The determinant of {±1}-matrices and oriented hypergraphs

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2024-08-20 DOI:10.1016/j.laa.2024.08.013

Lucas J. Rusnak , Josephine Reynes , Russell Li , Eric Yan , Justin Yu

引用次数: 0

Abstract

The determinants of ${\pm 1}$ -matrices are calculated via the oriented hypergraphic Laplacian and summing over incidence generalizations of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of 0 to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries of the incidence matrix are reclaimed using only the signs of an adjacency-minimal set of cycle-covers from an edge-monic family.

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{±1}矩阵和定向超图的行列式

{±1}矩阵的行列式是通过定向超图拉普拉卡和顶点循环覆盖的入射概括求和计算得出的。这些循环覆盖被签名，并根据它们的超边包含性划分为族。结果表明，每个非边缘单值族对拉普拉卡方的净贡献值为 0，而每个边缘单值族的总和为原始入射矩阵行列式的绝对值。我们还确定了简单的对称性及其与哈达玛最大行列式问题的关系。最后，只需使用边缘单子族循环覆盖的邻接最小集的符号，就能重新获得入射矩阵的条目。

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

Linear Algebra and its Applications 数学-应用数学

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.