journal6 ›› 2010, Vol. 31 ›› Issue (1): 4-6.

• Mathematics • Previous Articles     Next Articles

On the Cardinalities of Row Space of Some Special Boolean Matrices


  1. (1.Student Affair Office,Zhanjiang Normal University,Zhanjiang 524048,Guangdong China;2.Mathematics and Computational Science School,Zhanjiang Normal University,Zhanjiang 524048,Guangdong China)
  • Online:2010-01-25 Published:2012-04-19
  • About author:ZHONG Li-ping(1963-),female,was born in Meizhou City,Guangdong Province,associate professor of Zhanjiang Normal University,M.S.D;research area is algebra theory.
  • Supported by:

    Zhanjiang Normal University Science Foundation (L0701)

Abstract: Let Bm×n  be the set of all m×n Boolean matrices;R(A) denote the row space of A∈Bn,|R(A)| denote the cardinality of R(A),m,n  be positive integers,and k be non negative integers.In this paper,we prove the following three results:(1) let A∈Bn×n,m,(ⅰ) if A  is the idempotent matrix,i.e.,A2=A,then |R(Am)|=|R(A)|;(ⅱ) if A  is the involutory  matrix,i.e.,A2=I,then |R(Am)|=|R(A)|  when m is an odd number or |R(A)|=2n when m is an even number;(2) let  A∈Bm×nbe k of  the numbers  of  1,0≤k≤min{m,n},and each row and column is at most one of the numbers of 1 in A,then |R(A)|=2k; (3) let A∈Bn×n be the partitioned matrix as A=(O OO A1),A1=(aij)k×k,aij=0(i>j),aij=1(i≤j),i,j=1,2,…,k,then |R(A)|=k+1.

Key words: Boolean matrix, row space, cardinality of a row space, permutation matrix

WeChat e-book chaoxing Mobile QQ