Infinitesimal rigidity detects unique size-two factors in the presence of zeros
Let $\mathbb{S}^2_+$ be the cone of real symmetric positive semidefinite $2 \times 2$ matrices. Let $M \in \mathbb{R}_{\ge 0}^{p \times q}$ have rank $3$ and positive semidefinite rank $2$, and fix a size-two positive semidefinite factorization $M_{ij} = \operatorname{tr}(A_i B_j)$ with $A_1, \dots, A_p, B_1, \dots, B_q \in \mathbb{S}^2_+$; zero entries of $M$ are allowed. Call a tuple $(E_1, \dots, E_p, F_1, \dots, F_q)$ of real symmetric $2 \times 2$ matrices a feasible linear direction if $\operatorname{tr}(E_i B_j) + \operatorname{tr}(A_i F_j) = 0$ for all $i, j$ and, for some $h > 0$, $A_i + tE_i$ and $B_j + tF_j$ are positive semidefinite for all $i, j$ and all $t \in [0, h)$. The trace condition is imposed only to first order: the perturbed factors need not represent $M$ for $t > 0$. For every real $d$, $E_i = dA_i$, $F_j = -dB_j$ is a feasible linear direction, and for every $S \in GL(2, \mathbb{R})$ the matrices $S^{\mathsf T} A_i S$ and $S^{-1} B_j S^{-\mathsf T}$ form another size-two factorization of $M$. In the terms of Dawson, Hoşten, Kubjas and Metsälampi, condition (i) below is 2-infinitesimal rigidity and condition (ii) is global rigidity; they proved that the two are equivalent when every entry of $M$ is positive.
Problem. Are the following two conditions equivalent for every such $M$ and every size-two positive semidefinite factorization of it: (i) every feasible linear direction has the form $E_i = dA_i$, $F_j = -dB_j$ for a single real $d$ common to all factors; (ii) every size-two positive semidefinite factorization $(\widetilde A_i, \widetilde B_j)$ of $M$ satisfies $\widetilde A_i = S^{\mathsf T} A_i S$ and $\widetilde B_j = S^{-1} B_j S^{-\mathsf T}$ for a single $S \in GL(2, \mathbb{R})$?
Source: Dawson, Hoşten, Kubjas and Metsälampi (2024), the conjecture after Theorem 43 of arXiv:2410.18891v1 (after Theorem 8 in v2, 2026) that 2-infinitesimal, local and global rigidity coincide; stated by OpenProblemsInNLA without local rigidity, which their Lemma 42 (Lemma 19 in v2) shows equivalent to global rigidity in this class.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- MG, OC
- Keywords
- infinitesimal rigidity, OpenProblemsInNLA, positive semidefinite factorizations, positive semidefinite rank
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@misc{dawson2024,
author = {Dawson, Kristen and Ho{\c{s}}ten, Serkan and Kubjas, Kaie and Mets{\"a}lampi, Lilja},
title = {Uniqueness of size-2 positive semidefinite matrix factorizations},
year = {2024},
eprint = {2410.18891},
archivePrefix = {arXiv},
note = {Version 2, 31 August 2026, accepted for a Fields Communications volume; Definitions 1, 3 and 8, Remark 1, Theorems 3 and 8, Lemma 19 and the conjecture after Theorem 8},
}
@misc{openproblemsinnla-pf05,
author = {Townsend, Alex},
title = {{PF-05}: Infinitesimal rigidity detects unique size-two factors in the presence of zeros},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/nonnegative-and-positive-factorizations/PF-05/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
For real size-two positive semidefinite factorizations $M_{ij} = \operatorname{tr}(A_i B_j)$ of matrices $M$ of ordinary rank three, the paper proves that linear (2-infinitesimal) rigidity, meaning that every feasible straight-line first-order direction is a common scaling $E_i = dA_i$, $F_j = -dB_j$, is equivalent to uniqueness of the factorization up to the congruence action of $GL(2, \mathbb{R})$, also when $M$ has …
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-00262, "Infinitesimal rigidity detects unique size-two factors in the presence of zeros", clopen.solutions.