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Rank relations between a {0, 1}-matrix and its complement

Rank relations between a {0, 1}-matrix and its complement AbstractLet A be a {0, 1}-matrix and r(A) denotes its rank. The complement matrix of A is defined and denoted by Ac = J − A, where J is the matrix with each entry being 1. In particular, when A is a square {0, 1}-matrix with each diagonal entry being 0, another kind of complement matrix of A is defined and denoted by A = J − I − A, where I is the identity matrix. We determine the possible values of r(A) ± r(Ac) and r(A) ± r(A) in the general case and in the symmetric case. Our proof is constructive. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

Rank relations between a {0, 1}-matrix and its complement

Open Mathematics , Volume 16 (1): 6 – Mar 20, 2018

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Publisher
de Gruyter
Copyright
© 2018 Ma and Zhong, published by De Gruyter
ISSN
2391-5455
eISSN
2391-5455
DOI
10.1515/math-2018-0020
Publisher site
See Article on Publisher Site

Abstract

AbstractLet A be a {0, 1}-matrix and r(A) denotes its rank. The complement matrix of A is defined and denoted by Ac = J − A, where J is the matrix with each entry being 1. In particular, when A is a square {0, 1}-matrix with each diagonal entry being 0, another kind of complement matrix of A is defined and denoted by A = J − I − A, where I is the identity matrix. We determine the possible values of r(A) ± r(Ac) and r(A) ± r(A) in the general case and in the symmetric case. Our proof is constructive.

Journal

Open Mathematicsde Gruyter

Published: Mar 20, 2018

Keywords: {0, 1}-matrix; Complement matrix; Rank; 15A03; 15B36; 05B20; 05C50

References