Symmetric nonnegative trifactorization of pattern matrices

Kokol-Bukovšek, Damjana (author), Šmigoc, Helena (author)

A factorization of an $n$ x $n$ nonnegative symmetric matrix $A$ of the form $BCB^T$, where $C$ is a $k$ x $k$ symmetric matrix, and both $B$ and $C$ are required to be nonnegative, is called the Symmetric Nonnegative Matrix Trifactorization (SN-Trifactorization). The SNT-rank of $A$ is the minimal $k$ for which such factorization exists. The SNT-rank of a simple graph $G$ that allows loops is defined to be the minimal possible SNT-rank of all symmetric nonnegative matrices whose zero-nonzero pattern is prescribed by the graph $G$. We define set-join covers of graphs, and show that finding the SNT-rank of $G$ is equivalent to finding the minimal order of a set-join cover of $G$. Using this insight we develop basic properties of the SNT-rank for graphs and compute it for trees and cycles without loops. We show the equivalence between the SNT-rank for complete graphs and the Katona problem, and discuss uniqueness of patterns of matrices in the factorization.

• Language: English
• Keywords: mathematics, mathematical economy, matrix algebra, nonnegative matrix factorization, nonnegative symmetric matrices, symmetric nonnegative trifactorization, pattern matrices
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: EF - School of Economics and Business
• Publication status: Published
• Publication version: Version of Record
• Year: 2025
• Number of pages: Str. 310-338
• Numbering: Vol. 721
• PID: 20.500.12556/RUL-169499
• UDC: 330.4
• ISSN on article: 0024-3795
• DOI: 10.1016/j.laa.2024.05.017
• COBISS.SI-ID: 197444611
• Publication date in RUL: 30.05.2025

Record is a part of a journal

• Title: Linear algebra and its applications
• Shortened title: Linear algebra appl.
• Publisher: Elsevier
• ISSN: 0024-3795
• COBISS.SI-ID: 1119247

Licences

• License: CC BY-NC 4.0, Creative Commons Attribution-NonCommercial 4.0 International
• Link: http://creativecommons.org/licenses/by-nc/4.0/
• Description: A creative commons license that bans commercial use, but the users don’t have to license their derivative works on the same terms.

Secondary language

• Language: Slovenian
• Keywords: matematika, matematična ekonomija, matrična algebra

Projects

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: P1-0222
• Name: Algebra, teorija operatorjev in finančna matematika