The factor width rank of a matrix

IF 1 3区 数学 Q1 MATHEMATICS
Nathaniel Johnston , Shirin Moein , Sarah Plosker
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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 k×k 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.
如果一个矩阵可以被写成仅在一个 k×k 主子矩阵中不为零的正半inite 矩阵之和,则称该矩阵的因子宽度最多为 k。我们探讨了矩阵的 "因子宽度-k 级",即在这种因子宽度至多为 k 的分解中可以使用的秩 1 矩阵的最小数目。我们证明,带状矩阵或箭头矩阵的因子宽度秩等于其通常秩,但对于其他矩阵,它们可能不同。我们还建立了矩阵因子宽度秩的几个约束,包括因子宽度-k 秩与图的 k-clique 覆盖数之间的紧密联系,并讨论了在取哈达玛积和哈达玛幂时因子宽度和因子宽度秩如何变化。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.20
自引率
9.10%
发文量
333
审稿时长
13.8 months
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.
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