Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 178

(2) Since c ij = 1 if and only if A i =

, = , ,...,

A j , i j 1 2 m , it yields

c ij = ma k { min { c ik , c kj }} = 1 if and only if A i = A k = A j , k = 1 , 2 ,..., m (2.82)

(3) Since c ij = c ji , i , j = 1 , 2 ,..., m , we get

c ij = ma k { min { c ik , c kj }} = ma k { min { c ki , c jk }}

= ma k { min { c jk , c ki }} = c ji , i , j = 1 , 2 ,..., m (2.83)

which completes the proof of the theorem.

According to Theorem 2.8, we can derive the following conclusion:

= (

c ij ) m × m be an association matrix. Then for

Theorem 2.9 (Xu et al. 2008) Let C

any positive integer k ,wehave

C 2 k + 1 C 2 k C 2 k

= ◦

(2.84)

where the composition matrix C 2 k + 1 is also an association matrix.

Definition 2.11 (Xu et al. 2008) Let C = ( c ij ) m × m be an association matrix. If

C 2

⊆ C , i.e., for any i , j = 1 , 2 ,..., m , the following inequality holds:

ma k { min { c ik , c kj }} ≤ c ij (2.85)

Thus, C is called an equivalent association matrix.
By the transitivity principle of equivalent matrix (Wang 1983), we can easily
prove the following theorem:

Theorem 2.10 (Xu et al. 2008) Let C = ( c ij ) m × m be an association matrix. Then

after the finite times of compositions:

C 2 k C 2 C 4

C → → → ··· → → ··· (2.86)

there must exist a positive integer k , such that C 2 k C 2 ( k + 1 ) , and C 2 k = is also an

equivalent association matrix.
Based on the equivalent association matrix, we give the following useful concept:

Definition 2.12 (Xu et al. 2008) Let C = ( c ij ) m × m be an equivalent association

matrix. Then C

λ = ( λ

c ij ) m × m is called the λ -cutting matrix of C , where

0

, c ij <λ,

c ij =

c ij ≥ λ, ,

, = , ,...,

i j 1 2 m (2.87) λ

1 ,

λ λ ∈[ , ]

and is the confidence level with 0 1 .