The minimum rank problem: A counterexample

被引：12

作者：

Kopparty, Swastik ^{[2
]}

Rao, K. P. S. Bhaskara ^{[1
]}

机构：

[1] Indiana State Univ, Dept Math & Comp Sci, Terre Haute, IN 47809 USA

[2] MIT, Comp Sci & Artificial Intelligence Lab, Cambridge, MA 02139 USA

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2008年 / 428卷 / 07期

关键词：

sign pattern matrix; zero nonzero pattern; minimum rank; minimum rank of a graph;

D O I：

10.1016/j.laa.2007.10.016

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We provide a counterexample to a recent conjecture that the minimum rank over the reals of every sign pattern matrix can be realized by a rational matrix. We use one of the equivalences of the conjecture and some results from projective geometry. As a consequence of the counterexample we show that there is a graph for which the minimum rank of the graph over the reals is strictly smaller than the minimum rank of the graph over the rationals. We also make some comments on the minimum rank of sign pattern matrices over different subfields of R. (C) 2007 Elsevier Inc. All rights reserved.

引用

页码：1761 / 1765

页数：5

共 50 条

[1] The minimum rank problem for circulants
Deaett, Louis
Meyer, Seth A.
[J]. LINEAR ALGEBRA AND ITS APPLICATIONS, 2016, 491 : 386 - 418
[2] The minimum rank problem over the finite field of order 2: Minimum rank 3
Barrett, Wayne
Grout, Jason
Loewy, Raphael
[J]. LINEAR ALGEBRA AND ITS APPLICATIONS, 2009, 430 (04) : 890 - 923
[3] Matroids and the Minimum Rank Problem for Matrix Patterns
Deaett, Louis
[J]. AMERICAN MATHEMATICAL MONTHLY, 2020, 127 (09): : 771 - 788
[4] THE MINIMUM RANK PROBLEM OVER FINITE FIELDS
Grout, Jason
[J]. ELECTRONIC JOURNAL OF LINEAR ALGEBRA, 2010, 20 : 691 - 716
[5] Expected duration of the no-information minimum rank problem
Demers, Simon
[J]. STATISTICS & PROBABILITY LETTERS, 2021, 168
[6] A COUNTEREXAMPLE TO THE RANK-COLORING CONJECTURE
ALON, N
SEYMOUR, PD
[J]. JOURNAL OF GRAPH THEORY, 1989, 13 (04) : 523 - 525
[7] A COUNTEREXAMPLE TO A COMPLEMENTATION PROBLEM
BOURGAIN, J
[J]. COMPOSITIO MATHEMATICA, 1981, 43 (01) : 133 - 144
[8] A Note on a Lower Bound on the Minimum Rank of a Positive Semidefinite Hankel Matrix Rank Minimization Problem
Xu, Yi
Ren, Xiaorong
Yan, Xihong
[J]. MATHEMATICAL PROBLEMS IN ENGINEERING, 2021, 2021
[9] An upper bound on the minimum rank of a symmetric Toeplitz matrix completion problem
Xu, Yi
Yan, Xihong
Guo, Jiahao
Ma, Cheng
[J]. OPTIMIZATION, 2023, 72 (09) : 2399 - 2414
[10] A COUNTEREXAMPLE TO THE SINGULAR SEQUENCE PROBLEM
VOIGT, J
[J]. INDIANA UNIVERSITY MATHEMATICS JOURNAL, 1990, 39 (01) : 27 - 30

← 1 2 3 4 5 →