OP-00156 Partially resolved

Rank-two maximizers of spectral spread on an entry interval

For $n \ge 2$ and $-1 \le a < 1$, let $\mathcal S_n[a,1]$ be the set of real symmetric $n \times n$ matrices all of whose entries, diagonal ones included, lie in $[a, 1]$, and let $s(A) = \lambda_{\max}(A) - \lambda_{\min}(A)$ be the spread of $A$. Fallat and Xing reduced the question below for an arbitrary real interval to these intervals. Biborski showed that every matrix of maximal spread in $\mathcal S_n[a,1]$ has all its entries in $\{a, 1\}$. The case $a = -1$ follows from a theorem of Zhan, and Calkin, Corless, Gonzalez-Vega, Sendra and Sendra showed, partly by exact computation, that the answer is yes for $-1 < a < 1$ when $n \le 7$, and for $a = 0$ when $n \le 8$ or $3$ divides $n$.

Problem. For every $n \ge 2$ and every $a \in [-1, 1)$, is $\max_{A \in \mathcal S_n[a,1]} s(A)$ attained by a matrix of rank exactly two all of whose entries lie in $\{a, 1\}$?

Source: Fallat and Xing (2012), §2 (numbered Conjecture 2.7 in Biborski's citation); also Conjecture 3.4.3 of Xing's PhD thesis (2011).

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Classification

Primary: NA — Numerical Analysis
Secondary: SP
Keywords: Bohemian matrices, matrix spread, OpenProblemsInNLA
Literature checked: Oct. 7, 2026 by clemens-admin

References

@article{fallat2012,
  author  = {Fallat, Shaun M. and Xing, YongJun},
  title   = {On the spread of certain normal matrices},
  journal = {Linear Multilinear Algebra},
  volume  = {60},
  number  = {11--12},
  year    = {2012},
  pages   = {1391--1407},
  doi     = {10.1080/03081087.2012.703189},
}
@phdthesis{xing2011,
  author = {Xing, Yongjun},
  title  = {On the Spread of Certain Normal Matrices},
  school = {University of Regina},
  year   = {2011},
  url    = {https://ourspace.uregina.ca/bitstream/10294/14274/1/NR88593.pdf},
}
@article{zhan2005,
  author  = {Zhan, Xingzhi},
  title   = {Extremal eigenvalues of real symmetric matrices with entries in an interval},
  journal = {SIAM J. Matrix Anal. Appl.},
  volume  = {27},
  number  = {3},
  year    = {2005},
  pages   = {851--860},
  doi     = {10.1137/050627812},
}
@article{biborski2022,
  author        = {Biborski, Iwo},
  title         = {Note on the spread of real symmetric matrices with entries in fixed interval},
  journal       = {Linear Algebra Appl.},
  volume        = {632},
  year          = {2022},
  pages         = {246--257},
  doi           = {10.1016/j.laa.2021.09.017},
  eprint        = {2103.09088},
  archivePrefix = {arXiv},
}
@misc{calkin2025,
  author        = {Calkin, Neil J. and Corless, Robert M. and Gonzalez-Vega, Laureano and Sendra, J. Rafael and Sendra, Juana},
  title         = {On the maximal spread of symmetric {B}ohemian matrices},
  year          = {2025},
  eprint        = {2510.15919},
  archivePrefix = {arXiv},
}
@misc{openproblemsinnla-sp08,
  author       = {Townsend, Alex},
  title        = {{SP-08}: Rank-two maximizers of spectral spread on an entry interval},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/eigenvalues-and-inverse-problems/SP-08/README.md},
  note         = {Read at commit 7336302, 2026-10-06},
}

Resolution claims

Centered Rank-Two Sign Matrices v1 GitHub ↗

Sidney Holden

This report proves the spread of the best rank-two block matrix as an upper bound for every endpoint matrix whose centered sign matrix has rank at most two, in every dimension and throughout the normalized interval $-1\le a<1$. An analytic monotonicity lemma retains integer block sizes, avoiding a fractional-mass relaxation. The negative-parameter comparison is analytic; the nonnegative comparison is established …

Partial resolution · not verified · fidelity not established · posted on GitHub 14 Sep 2026

A signed-threshold reduction and three exact finite cases v1 GitHub ↗

Sidney Holden

For real $a<b$ and every $n\ge2$, the maximum spread over real symmetric matrices with entries in $[a,b]$ equals the maximum over the $2^{2n-1}$ signed-threshold endpoint matrices $\frac{a+b}2\mathbf 1\mathbf 1^T+\frac{b-a}2S$ with $S_{ij}=d_id_j\varepsilon_{\min(i,j)}$. The proof perturbs the linear functional $xx^T-yy^T$ given by extreme eigenvectors of a maximizer, orders coordinate ratios, and passes to a limit. Enumerating this family, computing exact integer characteristic …

Partial resolution · not verified · fidelity not established · posted on GitHub 12 Sep 2026

Status history

When	Change	By	Reason
2026-10-09	Open → Partially resolved	The registry's curators	A signed-threshold reduction and three exact finite cases v1 recorded from OpenProblemsInNLA @ eb7c9d6, SP-08/proof.md.

Citing this problem

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

OP-00156, "Rank-two maximizers of spectral spread on an entry interval", clopen.solutions.