模糊数学教案02.ppt

文档介绍：第 2 章模糊聚类分析
§ 模糊矩阵
定义1 设R = (rij)m×n,若0≤rij≤1,则称R为模糊矩阵. 当rij只取0或1时,称R为布尔(Boole)矩阵. 当模糊方阵R = (rij)n×n的对角线上的元素rii都为1时,称R为模糊自反矩阵.
定义2 设A=(aij)m×n,B=(bij)m×n都是模糊矩阵,
相等:A = B  aij = bij;
包含:A≤B  aij≤bij;
并:A∪B = (aij∨bij)m×n;
交:A∩B = (aij∧bij)m×n;
余:Ac = (1- aij)m×n.
模糊矩阵的并、交、余运算性质
幂等律:A∪A = A,A∩A = A;
交换律:A∪B = B∪A,A∩B = B∩A;
结合律:(A∪B)∪C = A∪(B∪C),
(A∩B)∩C = A∩(B∩C);
吸收律:A∪(A∩B) = A,A∩(A∪B) = A;
分配律:(A∪B)∩C = (A∩C )∪(B∩C);
(A∩B)∪C = (A∪C )∩(B∪C);
0-1律: A∪O = A,A∩O = O;
A∪E = E,A∩E = A;
还原律:(Ac)c = A;
对偶律: (A∪B)c =Ac∩Bc, (A∩B)c =Ac∪Bc.
模糊矩阵的合成运算与模糊方阵的幂
设A = (aik)m×s,B = (bkj)s×n,定义模糊矩阵A 与B 的合成为:
A ° B = (cij)m×n,
其中cij = ∨{(aik∧bkj) | 1≤k≤s} .
模糊方阵的幂
定义:若A为 n 阶方阵,定义A2 = A ° A,A3 = A2 ° A,…,Ak = Ak-1 ° A.
合成(° )运算的性质:
性质1:(A ° B) ° C = A ° (B ° C);
性质2:Ak ° Al = Ak + l,(Am)n = Amn;
性质3:A ° ( B∪C ) = ( A ° B )∪( A ° C );
( B∪C ) ° A = ( B ° A )∪( C ° A );
性质4:O ° A = A ° O = O,I ° A=A ° I =A;
性质5:A≤B,C≤D  A ° C ≤B ° D.
注:合成(° )运算关于(∩)的分配律不成立,即
( A∩B ) ° C ( A ° C )∩( B ° C )
( A∩B ) ° C
( A ° C )∩( B ° C )
( A∩B ) ° C ( A ° C )∩( B ° C )
模糊矩阵的转置
定义设A = (aij)m×n, 称AT = (aijT )n×m为A的转置矩阵,其中aijT = aji.
转置运算的性质:
性质1:( AT )T = A;
性质2:( A∪B )T = AT∪BT,
( A∩B )T = AT∩BT;
性质3:( A ° B )T = BT ° AT;( An )T = ( AT )n ;
性质4:( Ac )T = ( AT )c ;
性质5:A≤B  AT ≤BT .
证明性质3:( A ° B )T = BT ° AT;( An )T = ( AT )n .
证明:设A=(aij)m×s, B=(bij)s×n, A °B=C =(cij)m×n,
记( A ° B )T = (cijT )n×m , AT = (aijT )s×m , BT = (bijT )n×s ,
由转置的定义知,
cijT = cji , aijT = aji , bijT = bji .
BT ° AT= [∨(bikT∧akjT )]n×m
=[∨(bki∧ajk)]n×m
=[∨(ajk∧bki)]n×m = (cji)n×m
= (cijT )n×m= ( A ° B )T .
模糊矩阵的- 截矩阵
定义7 设A = (aij)m×n,对任意的∈[0, 1],称
A= (aij())m×n,
为模糊矩阵A的- 截矩阵, 其中
当aij≥时,aij() =1;当aij<时,aij() =0.
显然,A的- 截矩阵为布尔矩阵.
对任意的∈[0, 1],有
性质1:A≤B  A≤B;
性质2:(A∪B)= A∪B,(A∩B)= A∩B;
性质3:( A ° B )= A° B;
性质4:( AT )= ( A)T.
下面证明性质1: A≤B  A≤B和性质3.
性质1的证明: A≤B  aij≤bij;
当≤aij≤bij时, aij() =bij() =1;
当aij<≤bij时, aij() =0, bij() =1;
当aij≤bij<时, aij()