AI 中文总结
本文研究SNT-秩与克罗内克积、欧氏距离矩阵的关系，推导了欧氏距离矩阵SNT-秩的更紧上界，证明SNT-秩关于克罗内克积次可乘，还解决了相关秩的猜想。
AI 中文摘要
对称非负矩阵三重分解（SN-Trifactorizations）由Bukovšek-Šmigoc于《Linear Algebra Appl. 2023》中提出，是非负矩阵分解的对称类似物。对称非负矩阵A的SN-三重分解形式为A = BCBᵀ，其中B和C为非负矩阵，且C对称。A的相关SNT-秩定义为最小整数k，使得A存在满足C ∈ ℝ₊ᵏˣᵏ的此类分解。本文中，我们针对Shitov于《Linear Algebra Appl. 2025》及Bukovšek-Šmigoc于《Linear Algebra Appl. 2023》中考虑的欧氏距离矩阵，推导了SNT-秩的更紧上界；还建立了对称非负矩阵的秩与SNT-秩之间的若干新关系，证明SNT-秩关于克罗内克积是次可乘的。最后，受Dagstuhl Seminar Report 13082中提出的一个猜想的启发，我们在附加结构假设下证明了非负秩的一个可乘性结果，还部分解决了Vandaele-Gillis-Glineur-Tuyttens于《J. Global Optim. 2016》中提出的一个猜想。
英文摘要
Symmetric nonnegative matrix trifactorizations (SN-Trifactorizations) were introduced by Bukovšek-Šmigoc [Linear Algebra Appl. 2023] as a symmetric analogue of nonnegative matrix factorizations. A SN-Trifactorization of a symmetric nonnegative matrix $A$ is of the form $A = BCB^{T},$ where $B$ and $C$ are nonnegative matrices, with $C$ symmetric. The associated SNT-rank of $A$ is defined as the smallest integer $k$ for which $A$ admits such a factorization with $C \in \mathbb{R}_{+}^{k \times k}$. In this paper, we derive sharper upper bounds for the SNT-rank of the Euclidean distance matrices considered by Shitov [Linear Algebra Appl. 2025] and Bukovšek-Šmigoc [Linear Algebra Appl. 2023]. We also establish several new relationships between the rank and the SNT-rank of symmetric nonnegative matrices and show that the SNT-rank is submultiplicative with respect to the Kronecker product. Finally, motivated by a conjecture posed in the Dagstuhl Seminar Report 13082, we prove a multiplicativity result for the nonnegative rank under an additional structural assumption. We also partially resolve a conjecture of Vandaele-Gillis-Glineur-Tuyttens [J. Global Optim. 2016].
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