一些特殊结构的布尔矩阵行空间基数

设Bm×n是所有m×n布尔矩阵的集合，R(A)为A∈Bn的行空间，|R(A)|表示行空间R(A)的基数,m,n是正整数,k为非负整数.证明了如下3个结果:(1) 设A∈Bm×n,m,(ⅰ) 如果A是幂等矩阵,即A2=A,那么|R(Am)|=|R(A)| ；(ⅱ) 如果A是对合矩阵，即A2=I,那么当m是奇数时，|R(Am)|=|R(A)|，当m是偶数时|R(A)|=2n.(2) 设A∈Bm×n,A含1的元素个数为k,0≤k≤min{m,n},且A的每行每列元素中1的元素个数最多为1,那么|R(A)|=2k.(3) 若A∈Bm×n是形如A=(O OO A1)的分块矩阵,A1=(aij)k×k,aij=0(i>j),aij=1(i≤j),i,j=1,2,…,k,则|R(A)|=k+1.

On the Cardinalities of Row Space of Some Special Boolean Matrices"

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 i...