Critical exponent for generalized doubly nonnegative matrices
A matrix $A \in \mathbb{R}^{n \times n}$ is generalized doubly nonnegative if it is entrywise nonnegative and diagonalizable with only nonnegative real eigenvalues; symmetry is not assumed. For such $A = S \operatorname{diag}(\lambda_1, \dots, \lambda_n) S^{-1}$ and real $t > 0$ let $A^t = S \operatorname{diag}(\lambda_1^t, \dots, \lambda_n^t) S^{-1}$, with $0^t = 0$, and let $\mathrm{CE}_n$ be the infimum of the $c \ge 0$ such that $A^t$ is entrywise nonnegative for every generalized doubly nonnegative $A \in \mathbb{R}^{n \times n}$ and every real $t > c$. For the symmetric matrices in this class, the doubly nonnegative ones, the corresponding exponent is $n - 2$ (Guillot, Khare and Rajaratnam). Han, Johnson and Paparella proved that $\mathrm{CE}_n$ is finite, with $\mathrm{CE}_n \le (n^2 - 2n)/2$ for even $n$ and $\mathrm{CE}_n \le (n^2 - 3n + 4)/2$ for odd $n$, and that $\mathrm{CE}_3 = 2$.
Problem. Is $\mathrm{CE}_n = 2(n - 2)$ for every integer $n \ge 3$?
Source: Han, Johnson and Paparella (2017), Question 4.7.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- RA
- Keywords
- critical exponents, matrix powers, nonnegative matrices, OpenProblemsInNLA
- Literature checked
- Oct. 7, 2026 by clemens-admin
References
@article{han2017,
author = {Han, Xuchen and Johnson, Charles R. and Paparella, Pietro},
title = {The critical exponent for generalized doubly nonnegative matrices},
journal = {Linear Multilinear Algebra},
volume = {65},
number = {5},
year = {2017},
pages = {1035--1044},
doi = {10.1080/03081087.2016.1223009},
eprint = {1407.7059},
archivePrefix = {arXiv},
}
@article{guillot2013,
author = {Guillot, Dominique and Khare, Apoorva and Rajaratnam, Bala},
title = {The critical exponent conjecture for powers of doubly nonnegative matrices},
journal = {Linear Algebra Appl.},
volume = {439},
number = {8},
year = {2013},
pages = {2422--2427},
doi = {10.1016/j.laa.2013.06.031},
eprint = {1303.4701},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mf15,
author = {Townsend, Alex},
title = {{MF-15}: Critical exponent for generalized doubly nonnegative matrices},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-functions-and-stability/MF-15/README.md},
note = {Read at commit b8c7852, 2026-10-07},
}
Resolution claims
For every $n\ge3$, the conventional critical exponent of the $n\times n$ generalized doubly nonnegative matrices is at least $2n-4$. The proof uses a rational, nonnegative cyclic-bidiagonal family with two eigenvalues of order one and $n-2$ positive eigenvalues of order $\epsilon$. A particular entry of its real powers has a negative leading coefficient throughout the interval $(2n-5,2n-4)$. Together with the published …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-08 | Open → Partially resolved | The registry's curators | A linear lower bound for generalized doubly nonnegative critical exponents v1 recorded from OpenProblemsInNLA @ eb7c9d6, manuscripts/MF-15.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00166, "Critical exponent for generalized doubly nonnegative matrices", clopen.solutions.