{"title":"The factor width rank of a matrix","authors":"Nathaniel Johnston , Shirin Moein , Sarah Plosker","doi":"10.1016/j.laa.2025.03.016","DOIUrl":null,"url":null,"abstract":"<div><div>A matrix is said to have factor width at most <em>k</em> if it can be written as a sum of positive semidefinite matrices that are non-zero only in a single <span><math><mi>k</mi><mo>×</mo><mi>k</mi></math></span> principal submatrix. We explore the “factor-width-<em>k</em> rank” of a matrix, which is the minimum number of rank-1 matrices that can be used in such a factor-width-at-most-<em>k</em> decomposition. We show that the factor width rank of a banded or arrowhead matrix equals its usual rank, but for other matrices they can differ. We also establish several bounds on the factor width rank of a matrix, including a tight connection between factor-width-<em>k</em> rank and the <em>k</em>-clique covering number of a graph, and we discuss how the factor width and factor width rank change when taking Hadamard products and Hadamard powers.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"716 ","pages":"Pages 32-59"},"PeriodicalIF":1.0000,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379525001247","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A matrix is said to have factor width at most k if it can be written as a sum of positive semidefinite matrices that are non-zero only in a single principal submatrix. We explore the “factor-width-k rank” of a matrix, which is the minimum number of rank-1 matrices that can be used in such a factor-width-at-most-k decomposition. We show that the factor width rank of a banded or arrowhead matrix equals its usual rank, but for other matrices they can differ. We also establish several bounds on the factor width rank of a matrix, including a tight connection between factor-width-k rank and the k-clique covering number of a graph, and we discuss how the factor width and factor width rank change when taking Hadamard products and Hadamard powers.
期刊介绍:
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.