A new similarity measure between intuitionistic fuzzy sets and the positive definiteness of the similarity matrix

Abstract

As a generation of fuzzy set theory, intuitionistic fuzzy (IF) set theory has received considerable attention for its capability on dealing with uncertainty. Similarity measures of IF sets are used to indicate the degree of commonality between IF sets. Although several similarity measures for IF sets have been proposed in previous studies, some of those cannot satisfy the axiomatic definitions of similarity, or provide counter-intuitive cases. In this paper, a new similarity measure between IF sets is proposed. The definition of similarity matrix is also presented to depict the relations among more than two IF sets. It is proved that the proposed similarity measures satisfy the properties of the axiomatic definition for similarity measures. Comparison between the previous similarity measures and the proposed similarity measure indicates that the proposed similarity measure does not provide any counter-intuitive cases. Moreover, it is demonstrated that the proposed similarity measure can be applied to define a positive definite similarity matrix.

Appendix 1: Definitions and properties about positive definite matrix

When proving theorems concerning positive definite matrices, we use some properties of positive definite matrix. For ease of reference, some background knowledge related to positive definite matrix is shown in this part. Since all the results given below are well known, we mainly present definitions and theorems with an absence of their proofs.

Definition 9

An $n \times n$ real symmetric matrix A is positive semidefinite (PSD) if it holds that $\varvec{x}^{{\mathbf{T}}} \varvec{Ax} \ge \text{0}$ for every n × 1 column vector $\varvec{x} \ne {\mathbf{0}}$.It is (strictly) positive definite (PD) if additionally: $\varvec{x}^{{\mathbf{T}}} \varvec{Ax} = \text{0} \Rightarrow \varvec{x} = {\mathbf{0}}$.

Theorem 7

Eigenvalues of real symmetric matrix A are all real numbers.

Theorem 8

A is PSD iff its eigenvalues are nonnegative and A is PD iff its eigenvalues are strictly positive.

Theorem 9

If two matrices A and B are both PSD, A + B is also PSD.

Consequently, the sum of PSD matrices is PSD. The result is PD if there is at least one PD matrix among the PSD matrices.

Theorem 10

A set of necessary and sufficient conditions for an n × n symmetric matrix A to be PSD is that all the principal leading minors $\Delta_{pp} (p = 1,2, \ldots ,n)$ of A must be nonnegative. Additionally, A is PD iff $\Delta_{pp} (p = 1,2, \ldots ,n)$ is strictly positive.

Theorem 11

Let x be an n × 1 column vector, $\varvec{x} \ne {\mathbf{0}}$. Then $\varvec{A = xx}^{{\mathbf{T}}}$ is PSD.

Theorem 12

If A is square, symmetric and positive definite, then A can be uniquely factorized as $\varvec{A = U}^{{\mathbf{T}}} \varvec{U}$ where U is upper triangular with positive diagonal entries (called Cholesky decomposition).

Definition 10

Let $\varvec{A} = \text{(}a_{ij} \text{)}_{n \times n}$ denote an n × n real matrix, the area defined by $\left| {z - a_{ii} } \right| \le \sum\nolimits_{j = 1,j \ne i}^{n} {\left| {a_{ij} } \right|}$ is called the ith Gerschgorin circle of A.

Theorem 13

Let $\varvec{A} = \text{(}a_{ij} \text{)}_{n \times n}$ denote an n × n real matrix. Then all of its eigenvalues are in the union of its n Gerschgorin circles (known as Gerschgorin theorem).

Let λ be an arbitrary eigenvalue of A, we have:

$$\left| {\lambda - a_{ii} } \right| \le \sum\limits_{\begin{subarray}{l} j = 1, \\ j \ne i \end{subarray} }^{n} {\left| {a_{ij} } \right|} ,\quad \exists i \in \{ 1,2, \ldots ,n\} .$$