Rational and Integral Completely Positive Factorizations

Definitions and mathematical statement

For each integer $n\geq 1$, let $\mathbb S^n$ denote the real symmetric $n\times n$ matrices. A matrix $A\in\mathbb S^n$ is completely positive if there exist an integer $r\geq 1$ and vectors $v_1,\ldots,v_r\in{\mathbb{R}}_{\geq 0}^n$ such that \[ A=\sum_{k=1}^{r}v_kv_k^{\mathsf T}. \] Equivalently, $A=BB^{\mathsf T}$ for an entrywise nonnegative real matrix $B\in{\mathbb{R}}_{\geq 0}^{n\times r}$. Write $\mathcal{CP}_n$ for the cone of these matrices. A rational completely positive factorization requires $B\in{\mathbb{Q}}_{\geq 0}^{n\times r}$; an integral completely positive factorization requires $B\in{\mathbb{Z}}_{\geq 0}^{n\times r}$. The number $r$ may depend on $A$, and no bound on it is imposed.

The supplied record asks the following two questions.

1. Is it true that \[ \forall n\geq 1\;\forall A\in\mathcal{CP}_n\cap{\mathbb{Q}}^{n\times n},\quad \exists r\geq 1\;\exists B\in{\mathbb{Q}}_{\geq 0}^{n\times r} \quad A=BB^{\mathsf T}? \]
2. Is it true that \[ \forall A\in\mathcal{CP}_2\cap{\mathbb{Z}}^{2\times 2},\quad \exists r\geq 1\;\exists B\in{\mathbb{Z}}_{\geq 0}^{2\times r} \quad A=BB^{\mathsf T}? \]

The first question includes matrices on the boundary of $\mathcal{CP}_n$, and both questions include singular matrices and the zero matrix.

Supplied dated qualification. The record's literature assessment dated 2026-08-21 reports an affirmative answer to the first question for rational matrices in $\operatorname{int}_{\mathbb S^n}(\mathcal{CP}_n)$, and an affirmative answer to the second question for every integral $2\times 2$ completely positive matrix. Here the interior is taken in the usual Euclidean topology on $\mathbb S^n$. The assessment reports that the first question remains unresolved for arbitrary rational completely positive matrices on the boundary. Accordingly, the record labels the combined problem partially solved. These are supplied dated curation claims, not independently verified mathematical or current-status conclusions.

Short English statement

A completely positive matrix is a finite sum of outer products of nonnegative real vectors. The questions ask whether rational entries always permit rational vectors, and whether integral entries in dimension $2$ always permit integral vectors. The supplied assessment dated 2026-08-21 reports affirmative results for rational interior matrices and for the integral $2\times 2$ question, while leaving the general rational boundary case unresolved; this assessment has not been independently verified.

Sources

• Local UnsolvedMath registry: supplied complete source record for permanent ID $30003649$, titled Rational and Integral Completely Positive Factorizations, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2017 and names no individual proposer. Its supplied statement status is corrected_verified, with an explanation that mathematical content was preserved while notation, typography, or omitted definitions were repaired; this is registry metadata, not independent verification here.
• Supplied workshop source citation: Copositivity and Complete Positivity (2018), Oberwolfach Reports, pp. 3071--3120, DOI 10.4171/owr/2017/52. https://doi.org/10.4171/owr/2017/52. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied 2018 citation label is retained alongside the 2017 DOI component and proposed year, without resolving their relationship.
• Supplied literature reference: Dutour Sikirić--Schürmann--Vallentin, rational cp factorizations. https://doi.org/10.1016/j.laa.2017.02.017.
• Supplied literature reference: A simplex algorithm for rational cp-factorization. https://doi.org/10.1007/s10107-020-01467-4.
• Supplied literature reference: Oberwolfach Report 52, open questions. https://doi.org/10.4171/owr/2017/52.
• Literature assessment date: 2026-08-21. The supplied dated curation reports rational factorizations for rational interior completely positive matrices, integral factorizations for every integral $2\times 2$ completely positive matrix, and an unresolved general rational boundary case. The background explicitly describes this as dated literature triage derived from the supplied OWR curation and calls for independent verification of time-sensitive status and cited claims. No source visit or independent literature verification is claimed here.