-2016-0030},
  eprint        = {1608.02844},
  archivePrefix = {arXiv},
}
@article{drew1989counterexample,
  author  = {Drew, John H. and Johnson, Charles R.},
  title   = {Counterexample to a conjecture of {M}ehta regarding permanental maximization},
  journal = {Linear Multilinear Algebra},
  volume  = {25},
  number  = {3},
  year    = {1989},
  pages   = {253--254},
  doi     = {10.1080/03081088908817948},
}
@article{drew1989maximum,
  author  = {Drew, John H. and Johnson, Charles R.},
  title   = {The maximum permanent of a $3$-by-$3$ positive semidefinite matrix, given the eigenvalues},
  journal = {Linear Multilinear Algebra},
  volume  = {25},
  number  = {3},
  year    = {1989},
  pages   = {243--251},
  doi     = {10.1080/03081088908817947},
}
@article{hu1994,
  author  = {Hu, Shu-An},
  title   = {On a conjecture of {D}rew and {J}ohnson: maximizing the permanent},
  journal = {Linear Multilinear Algebra},
  volume  = {36},
  number  = {4},
  year    = {1994},
  pages   = {279--285},
  doi     = {10.1080/03081089408818302},
}
@article{hu1996,
  author  = {Hu, Shu-An},
  title   = {Maximum permanent and {H}ermitian matrices},
  journal = {Linear Algebra Appl.},
  volume  = {235},
  year    = {1996},
  pages   = {93--105},
  doi     = {10.1016/0024-3795(94)00121-9},
}
@misc{openproblemsinnla-mi16,
  author       = {Townsend, Alex},
  title        = {{MI-16}: The maximum permanent on a positive-semidefinite unitary orbit},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-16/README.md},
  note         = {Read at commit eb7c9d6, 2026-10-07},
}

Resolution claims

An exact algebraic determination and a sharp exceptional-spectrum formula v1 GitHub ↗

Sidney Holden

For every order $n$ and every nonnegative spectrum $\lambda = (\lambda_1, \dots, \lambda_n)$, the dossier gives an exact finite algebraic prescription for the maximum $M_n(\lambda)$ of $\operatorname{per}(U^* \operatorname{diag}(\lambda) U)$ over unitary $U$. A maximizer is a critical point of the permanent on the similarity orbit, where $A$ commutes with its permanental adjoint; Gröbner elimination of these equations, with the spectrum …

Partial resolution · not verified · fidelity not established · posted on GitHub 13 Sep 2026

Status history

When	Change	By	Reason
2026-10-08	Open → Partially resolved	The registry's curators	An exact algebraic determination and a sharp exceptional-spectrum formula v1 recorded from OpenProblemsInNLA @ eb7c9d6, MI-16/solution.pdf.

Citing this problem

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

OP-00263, "The maximum permanent on a positive-semidefinite unitary orbit", clopen.solutions.