Ghabries's determinant comparison for arbitrary nonnegative base powers
For $X \in \mathbb{C}^{n \times n}$ let $|X| = (X^*X)^{1/2}$; real powers of positive definite matrices are taken by functional calculus, with $X^0 = I$. The matrix $A^k + A^pB^p$ need not be Hermitian, but its determinant is positive, since $A^{p-k}B^p$, a product of two positive definite matrices, has positive eigenvalues. For $k = 2$ the inequality below is a conjecture of Lin. Abbas, Ghabries and Mourad proved it for $p = 2$ and every $k \ge 0$, and Ghabries, Abbas, Mourad and Assi for every $k \ge 2$. Ghabries also proved it when $k \ge 2p$, and when $1 \le p \le 2$ and $k \ge 3p - 2$.
Problem. Is $$\det(A^k + |AB|^p) \ge \det(A^k + A^pB^p)$$ for every $n \ge 1$, all positive definite $A, B \in \mathbb{C}^{n \times n}$, every real $k \ge 0$ and every $p \in [0, 2]$?
Source: Ghabries (2022), PhD thesis, Conjecture 2.3 and Open Problems, Problem 1, for positive semidefinite A and B; stated for positive definite A and B by OpenProblemsInNLA. The case k = 2 is Lin's (2017) conjecture, and the base powers kp with k ≥ 1 are Abbas, Ghabries and Mourad's (2021) Conjecture 2, with A^k on the right where the thesis prints A^2.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- NA
- Keywords
- determinantal inequalities, OpenProblemsInNLA, positive definite matrices
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@phdthesis{ghabries2022,
author = {Ghabries, Mohammad Mahdi},
title = {Contributions to Matrix Inequalities and Some Applications},
school = {Universit{\'e} d'Angers and Lebanese University},
year = {2022},
doi = {10.70675/32bff73dzcb5az430ezbda8ze209c4b1c319},
note = {HAL tel-03936351},
url = {https://theses.hal.science/tel-03936351},
}
@article{lin2017,
author = {Lin, Minghua},
title = {On a determinantal inequality arising from diffusion tensor imaging},
journal = {Commun. Contemp. Math.},
volume = {19},
number = {5},
year = {2017},
pages = {1650044},
doi = {10.1142/S0219199716500449},
eprint = {1604.04141},
archivePrefix = {arXiv},
}
@article{abbas2021,
author = {Abbas, Hassane and Ghabries, Mohammad M. and Mourad, Bassam},
title = {New determinantal inequalities concerning {H}ermitian and positive semi-definite matrices},
journal = {Oper. Matrices},
volume = {15},
number = {1},
year = {2021},
pages = {105--116},
doi = {10.7153/oam-2021-15-07},
}
@article{ghabries2020proof,
author = {Ghabries, Mohammad M. and Abbas, Hassane and Mourad, Bassam and Assi, Abdallah},
title = {A proof of a conjectured determinantal inequality},
journal = {Linear Algebra Appl.},
volume = {605},
year = {2020},
pages = {21--28},
doi = {10.1016/j.laa.2020.07.013},
}
@misc{openproblemsinnla-mi28,
author = {Townsend, Alex},
title = {{MI-28}: Ghabries's determinant comparison for arbitrary nonnegative base powers},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-28/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
For positive definite matrices $A, B$, every $k \ge 0$ and $0 \le p \le 2$, we prove $$\det(A^k + |AB|^p) \ge \det(A^k + A^pB^p).$$ In fact, a log-majorization of the corresponding normalized matrices holds: $\lambda(A^{(p-k)/2} B^p A^{(p-k)/2}) \prec_{\log} \lambda(A^{-k/2} |AB|^p A^{-k/2})$. The previously proved range $k \ge 2$ is due to Ghabries, Abbas, Mourad and Assi. We treat the …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-08 | Needs verification → Solution claimed | clemens-admin | Listed with the record of its solution, unreviewed. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00250, "Ghabries's determinant comparison for arbitrary nonnegative base powers", clopen.solutions.