Minimum number of orthogonal conjugations for diagonal pinching
For $X \in \mathbb{R}^{d \times d}$ let $\Delta(X)$ be its diagonal part, and let $\varphi(d)$ be the least $q \ge 1$ for which there are $U_1, \dots, U_q \in O(d)$ with $$\Delta(X) = \frac1q \sum_{i=1}^q U_i X U_i^{T} \quad \text{for every } X \in \mathbb{R}^{d \times d}.$$ The same $U_1, \dots, U_q$ must serve every $X$. Taking $q = 2^{\lceil \log_2 d \rceil}$ always works (Bourin and Lee).
Problem. Determine $\varphi(d)$ for every $d \ge 1$.
Source: Bourin and Lee (2026), Question 4.8.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- NA
- Keywords
- OpenProblemsInNLA, orthogonal matrices, pinching
- Literature checked
- Oct. 2, 2026 by clemens-admin
References
@misc{bourin2026,
author = {Bourin, Jean-Christophe and Lee, Eun-Young},
title = {Averages over matrix unitary orbits and spectral order},
year = {2026},
eprint = {2606.15624},
archivePrefix = {arXiv},
}
@article{bhatia2000,
author = {Bhatia, Rajendra},
title = {Pinching, trimming, truncating, and averaging of matrices},
journal = {Amer. Math. Monthly},
volume = {107},
number = {7},
year = {2000},
pages = {602--608},
doi = {10.2307/2589115},
}
@misc{openproblemsinnla-mi08,
author = {Townsend, Alex},
title = {{MI-08}: Minimum number of orthogonal conjugations for diagonal pinching},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-08/README.md},
note = {Read at commit e375a6f, 2026-09-30},
}
Resolution claims
Let $\varphi(d)$ be the least $q$ for which fixed orthogonal matrices $O_1,\dots,O_q$ satisfy $\Delta(X)=\frac1q\sum_iO_iXO_i^T$ for every real $d\times d$ matrix $X$. Let $a(d)$ be the corresponding length when the $O_i$ may depend on $X$. Both equal $h(d)$, the least number of rows of a $\pm1$ matrix $H$ with $H^TH=h(d)I_d$. The upper bound uses the diagonal sign matrices given by the …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-08 | Open → Partially resolved | The registry's curators | Fixed and adaptive orthogonal pinching have the same length v1 recorded from OpenProblemsInNLA @ eb7c9d6, MI-08/solution.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00072, "Minimum number of orthogonal conjugations for diagonal pinching", clopen.solutions.