一些特殊结构的布尔矩阵行空间基数

R ( A) denote t he row space of A ∈B n , | R( A) | denote t he cardi nali t y of R ( A) . Then t he cardinalit y is in
[1 ,2 n ]. The re search of t he di st ribution of ca rdinalit y of row space of mat ri x A ∈ B n has a long ti me. In 1992 ,ref. [ 1 ] gave t he dist ribut io n of cardi nali t y of row space of mat rix A ∈B n i n interval ( 2 n - 1 , 2 n ] . Sim ult aneousl y ,he conject ured t hat t here exi st s A ∈B n wi t h | R ( A) | = m for any m ∈[ 1 , 2 n- 1 ] . In 1995 , ref . [2 - 6 ] showed some re sul t s on t he cardi nali ties of row space of Boolean mat rices . In t hi s paper ,by ) if A is t he st udyi ng some special Boolea n mat rices ,we obt ai n t hree res ul t s : ( 1 ) l et , A ∈B n ×n , Π m , ( ⅰ ) if A i s t he i nvol ut ory mat ri x ,i . e. , A2 = I , ide mpot ent mat ri x ,i . e . , A2 = A ,t hen | R ( Am ) | = R ( A) | ; ( ⅱ t hen | R ( Am ) | = | R ( A) | whe n m i s an odd numbe r or | R ( A) | = 2 n when m i s a n even number ; ( 2 ) let A ∈
O O O A1 ,A1 = ( a ij ) k ×k , a ij = 0 ( i > j) , a ij = 1 ( i ≤j) , i , j = 1 , 2 ,
1 In tr oduct ion
For any positi ve m , n ,let B m ×n denote t he set of all m ×n Boolean mat rice s. While m = n ,we wri te B n . Then B n i s a semi2group wit h t he ordinary mat rix m ultiplicat ion and ent ries usi ng Boolean operation. Let
第 31 卷 第 1期 2010 年 1 月
吉首大学学报 ( 自然科学版)
J ournal of Jral Sci ence Editio n)
Vol . 31 No. 1 J an. 2010
Ar t icle ID :1007 - 2985 (2010) 01 - 0004 - 03
On t he Car din al it ies of Ro w Spa ce of Some Special Boolea n Mat r ices
ZHON G Li2pi ng1 ,DEN G 2jian2
3
( 1. Student Affair Office , Zhanjia ng Normal Univer sity , Zha njiang 524048 , Gua ngdong China ;2. Mathe ma tics a nd Computational Science Sc hool , Zhanjiang Normal Univer sity , Zha njiang 524048 , Gua ngdong China)
Abstract :Let B m ×n be t he set of all m ×n Boolean mat rices ; R (A) de not e t he row space of A ∈B n , | R( A) | denot e t he cardi nalit y of R (A) , m , n be po sit ive integer s ,and k be non negat ive i nt egers . In t hi s paper ,we prove t he followi ng t hree re sult s : (1) let A ∈ B n ×n , Πm , ( ⅰ ) if A i s t he idempot ent mat ri x ,i. e. , A2 = A , ) if A i s t he invol utory mat ri x ,i . e. , A2 = I ,t he n | R ( Am ) | = | R ( A) | when m t hen | R ( Am ) | = | R ( A) | ; ( ⅱ i s an odd number or | R ( A) | = 2 n w hen m is an even number ; ( 2) l et A ∈B m ×n be k of t he num bers of 1 ,0 ≤k ≤mi n{ m, n} ,and each row and col um n i s at mo st one of t he numbe rs of 1 i n A ,t hen | R ( A) | = 2 k ; ( 3) let A ∈B n×n be t he part itioned mat ri x a s A = …, k , t hen | R (A) | = k + 1 . Key wor ds :Boolean mat ri x ; row space ;cardinalit y of a row space ;perm ut at io n mat ri x CL C number :O151. 21 Document code :A