Similarity relations,eigenvalues and eigenvectors of bipolar fuzzy matrix

Abstract

Eigenvalues and eigenvectors are one of the important topics over bipolar fuzzy linear algebra. In order to develop the bipolar fuzzy linear space we introduce in this article, the similarity relations, eigenvalues and eigenvectors of bipolar fuzzy matrices (BFMs). Idempotent, diagonally dominant and spectral radius of BFMs are considered here. Also, some properties and results of eigenvalues and eigenvectors for BFMs are investigated.

Keywords

Relation bipolar fuzzy matrix vector eigenvalue eigenvector spectral radius

1 Introduction

Every matter has two sides one is called positive side and another is called negative side due to the observer’s point of view, which can be measured with certain degree of membership levels. In this article, we focus on problems that present positive and negative preferences and it is called bipolar preference problem.

Bipolar is an important topic in several domains, for example psychology, multi-criteria decision making, artificial intelligence, qualitative reasoning, etc. In a real life situation both positive and negative preferences are useful to handle the problem, in this aspect this topic is our next issue.

Some times, the membership degree means the satisfaction degree of elements to some property or constraint corresponding to a fuzzy set. So, some elements have irrelevant characteristics to the property corresponding to a fuzzy set and the others have contrary elements in fuzzy sets where the membership degrees ranged only on the interval [0, 1]. If a set representation could express this kind of difference it would be more informative than the traditional fuzzy set representation. Based on this observations Zhang [54] introduced an extension of fuzzy set, called it as bipolar fuzzy set (BFS).

Like classical (crisp) matrices, fuzzy matrices (FMs) are now a very rich topic, in modeling uncertain situations occurred in science, engineering, automata theory, logic of binary relations, medical diagnosis, etc. FMs defined first time by Thomson in 1977 [51] and discussed about the convergence of the powers of a fuzzy matrix. The theories of fuzzy matrices were developed by Kim and Rosh [27] as an extension of Boolean matrices. With max-min operation the fuzzy algebra and its matrix theory are considered by many authors [6 , 47]. Xin [53] studied controllable fuzzy matrices. The transitivity of matrices over path algebra (i.e. additively idempotent semiring) is discussed by Hashimoto [21 –23]. Generalized fuzzy matrices, matrices over an incline and some results about the transitive closer, determinant, adjoint matrices, convergence of powers and conditions for nilpotency are considered by Duan [17] and Lur et al. [30]. In FMs, row and columns are taken as certain. But, they may be uncertain. Pal [41] introduced this concept, i.e. rows and columns are taken as uncertain. He also investigated different properties of these type of matrices along with applications.

There are some limitations in dealing with uncertainties by fuzzy set. To overcome these difficulties, Atanassov [4] introduced theory of intuitionistic fuzzy set in 1983 as a generalization of fuzzy set. Based on this concept Pal et al. have defined intuitionistic fuzzy determinant in 2001 [39] and intuitionistic fuzzy matrices (IFMs) in 2002 [40]. Bhowmik and Pal [6 –10] introduced some results on IFMs, intuitionistic circulant fuzzy matrix and generalized intuitionistic fuzzy matrix. Shyamal and Pal [46, 48] defined the distances between IFMs and hence defined a metric on IFMs. They also cited few applications of IFMs. In [33], the similarity relations, invertibility conditions and eigenvalues of IFMs are studied. Idempotent, regularity, permutation matrix and spectral radius of IFMs are also discussed. Also, intuitionistic fuzzy incline matrix and determinant are studied in [34]. The parameterizations tool of IFM enhances the flexibility of its applications. For other works on IFMs see [1–3 , 43–45].

The concept of interval-valued fuzzy matrix (IVFM) as a generalization of FM was introduced and developed in 2006 by Shaymal and Pal [49] by extending the max-min operation in fuzzy algebra. We introduced interval-valued fuzzy vector space [31], rank and its associated properties on IVFMs [35]. Pal [42], defined a new type of IVFM whose rows and column are uncertain along with uncertain elements.

Combining IFMs and IVFMs, a new fuzzy matrix called interval-valued intuitionistic fuzzy matrices (IVIFMs) is defined [25]. For other works on IVIFMs, see [11, 12].

After the invention of BFSs [14 , 55] Zhang introduced fuzzy equilibrium relations and bipolar fuzzy clustering [56], bipolar logic and bipolar fuzzy partial ordering for clustering and coordination [57], bipolar logic and bipolar fuzzy logic [58] and Yin Yang bipolar logic, bipolar fuzzy logic [59]. The bipolar-valued fuzzy sets was introduced by Lee in [28, 29]. After that many author’s [5 , 19] are working on this topic till now. The bipolar fuzzy matrices was first introduce in [36]. Here we investigated the transitive closure and power convergent of BFMs. In [37] we introduce the bipolar fuzzy vector space (BFVS), rank and its associate properties are also investigated here.

In this article, first time we introduce the bipolar fuzzy similarity relations over BFMs. Over some special type of BFMs (e.g. diagonally dominant matrix etc.) we investigate some properties to find the eigenvalues and eigenvectors of the matrices and illustrated some suitable examples. Also some result about spectral radius are investigated here.

2 Preliminaries

The BFS is one of the extension of fuzzy set with positive and negative membership values. In this section, some basic notions of BFS are introduced. Also, some basic operations both binary and unary, viz. +, · , × , - , ¬ , ⇒ on BFS are given.

Definition 1. [Bipolar fuzzy set] A BFS ℬ_F in X (universe of discourse) is an object having the form $ℬ_{F} = {(x, μ_{\underline{n}} (x), μ_{\bar{p}} (x))}$ where $μ_{\underline{n}} : X \to [- 1, 0]$ and $μ_{\bar{p}} : X \to [0, 1]$ are two mappings.

The positive membership degree $μ_{\bar{p}} (x)$ denotes the satisfaction degree of an element x to the property corresponding to a BFS ℬ_F and the negative membership degree $μ_{\underline{n}} (x)$ denotes the satisfaction degree of x to some implicit counter-property of ℬ_F.

If $μ_{\bar{p}} (x) \neq 0$ and $μ_{\underline{n}} (x) = 0$ , it is the situation that x is regarded as having only positive satisfaction for ℬ_F. If $μ_{\bar{p}} (x) = 0$ and $μ_{\underline{n}} (x) \neq 0$ , it is the situation that x does not satisfy the property of ℬ_F but some what satisfied the counter property of ℬ_F. There is a possibility that for an element x, $μ_{\bar{p}} (x) \neq 0$ and $μ_{\underline{n}} (x) \neq 0$ , when the membership function of the property overlaps that of its counter property over some portion of the domain [29].

In arithmetic operations (such as addition, multiplication, etc.) only the membership values of BFS are needed so from now we represents a BFS as $ℬ_{F} = {x = (- x_{n}, x_{p}) | x \in X}$ where -x_n ∈ [-1, 0], i.e. x_n ∈ [0, 1] and x_p ∈ [0, 1] are the respectively negative and positive membership degree of x ∈ X in ℬ_F.

Definition 2. [Equality] Let x, y ∈ ℬ_F where x = (- x_n, x_p) and y = (- y_n, y_p), then the equality of two elements x and y is denoted by x = y and is defined by, x = y if and only if x_n = y_n and x_p = y_p.

Definition 3. Let x, y ∈ ℬ_F where x = (- x_n, x_p) , y = (- y_n, y_p) and x_n, x_p, y_n, y_p ∈ [0, 1] then the following operations are defined by Zhang and Zhang [59].

The disjunction of x and y is denoted by x + y and is defined by $\begin{matrix} x + y & = & (- x_{n}, x_{p}) + (- y_{n}, y_{p}) \\ = & (- max {x_{n}, y_{n}}, max {x_{p}, y_{p}}) \\ = & (- {x_{n} \lor y_{n}}, {x_{p} \lor y_{p}}) . \end{matrix}$

The parallel conjunction of x and y is denoted by x · y and is defined by $\begin{matrix} x \cdot y & = & (- x_{n}, x_{p}) \cdot (- y_{n}, y_{p}) \\ = & (- min {x_{n}, y_{n}}, min {x_{p}, y_{p}}) \\ = & (- {x_{n} \land y_{n}}, {x_{p} \land y_{p}}) . \end{matrix}$

The serial conjunction of x and y is denoted by x × y and is defined by $\begin{matrix} x \times y & = & (- x_{n}, x_{p}) \times (- y_{n}, y_{p}) \\ = & (- {(x_{n} \land y_{p}) \lor (x_{p} \land y_{n})}, \\ {(x_{n} \land y_{n}) \lor (x_{p} \land y_{p})}) . \end{matrix}$

The negation of x is denoted by -x and is defined by $- x = - (- x_{n}, x_{p}) = (- x_{p}, x_{n}) .$

The complement of x is denoted by ¬x and is defined by $\begin{matrix} \neg x & = & \neg (- x_{n}, x_{p}) = (\neg (- x_{n}), \neg x_{p}) \\ = & (- 1 + x_{n}, 1 - x_{p}) . \end{matrix}$

The implication of x to y is denoted by x ⇒ y and is defined by $(x \Rightarrow y) = \neg x + y .$

Definition 4. [Zero element] The zero element of a BFS is denoted by o_b and is defined by o_b = (0, 0).

Definition 5. [Unit element] The unit element of a BFS is denoted by i_b and is defined by i_b = (-1, 1).

2.1 Bipolar fuzzy relation and matrix

Definition 6. [Cartesian product of BFSs] Let X₁ and X₂ be two universe of discourses and let A = {x = (- x_n, x_p) |x ∈ X₁}, B = {y = (- y_n, y_p) |y ∈ X₂} be two BFSs. The Cartesian product of A and B is denoted by A × B and is defined by $A \times B = {(x, y) | x \in X_{1} and y \in X_{2}} .$

Definition 7. [Bipolar fuzzy relation] A bipolar fuzzy relation (BFR) between two BFSs A and B is defined as a BFS in A × B. If R is a relation between A and B, x ∈ A and y ∈ B, and if -r_n (x, y), r_p (x, y) denote the negative and positive membership values to which x is in relation R with y, then r = (- r_n, r_p) ∈ R.

Now, we define an order relation ‘≤’ below.

Definition 8. [Inclusion] Let ℬ_F be a BFS over X and let x, y ∈ ℬ_F where x = (- x_n, x_p) and y = (- y_n, y_p), then x ≤ y if and only if x_n ≤ y_n and x_p ≤ y_p. That is, x ≤ y if and only if x + y = y.

Definition 9. Let ℬ_F be a BFS over X and let x, y ∈ ℬ_F, where x = (- x_n, x_p) and y = (- y_n, y_p), then x < y if and only if x ≤ y and x ≠ y.

In order to develop the theory of BFM, we begin with the concept of bipolar fuzzy algebra (BFA). A BFA is a mathematical system (ℬ_F, + , ·) with two binary operations + and · defined on a set ℬ_F satisfying the following properties.

Idempotent: x + x = x, x · x = x

Commutativity: x + y = y + x, x · y = y · x

Associativity: x + (y + z) = (x + y) + z, x · (y · z) = (x · y) · z

Absorption: x + (x · y) = x, x · (x + y) = x

Distributivity: x · (y + z) = (x · y) + (x · z), x + (y · z) = (x + y) · (x + z)

Universal bounds: x + o_b = x, x + i_b = i_b, x · o_b = o_b, x · i_b = x

where x = (- x_n, x_p) , y = (- y_n, y_p) and z = (- z_n, z_p) ∈ ℬ_F.

Definition 10. [Bipolar fuzzy matix] A BFM A = [a_ij] is a matrix on the BFA. The zero matrix O_m of order m × m is the matrix where all the elements are o_b = (0, 0) and the identity matrix I_m of order m × m is the matrix where all the diagonal entries are i_b = (-1, 1) and the other entries are o_b = (0, 0).

The set of all rectangular bipolar fuzzy matrices of order l × m is denoted by _Mlm and that of square matrices of order m × m is denoted by _Mm.

From the definition we conclude that if A = (a_ij) _l×m ∈ _Mlm, then a_ij = (- a_ijn, a_ijp) ∈ ℬ_F, where a_ijn, a_ijp ∈ [0, 1] are the negative and positive membership values of the element a_ij respectively.

The operations on BFM are as follows:

Definition 11. Let A = (a_ij) , B = (b_ij) ∈ _Mlm be two BFMs. Therefore, a_ij, b_ij ∈ ℬ_F, then $\begin{matrix} A + B & = & (a_{ij} + b_{ij})_{l \times m} \\ = & {(- max {a_{ijn}, b_{ijn}}, max {a_{ijp}, b_{ijp}})}_{l \times m} \end{matrix}$ and $\begin{matrix} A \cdot B & = & (a_{ij} \cdot b_{ij})_{l \times m} \\ = & {(- min {a_{ijn}, b_{ijn}}, min {a_{ijp}, b_{ijp}})}_{l \times m} . \end{matrix}$

Definition 12. Let A = (a_ij) ∈ _Mlm and B = (b_ij) ∈ _Mmq be two BFMs. Therefore, a_ij, b_ij ∈ ℬ_F, then $\begin{matrix} A ⊙ B & = & {(\sum_{k = 1}^{m} a_{ik} \cdot b_{kj})}_{l \times q} \\ = & (- {max}_{k = 1}^{m} [min {a_{ikn}, b_{kjn}}], \\ {{max}_{k = 1}^{m} [min {a_{ikp}, b_{kjp}}])}_{l \times q} \end{matrix}$ and $\begin{matrix} A \otimes B & = & {(\prod_{k = 1}^{m} {a_{ik} + b_{kj}})}_{l \times q} \\ = & (- {min}_{k = 1}^{m} [max {a_{ikn}, b_{kjn}}], \\ {{min}_{k = 1}^{m} [max {a_{ikp}, b_{kjp}}])}_{l \times q} \end{matrix}$

3 Bipolar fuzzy vector space

Katsaras and Liu [24] first introduced the concept of fuzzy vector space. Here we present some basic notions of bipolar fuzzy vector space (BFVS) with respect to BFA.

Definition 13. [Bipolar fuzzy vector] A bipolar fuzzy vector (BFV) is an m-tuple of elements from a BFA. That is, a BFV is of the form [x₁, x₂, x₃, …, x_m], where each element x_i ∈ ℬ_F, i = 1, 2, 3, …, m.

Definition 14. [Bipolar fuzzy vector space] A bipolar fuzzy vector space (BFVS) is a pair (E, B (x)), where E is a vector space in crisp sense over the real field R and B: E → ([-1, 0] × [0, 1]) is the bipolar fuzzy membership function with the property that for all a, b ∈ R and x, y ∈ E we have $B_{n} (ax + by) \geq B_{n} (x) \land B_{n} (y)$ and $B_{p} (ax + by) \geq B_{p} (x) \land B_{p} (y) .$

Example 1. Let V_m denote the set of all m-tuples [x₁, x₂, x₃, …, x_m] over ℬ_F. An element of V_m is called a BFV of dimension m. For x = [x₁, x₂, x₃, …, x_m] and y = [y₁, y₂, y₃, …, y_m] in V_m, the following operations addition (+) and multiplication (·) are defined as $\begin{matrix} x + y & = & [x_{1} + y_{1}, x_{2} + y_{2}, x_{3} + y_{3}, \\ \dots, x_{m} + y_{m}] \in V_{m} \end{matrix}$ and for any a ∈ ℬ_F $ax = [{ax}_{1}, {ax}_{2}, {ax}_{3}, \dots, {ax}_{m}] \in V_{m} .$

The set V_m together with these operations of componentwise addition and scalar multiplication is a BFVS over ℬ_F, as the scalars are restricted in ℬ_F.

Definition 15. Let V^m = {x^t | x ∈ V_m} where x^t is the transpose of the vector x. For u, v ∈ V^m and a ∈ ℬ_F we define av = (av^t) ^t, u + v = (u^t + v^t) ^t. Then V^m is a BFVS. If we write an element of V_m as 1 × m matrix, it is called a row vector. The element of V^m are column vectors.

For any result of V_m there exists a similar result on V^m. Thus V^m is isomorphic to V_m, so in general we denote the vector space as V.

4 Similarity relation on BFS

The BFMs satisfy the properties of BFRs such as reflexive, symmetric and transitive. These relations are defined and investigated here.

Let R (A, A) be a BFR on a set A. Let r_n : A → [0, 1] be the negative membership function and r_p : A → [0, 1] be the positive membership function and M_R be the BFM with respect to the relation R.

Definition 16. [Reflexive relation] The relation R (A, A) is reflexive if the diagonal entries of M_R are all i_b = (-1, 1),

i.e. r_n (x, x) = r_p (x, x) =1 for all x ∈ A.

Definition 17. [Symmetric relation] The relation R (A, A) is symmetric if $M_{R} = M_{R}^{T}$ , where $M_{R}^{T}$ is the transpose of M_R,

i.e. r_n (x, y) = r_n (y, x) and r_p (x, y) = r_p (y, x) for all x, y ∈ A.

Definition 18. [Transitive relation] The relation R (A, A) is transitive if $M_{R} \geq M_{R}^{2}$ ,

i.e. $r_{n} (x, z) \geq max_{y \in X} {min {r_{n} (x, y), r_{n} (y, z)}}$ and $r_{p} (x, z) \geq min_{y \in X} {max {r_{p} (x, y), r_{p} (y, z)}}$ for all pair (x, z) ∈ A × A.

Definition 19. [Similarity relation] The relation R (A, A) is a similarity relation if and only if R (A, A) is reflexive, symmetric and transitive.

Proposition 1. For a BFM P ∈ _Mm, P is reflexive if P ≥ I_m.

Proof. Since P ≥ I_m therefore all diagonal entries of the matrix P are i_b = (-1, 1). Therefore P is a reflexive matrix. □

Definition 20. For a BFM P = [p_ij] = [(- p_ijn, p_ijp)] ∈ _Mm, define the following bipolar fuzzy matrices:

Type of P	Definition
Reflexive	P ≥ I_m.
Weakly reflexive	p_ii ≥ p_ij for all i, j ∈ {1, 2, …, m}.
Symmetric	P = P^T.
Idempotent	P = P².
Transitive	P² ≤ P.

Proposition 2.LetP ∈ _Mm be a reflexive BFM. Then the following holds.

P^T is a reflexive BFM,

P^k is a reflexive BFM for some positive integer k,

PQ ≥ Q for Q ∈ _Mm,

QP ≥ Q for Q ∈ _Mm,

PQ and QP are reflexive BFMs if Q is reflexive,

PP^T and P^TP are reflexive BFMs.

Proof. (i) Since P is reflexive, its diagonal entries are all i_b = (-1, 1). Therefore, the diagonal entries of P^T are also i_b = (-1, 1). Hence P^T is reflexive.

(ii) Since P is reflexive, P ≥ I_m. Then P² ≥ P ≥ I_m [Multiplying by P in both side]. Proceeding in this way we get P^k ≥ P^k-1 ≥ ⋯ ≥ P² ≥ P ≥ I_m for any positive integer k. Hence P^k is reflexive.

(iii)P ≥ I_m then PQ ≥ I_mQ this implies that PQ ≥ Q.

(iv) Also QP ≥ QI_m or QP ≥ Q.

(v) Since Q is reflexive Q ≥ I_m. Then PQ ≥ Q ≥ I_m and QP ≥ Q ≥ I_m. Hence PQ and QP are also reflexive.

(vi) From (i) and (v) it follows that PP^T and P^TP are reflexive. □

Proposition 3. If P ∈ _Mm be transitive and reflexive, then P is idempotent.

Proof. Since P is reflexive P ≥ I_m. Therefore $P^{2} \geq P \geq I_{m} .$ (1) Also, P is transitive, then $P^{2} \leq P .$ (2) Combining (1) and (2) we get P² = P.

Hence P is idempotent. □

The condition is not sufficient which can be shown by the following example.

Example 2. Let $P = [\begin{matrix} (- 0.8, 0.7) & (- 0.8, 0.7) \\ (- 0.8, 0.7) & (- 0.8, 0.7) \end{matrix}] ≱ I_{2} .$ Hence P is not reflexive. But $\begin{matrix} P^{2} & = & P ⊙ P \\ = & [\begin{matrix} (- 0.8, 0.7) & (- 0.8, 0.7) \\ (- 0.8, 0.7) & (- 0.8, 0.7) \end{matrix}] \\ ⊙ [\begin{matrix} (- 0.8, 0.7) & (- 0.8, 0.7) \\ (- 0.8, 0.7) & (- 0.8, 0.7) \end{matrix}] \\ = & [\begin{matrix} (- 0.8, 0.7) & (- 0.8, 0.7) \\ (- 0.8, 0.7) & (- 0.8, 0.7) \end{matrix}] \\ = & P . \end{matrix}$

That is, P is idempotent.

The proof of the following result is straight forward.

Proposition 4. If P and Q be two symmetric BFMs in _Mm such that PQ = QP, then PQ is symmetric.

Remark 1. If P is a symmetric BFM in _Mm, then P^k is symmetric for any positive integer k.

Proposition 5. If P and Q are transitive BFMs in _Mm such that PQ = QP, then PQ is transitive.

Proof. Since P and Q are transitive, P² ≤ P and Q² ≤ Q. Now $\begin{matrix} (PQ)^{2} & = & (PQ) (PQ) \\ = & P (QP) Q [by associative property] \\ = & P (PQ) Q [since PQ = QP] \\ = & (PP) (QQ) = P^{2} Q^{2} \\ That is (PQ)^{2} & \leq & PQ . \end{matrix}$ Hence PQ is transitive.

Remark 2. If P is transitive in _Mm, then P^k is transitive for any positive integer k.

Proposition 6. If P = [p_ij] = [(- p_ijn, p_ijp)] ∈ _Mm is symmetric and transitive, then p_ij ≤ p_ii for i, j ∈ {1, 2, 3, …, m}.

Proof. Since P is symmetric p_ij = p_ji for all i, j ∈ {1, 2, 3, …, m}. Also since P is transitive P² ≤ P, i.e. P ≥ P². Thus for k ∈ {1, 2, 3, …, m} $\begin{matrix} p_{ij} & \geq & max_{k} {min (p_{ik}, p_{kj})} for all i, j, \\ i . e . p_{ii} & \geq & max_{k} {min (p_{ik}, p_{ki})} for i = j and each k \\ \geq & min (p_{ij}, p_{ji}) for k = j for each i \end{matrix}$ This gives p_ii ≥ p_ij [Since p_ij = p_ji] . □

5 Eigenvalues and eigenvectors of BFMs

Eigenvalue problems are very important in many fields. These are formulated when modeling real cases into mathematical models. For example, the natural frequencies and mode shapes in vibration problems, the principal axes in elasticity and dynamics, the Markov chain in stochastic modeling and queueing theory, and the analytical hierarchy process for decision making, etc. all come up with eigenvalue problems.

A very few works are available for eigenvalues and eigenvectors for fuzzy matrices [13 , 16]. But, their methods are not applicable for all kinds of matrices and these are very laborious methods. Actually, it is very difficult task to find eigenvalues and eigenvectors for a fuzzy matrix. Some authors are tried to find out the eigenvalues and eigenvectors as per rules of scrips matrices introducing α-cut approach [50, 52]. No authors used the max-min operation. And hence it is seen that the eigenvalues/ eigenvectors are also negative. But, in fuzzy content negative value is not acceptable. Keeping this situation in mind we are trying to find out that eigenvalue and eigenvectors those are positive and lies in [0, 1]. Also, we have use the max-min operation in the equation AX = λX or XA = λX to find λ and X. This is realistic and natural in fuzzy context. This is the first attempt to find λ and X using max-min operation. In this section, some particular cases are consider to find λ and X for a BFM.

Definition 21. Let A ∈ _Mm and a scalar λ = (- λ_n, λ_p) ∈ ℬ_F be called an eigenvalue of A and a non-zero vector X is called a row (column) eigenvector of A if XA = λX (AX = λX), X is called an eigenvector associated with the eigenvalue λ.

Theorem 1. If A = [a_ij] = [(- a_ijn, a_ijp)] be a BFM of order m × m, such that a_1i = a_2i = ⋯ = a_i-1,i = a_i+1,i = ⋯ = a_mi = o_b (say) where i ∈ {1, 2, 3, …, m}. Then a_ii is an eigenvaluecorresponding to the column eigenvector [o_b, o_b, o_b, …, i_b, …, o_b] ^T ∈ V^m, where i_b = (-1, 1) be the ith entry.

Proof. Here X = [o_b, o_b, o_b, …, i_b, …, o_b] ^T = (x_i1) ∈ V^m (say). Then $AX = [\begin{matrix} \sum_{k = 1}^{m} a_{1 k} x_{k 1} \\ \sum_{k = 1}^{m} a_{2 k} x_{k 2} \\ ⋮ \\ \sum_{k = 1}^{m} a_{mk} x_{km} \end{matrix}] = [\begin{matrix} o_{b} \\ o_{b} \\ ⋮ \\ a_{ii} \\ ⋮ \\ o_{b} \end{matrix}] = a_{ii} [\begin{matrix} o_{b} \\ o_{b} \\ ⋮ \\ i_{b} \\ ⋮ \\ o_{b} \end{matrix}]$ [Since the ith entry $\sum_{k = 1}^{m} a_{ik} x_{ki} = a_{i 1} \cdot o_{b} + a_{i 2} \cdot o_{b} + \dots + a_{ii} \cdot i_{b} + \dots + a_{in} \cdot o_{b} = a_{ii}$ .] Therefore, AX = a_iiX.

Hence, a_ii is the eigenvalue corresponding to the column eigenvector X = [o_b, o_b, …, i_b, …, o_b] ^T ∈ V^m. □

Example 3. Let $A = [\begin{matrix} (- 0.5, 0.4) & (0, 0) & (- 0.6, 0.7) \\ (- 0.6, 0.5) & (- 0.4, 0.3) & (- 0.7, 0.6) \\ (- 0.7, 0.8) & (0, 0) & (- 0.8, 0.7) \end{matrix}]$ and X = [(0, 0) (-1, 1) (0, 0)] ^T. Then $\begin{matrix} AX & = & [\begin{matrix} (- 0.5, 0.4) & (0, 0) & (- 0.6, 0.7) \\ (- 0.6, 0.5) & (- 0.4, 0.3) & (- 0.7, 0.6) \\ (- 0.7, 0.8) & (0, 0) & (- 0.8, 0.7) \end{matrix}] \\ ⊙ [\begin{matrix} (0, 0) \\ (- 1, 1) \\ (0, 0) \end{matrix}] \\ = & [\begin{matrix} (0, 0) \\ (- 0.4, 0.3) \\ (0, 0) \end{matrix}] = (- 0.4, 0.3) [\begin{matrix} (0, 0) \\ (- 1, 1) \\ (0, 0) \end{matrix}] \\ = & (- 0.4, 0.3) X . \end{matrix}$ Thus, (-0.4, 0.3) is the eigenvalue of A corresponding to the column eigenvector X.

From above Theorem (1) and calculation of above Example (3) we observe that,

Note 1. Let a_ii = (- a_iin, a_iip) and α = (- α_n, α_p) ∈ ℬ_F and if a_ii ≤ α, i.e. if a_iin ≤ α_n and a_iip ≤ α_p,then [o_b, o_b, …, α, …, o_b] ^T = α [o_b, o_b, …, i_b, …, o_b] ^T ∈ V^m are also eigenvectors corresponding to the same eigenvalue a_ii, for any scaler α ∈ ℬ_F. And it is noted that eigenvectors are not unique corresponding to that eigenvalue.

Theorem 2. Let A = [a_ij] = [(- a_ijn, a_ijp)] ∈ _Mm. If a_i1 = a_i2 = ⋯ = a_i,i-1 = a_i,i+1 = ⋯ = a_im = o_b (say) where i ∈ {1, 2, 3, …, m}. Then, a_ii is an eigenvalue corresponding to the row eigenvector (o_b, o_b, o_b, …, i_b, …, o_b) ∈ V_m, where i_b be the ith entry. In addition, if a_ii ≤ α for some α ∈ ℬ_F, then α (o_b, o_b, o_b, …, i_b, …, o_b) ∈ V_m are also eigenvectors corresponding to the same eigenvalue a_ii.

Proof. The proof is similar to Theorem 1. □

Example 4. Let $A = [\begin{matrix} (- 0.6, 0.7) & (- 0.8, 0.7) & (- 0.7, 0.6) \\ (0, 0) & (- 0.6, 0.4) & (0, 0) \\ (- 0.7, 0.8) & (- 0.5, 0.4) & (- 0.8, 0.7) \end{matrix}]$ and X = ((0, 0) (-1, 1) (0, 0)). Then $\begin{matrix} XA & = & ((0, 0) (- 1, 1) (0, 0)) \\ ⊙ [\begin{matrix} (- 0.6, 0.7) & (- 0.8, 0.7) & (- 0.7, 0.6) \\ (0, 0) & (- 0.6, 0.4) & (0, 0) \\ (- 0.7, 0.8) & (- 0.5, 0.4) & (- 0.8, 0.7) \end{matrix}] \end{matrix}$

$\begin{matrix} = & ((0, 0) (- 0.6, 0.4) (0, 0)) \\ = & (- 0.6, 0.4) ((0, 0) (- 1, 1) (0, 0)) . \end{matrix}$ Therefore, XA = (-0.6, 0.4) X.

Hence, (-0.6, 0.4) is the eigenvalue of A corresponding to the row eigenvector X.

Theorem 3. If A = [a_ij] = [(- a_ijn, a_ijp)] ∈ _Mm such that a_1i = a_2i = a_3i = ⋯ = a_mi = λ ≥ a_ij for all i, j ∈ {1, 2, 3, …, m}. Then, λ is an eigenvalue corresponding to the column eigenvector [i_b, i_b, i_b, …, i_b] ^T ∈ V^m. In addition, if λ ≤ α for some α ∈ ℬ_F then α [i_b, i_b, i_b, …, i_b] ^T ∈ V^m are also eigenvectors corresponding to the same eigenvalue λ.

Proof. Since a_1i = a_2i = a_3i = ⋯ = a_mi = λ ≥ a_ij for all i, j ∈ {1, 2, 3, …, m}.

Therefore, $\sum_{j = 1}^{m} a_{ij} = λ$ . Also X = [i_b, i_b, i_b, …, i_b] ^T ∈ V^m. Then $\begin{matrix} AX & = & [\begin{matrix} \sum_{j = 1}^{m} a_{1 j} i_{b} \\ \sum_{j = 1}^{m} a_{2 j} i_{b} \\ ⋮ \\ \sum_{j = 1}^{m} a_{mj} i_{b} \end{matrix}] = [\begin{matrix} \sum_{j = 1}^{m} a_{1 j} \\ \sum_{j = 1}^{m} a_{2 j} \\ ⋮ \\ \sum_{j = 1}^{m} a_{mj} \end{matrix}] \\ = & [\begin{matrix} λ \\ λ \\ ⋮ \\ λ \end{matrix}] = λ [\begin{matrix} i_{b} \\ i_{b} \\ ⋮ \\ i_{b} \end{matrix}] = λ X \end{matrix}$ This shows, λ is an eigenvalue of A corresponding to the column eigenvector X. □

Example 5. Let $A = [\begin{matrix} (- 0.5, 0.4) & (- 0.8, 0.9) & (- 0.6, 0.7) \\ (- 0.7, 0.8) & (- 0.8, 0.9) & (- 0.5, 0.6) \\ (- 0.6, 0.5) & (- 0.8, 0.9) & (- 0.7, 0.7) \end{matrix}]$ and X = [(-1, 1) (-1, 1) (-1, 1)] ^T. Then $\begin{matrix} AX & = & [\begin{matrix} (- 0.5, 0.4) & (- 0.8, 0.9) & (- 0.6, 0.7) \\ (- 0.7, 0.8) & (- 0.8, 0.9) & (- 0.5, 0.6) \\ (- 0.6, 0.5) & (- 0.8, 0.9) & (- 0.7, 0.7) \end{matrix}] \end{matrix}$

$\begin{matrix} ⊙ [\begin{matrix} (- 1, 1) \\ (- 1, 1) \\ (- 1, 1) \end{matrix}] \\ = & [\begin{matrix} (- 0.8, 0.9) \\ (- 0.8, 0.9) \\ (- 0.8, 0.9) \end{matrix}] = (- 0.8, 0.9) [\begin{matrix} (- 1, 1) \\ (- 1, 1) \\ (- 1, 1) \end{matrix}] . \end{matrix}$

Hence, AX = (-0.8, 0.9) X.

Thus, (-0.8, 0.9) is the column eigenvalue of A corresponding to the eigenvector X.

Theorem 4. If A = [a_ij] = [(- a_ijn, a_ijp)] ∈ _Mm such that a_i1 = a_i2 = a_i3 = ⋯ = a_im = λ ≥ a_ij for all i, j ∈ {1, 2, 3, …, m}. Then, λ is an eigenvalue of A corresponding to the row eigenvector (i_b, i_b, i_b, …, i_b) ∈ V_m. Also, if λ ≤ α for some α ∈ ℬ_F then α (i_b, i_b, i_b, …, i_b) ∈ V_m are also eigenvectors corresponding to the same eigenvalue λ.

Proof. The proof is similar to the Theorem 3. □

Example 6. Let $A = [\begin{matrix} (- 0.5, 0.4) & (- 0.6, 0.7) & (- 0.7, 0.8) \\ (- 0.8, 0.7) & (- 0.5, 0.6) & (- 0.6, 0.5) \\ (- 0.9, 0.8) & (- 0.9, 0.8) & (- 0.9, 0.8) \end{matrix}]$ and X = ((-1, 1) (-1, 1) (-1, 1)) . Then $\begin{matrix} XA & = & ((- 1, 1) (- 1, 1) (- 1, 1)) \\ ⊙ [\begin{matrix} (- 0.5, 0.4) & (- 0.6, 0.7) & (- 0.7, 0.8) \\ (- 0.8, 0.7) & (- 0.5, 0.6) & (- 0.6, 0.5) \\ (- 0.9, 0.8) & (- 0.9, 0.8) & (- 0.9, 0.8) \end{matrix}] \\ = & ((- 0.9, 0.8) (- 0.9, 0.8) (- 0.9, 0.8)) \\ = & (- 0.9, 0.8) ((- 1, 1) (- 1, 1) (- 1, 1)) . \end{matrix}$ Therefore, XA = (-0.9, 0.8) X.

Hence, (-0.9, 0.8) is the eigenvalue of A corresponding to the row eigenvector X.

Definition 22. (Diagonally dominant) Let A = [a_ij] ∈ _Mm be a BFM. A is called row diagonally dominant if $a_{ii} \geq \sum_{j \neq i, j = 1}^{m} a_{ij}$ . A is called column diagonally dominant if $a_{ii} \geq \sum_{i \neq j, i = 1}^{m} a_{ij}$ . A is called diagonally dominant if it is both row as well as column diagonally dominant.

Theorem 5. If A = [a_ij] = [(- a_ijn, a_ijp)] ∈ _Mm such that a₁₁ = a₂₂ = a₃₃ = ⋯ = a_mm = c (say)and if A is diagonally dominant, then c is an eigenvalue corresponding to the row (column)eigenvectors α (i_b, i_b, i_b, …, i_b) ∈ V_m (α [i_b, i_b, i_b, …, i_b] ^T ∈ V^m) for some α ∈ ℬ_F with c ≤ α.

Proof. Since the BFM A = [a_ij] is diagonally dominant, therefore $\sum_{j = 1}^{m} a_{ij} = a_{ii} = c$ and $\sum_{i = 1}^{m} a_{ij} = a_{jj} = c$ . Also X = α [i_b, i_b, i_b, …, i_b] ^T ∈ V^m. Then $\begin{matrix} AX & = & [\begin{matrix} α \sum_{j = 1}^{m} a_{1 j} i_{b} \\ α \sum_{j = 1}^{m} a_{2 j} i_{b} \\ ⋮ \\ α \sum_{j = 1}^{m} a_{mj} i_{b} \end{matrix}] = [\begin{matrix} α \sum_{j = 1}^{m} a_{1 j} \\ α \sum_{j = 1}^{m} a_{2 j} \\ ⋮ \\ α \sum_{j = 1}^{m} a_{mj} \end{matrix}] \\ = & [\begin{matrix} α c \\ α c \\ ⋮ \\ α c \end{matrix}] = c α [\begin{matrix} i_{b} \\ i_{b} \\ ⋮ \\ i_{b} \end{matrix}] = cX . \end{matrix}$ Thus, c is an eigenvalue of the BFM A corresponding to the column eigenvectors X.

Similarly, we can prove the theorem for row eigenvectors. □

Example 7. Let $A = [\begin{matrix} (- 0.8, 0.7) & (- 0.5, 0.4) & (- 0.6, 0.7) \\ (- 0.7, 0.6) & (- 0.8, 0.7) & (- 0.5, 0.6) \\ (- 0.5, 0.5) & (- 0.7, 0.6) & (- 0.8, 0.7) \end{matrix}]$ and X = ((-0.9, 0.8) (-0.9, 0.8) (-0.9, 0.8)). Then $\begin{matrix} XA & = & ((- 0.9, 0.8) (- 0.9, 0.8) (- 0.9, 0.8)) \\ ⊙ [\begin{matrix} (- 0.8, 0.7) & (- 0.5, 0.4) & (- 0.6, 0.7) \\ (- 0.7, 0.6) & (- 0.8, 0.7) & (- 0.5, 0.6) \\ (- 0.5, 0.5) & (- 0.7, 0.6) & (- 0.8, 0.7) \end{matrix}] \\ = & ((- 0.8, 0.7) (- 0.8, 0.7) (- 0.8, 0.7)) \\ = & (- 0.8, 0.7) ((- 0.9, 0.8) (- 0.9, 0.8) (- 0.9, 0.8)) . \end{matrix}$ That is, XA = (-0.8, 0.7) X.

Hence, (-0.8, 0.7) is the eigenvalue of A corresponding to the row eigenvector X.

Theorem 6. If A = [a_ij] = [(- a_ijn, a_ijp)] ∈ _Mm then λ = (- λ_n, λ_p) ∈ ℬ_F be an eigenvaluecorresponding to the column eigenvectors α [i_b, i_b, i_b, …, i_b] ^T ∈ V^m if max {a_k1n, a_k2n, a_k3n, …, a_kmn} = λ_n and max {a_k1p, a_k2p, a_k3p, …, a_kmp} = λ_p for every k ∈ {1, 2, 3, …, m} and for some α ∈ ℬ_F with λ ≤ α.

Proof. Since max {a_k1n, a_k2n, a_k3n, …, a_kmn} = λ_n and max {a_k1p, a_k2p, a_k3p, …, a_kmp} = λ_p forevery k ∈ {1, 2, 3, …, m}. Therefore $\sum_{j = 1}^{m} a_{kj} = (- \sum_{j = 1}^{m} a_{kjn}, \sum_{j = 1}^{m} a_{kjp}) = (- λ_{n}, λ_{p}) = λ$ forevery k ∈ {1, 2, 3, …, m}. Also X = α [i_b, i_b, i_b, …, i_b] ^T ∈ V^m. Then $\begin{matrix} AX & = & [\begin{matrix} α \sum_{j = 1}^{m} a_{1 j} i_{b} \\ α \sum_{j = 1}^{m} a_{2 j} i_{b} \\ ⋮ \\ α \sum_{j = 1}^{m} a_{mj} i_{b} \end{matrix}] = [\begin{matrix} α \sum_{j = 1}^{m} a_{1 j} \\ α \sum_{j = 1}^{m} a_{2 j} \\ ⋮ \\ α \sum_{j = 1}^{m} a_{mj} \end{matrix}] \\ = & [\begin{matrix} α λ \\ α λ \\ ⋮ \\ α λ \end{matrix}] = λ α [\begin{matrix} i_{b} \\ i_{b} \\ ⋮ \\ i_{b} \end{matrix}] = λ X . \end{matrix}$

Thus, λ is an eigenvalue of the BFM A corresponding to the column eigenvectors X. □

Example 8. Let $A = [\begin{matrix} (- 0.5, 0.5) & (- 0.8, 0.6) & (- 0.6, 0.7) \\ (- 0.8, 0.7) & (- 0.6, 0.7) & (- 0.5, 0.6) \\ (- 0.6, 0.7) & (- 0.3, 0.4) & (- 0.8, 0.5) \end{matrix}]$ and X = [(-1, 1) (-1, 1) (-1, 1)] ^T. Then $\begin{matrix} AX & = & [\begin{matrix} (- 0.5, 0.5) & (- 0.8, 0.6) & (- 0.6, 0.7) \\ (- 0.8, 0.7) & (- 0.6, 0.7) & (- 0.5, 0.6) \\ (- 0.6, 0.7) & (- 0.3, 0.4) & (- 0.8, 0.5) \end{matrix}] \\ ⊙ [\begin{matrix} (- 1, 1) \\ (- 1, 1) \\ (- 1, 1) \end{matrix}] \\ = & [\begin{matrix} (- 0.8, 0.7) \\ (- 0.8, 0.7) \\ (- 0.8, 0.7) \end{matrix}] = (- 0.8, 0.7) [\begin{matrix} (- 1, 1) \\ (- 1, 1) \\ (- 1, 1) \end{matrix}] . \end{matrix}$ Hence, AX = (-0.8, 0.7) X.

Thus, (-0.8, 0.7) is the column eigenvalue of A corresponding to the eigenvector X.

Theorem 7. If A = [a_ij] = [(- a_ijn, a_ijp)] ∈ _Mmthen λ = (- λ_n, λ_p) ∈ ℬ_F be an eigenvaluecorresponding to the row eigenvectorsα (i_b, i_b, i_b, …, i_b) ∈ V_m if max {a_1kn, a_2kn, a_3kn, …, a_mkn} = λ_n and max {a_1kp, a_2kp, a_3kp, …, a_mkp} = λ_p for every k ∈ {1, 2, 3, …, m} and for some α ∈ ℬ_F with λ ≤ α.

Proof. The proof is similar to the Theorem 6. □

Example 9. Let $A = [\begin{matrix} (- 0.8, 0.7) & (- 0.5, 0.4) & (- 0.6, 0.9) \\ (- 0.7, 0.9) & (- 0.8, 0.9) & (- 0.5, 0.6) \\ (- 0.5, 0.6) & (- 0.7, 0.8) & (- 0.8, 0.6) \end{matrix}]$ and X = (-0.9, 0.9) ((-1, 1) (-1, 1) (-1, 1)). Therefore, $\begin{matrix} XA & = & (- 0.9, 0.9) ((- 1, 1) (- 1, 1) (- 1, 1)) \\ ⊙ [\begin{matrix} (- 0.8, 0.7) & (- 0.5, 0.4) & (- 0.6, 0.9) \\ (- 0.7, 0.9) & (- 0.8, 0.9) & (- 0.5, 0.6) \\ (- 0.5, 0.6) & (- 0.7, 0.8) & (- 0.8, 0.6) \end{matrix}] \\ = & (- 0.9, 0.9) ((- 0.8, 0.9) (- 0.8, 0.9) (- 0.8, 0.9)) \\ = & (- 0.8, 0.9) (- 0.9, 0.9) ((- 1, 1) (- 1, 1) (- 1, 1)) . \end{matrix}$ That is, XA = (-0.8, 0.9) X.

Hence, (-0.8, 0.9) is the eigenvalue of A corresponding to the row eigenvector X.

Corollary 1. Let A = [a_ij] = [(- a_ijn, a_ijp)] ∈ _Mm. If $\sum_{j = 1}^{m} a_{1 j} = \sum_{j = 1}^{m} a_{2 j} = \sum_{j = 1}^{m} a_{3 j} = \dots = \sum_{j = 1}^{m} a_{mj} = c$ (say). Then, c is an eigenvalue of A corresponding to the column eigenvectors α [i_b, i_b, i_b, …, i_b] ^T ∈ V^m for some α ∈ ℬ_F with c ≤ α.

Corollary 2. Let A = [a_ij] = [(- a_ijn, a_ijp)] ∈ _Mm. If $\sum_{i = 1}^{m} a_{i 1} = \sum_{i = 1}^{m} a_{i 2} = \sum_{i = 1}^{m} a_{i 3} = \dots = \sum_{i = 1}^{m} a_{im} = c$ (say). Then, c is an eigenvalue of A corresponding to the row eigenvectors α (i_b, i_b, i_b, …, i_b) ∈ V_m for some α ∈ ℬ_F with c ≤ α.

Theorem 8. Let A ∈ _Mm, then A has a zero column if and only if o_b ∈ σ (A) (set of all eigenvalues of A).

Proof.Condition is necessary: Let ith column of A is zero, we take X = [o_b, o_b, …, i_b, …, o_b] ^T ∈ V^m, where i_b is the ith entry, then X is a non-zero vector satisfying the equation AX = o_bX. Hence, X is an column eigenvector corresponding to the eigenvalue o_b.

Condition is sufficient: Let X = [x₁, x₂, x₃, …, x_m] ^T ∈ V^m be a column eigenvector corresponding to the eigenvalue o_b, then AX = o_bX. We assume that x_i ≠ o_b for i ∈ {1, 2, 3, …, m}. Then AX = o_bX implies that $\sum_{k = 1}^{m} a_{jk} x_{k} = o_{b}$ for each j ∈ {1, 2, 3, …, m}. This implies a_jkx_k = o_b for each j and k. Since x_i ≠ o_b, a_ij = o_b for each j, then ith column of A is zero. □

Definition 23. Let σ (A) be the set of all eigenvalues of A. Then δ (A) = sup {λ | λ ∈ σ (A)} is called the spectral radius of A.

Theorem 9. Let A ∈ _Mm. Then δ (A) is either o_b or i_b.

Proof. If σ (A) = {o_b}, then δ (A) = o_b, otherwise, if there exist λ ∈ σ (A) (λ ≠ o_b) then there is a non-zero eigenvector X ∈ V^m (set of column vectors of order m) such that AX = λX. Also we know that for any β with λ ≤ β ≤ i_b, β · λ = λ and λ · λ = λ.

Therefore, λX = (β · λ) X = β (λX)

⇒A (λX) = λ (AX) = λ (λX) = (λ · λ) X = lambdaX = β (λX). Hence, β ∈ σ (A).

Since β is arbitrary, i_b ∈ σ (A). Therefore δ (A) = i_b. □

Theorem 10. For any A, B ∈ _Mm if A ≤ B then δ (A) ≤ δ (B).

Proof. From Theorem 9, δ (A) is either o_b or i_b.

If δ (A) = o_b, then δ (A) ≤ δ (B) holds trivially.

If δ (A) = i_b, we have to prove that δ (B) = i_b.

Since δ (A) = i_b, then by definition i_b ∈ σ (A) and AX = i_bX = X for some non-zero column vector X. We consider e = [i_b, i_b, i_b, …, i_b] ^T ∈ V^m then X ≤ e.

Also A^mX = A^m-1AX = A^m-1X = A^m-2X = ⋯ = A²X = AX = X,

i.e. X = A^mX ≤ A^me ≤ B^me.

[Since X ≤ e and A ≤ B.]

Since X is non-zero hence B^me is non-zero.

Now, if Y = B^me, then BY = B^m+1e = B^me = Y = i_bY. Hence i_b ∈ σ (B).

Thus, δ (B) = i_b.

Therefore δ (A) ≤ δ (B). □

6 Conclusion

We study the properties of similarity relations eigenvalues and eigenvectors of bipolar fuzzy matrices. A very few works are available to find the eigenvalues and eigenvectors of a fuzzy matrix. In this paper, first time we investigate the properties of eigenvalues and eigenvectors of a BFM and illustrated with suitable examples and it is noted that eigenvectors are not unique corresponding to an eigenvalue of a BFM. However the proposed results are not established for general cases. Now, we are trying to find out the eigenvalues and eigenvectors for all types of BFMs using max-min operation.

References

Adak

A.K.

, Bhowmik

and Pal

, Some properties of generalized intuitionistic fuzzy nilpotent matrices over distributive lattice, Fuzzy Inf and Eng4(4) (2012), 371–387.

Adak

A.K.

, Bhowmik

and Pal

, Intuitionistic fuzzy block matrix and its some properties, Annals of Pure and Applied Mathematics1(1) (2012), 13–31.

Adak

A.K.

, Pal

and Bhowmik

, Distributive lattice over intuitionistic fuzzy matrices, The Journal of Fuzzy Mathematics21(2) (2013), 401–416.

Atanassov

K.T.

, Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, Bulgarian, 1983.

Benferhat

, Dubois

, Kaci

and Prade

, Bipolar possibility theory in preference modelling: Representation, fusion and optimal solutions, Inf Fusion7(1) (2006), 135–150.

Bhowmik

and Pal

, Generalized intuitionistic fuzzy matrices, Far-East Journal of Mathematical Sciences29(3) (2008), 533–554.

Bhowmik

and Pal

, Some results on intuitionistic fuzzy matrices and intuitionistic circulant fuzzy matrices, International Journal of Mathematical Sciences7 (2008), 81–96.

Bhowmik

, Pal

and Pal

, Circulant triangular fuzzy number matrices, Journal of Physical Sciences12 (2008), 141–154.

Bhowmik

and Pal

, Intuitionistic neutrosophic set, Journal of Information and Computing Science4(2) (2009), 142–152.

10.

Bhowmik

and Pal

, Intuitionistic neutrosophic set relations and some of its properties, Journal of Information and Computing Science5(3) (2010), 183–192.

11.

Bhowmik

and Pal

, Generalized interval-valued intuitionistic fuzzy sets, The Journal of Fuzzy Mathematics18(2) (2010), 357–371.

12.

Bhowmik

and Pal

, Some results on generalized intervalvalued intuitionistic fuzzy sets, International Journal of Fuzzy Systems14(2) (2012), 193–203.

13.

Buckley

J.J.

, Feuring

and Hayashi

, Fuzzy eigenvalues, The Journal of Fuzzy Mathematics13(4) (2005), 757–773.

14.

Cacioppo

J.T.

, Gardner

W.L.

and Berntson

G.G.

, Beyond bipolar conceptualization and measures: The case of attitudes and evaluative spaces, Pers Soc Psychol Rev1 (1997), 3–25.

15.

Chiao

K.P.

, Generalized fuzzy eigenvalue problems, Tamsui Oxford Journal of Mathematical Sciences14 (1998), 31–37.

16.

Das

and Baruah

H.K.

, Iterative computation of eigenvalues and corresponding eigenvectors of a fuzzy matrix, The Journal of Fuzzy Mathematics12(2) (2004), 285–294.

17.

Duan

J.-S.

, The transitive closure, convergence of powers and adjoint of generalized fuzzy matrices, Fuzzy Sets and Systems145 (2004), 301–311.

18.

Dubois

and Prade

, An introduction to bipolar representations of information and preference, Int J Intell Syst23(8) (2008), 866–877.

19.

Dudziak

and Pekala

, Equivalent bipolar fuzzy relations, Fuzzy Sets and Systems161 (2010), 234–253.

20.

Gavalec

, Computing matrix period in max-min algebra, Discrete Appl Math75 (1997), 63–70.

21.

Hashimoto

, Convergence of power of a fuzzy transitive matrix, Fuzzy Sets and Systems9 (1983), 153–160.

22.

Hashimoto

, Canonical form of a transitive fuzzy matrix, Fuzzy Sets and Systems11 (1983), 157–162.

23.

Hashimoto

, Transitivity of generalized fuzzy matrix, Fuzzy Sets and Systems17 (1985), 83–90.

24.

Katsaras

A.K.

and Liu

D.B.

, Fuzzy vector spaces and fuzzy topological vector spaces, Journal of Mathematical Analysis and Applications58 (1977), 135–146.

25.

Khan

S.K.

and Pal

, Interval-valued intuitionistic fuzzy matrices, Notes on Intuitionistic Fuzzy Sets11(1) (2005), 16–27.

26.

Khan

S.K.

and Pal

, The generalized inverse of intuitionistic fuzzy matrix, Journal of Physical Sciences11 (2007), 62–67.

27.

Kim

K.H.

and Roush

F.W.

, Generalized fuzzy matrices, Fuzzy Sets and Systems4 (1980), 293–315.

28.

Lee

K.M.

, Bipolar-valued fuzzy sets and their operations, Proc Int Conf on Intelligent Technologies, Bangkok, Thailand, 2000, pp. 307–312.

29.

Lee

K.M.

, Comparison of interval-valued fuzzy set, intuitionistic fuzzy sets and bipolar-valued fuzzy sets, J Fuzzy Logic Intelligent Systems14(2) (2004), 125–129.

30.

Lur

Y.-Y.

, Pang

C.-T.

and Guu

S.-M.

, On nilpotent fuzzy matrices, Fuzzy Sets and Systems145 (2004), 287–299.

31.

Mondal

, Interval-valued fuzzy vector space, Annals of Pure and Applied Mathematics2(1) (2012), 86–95.

32.

Mondal

and Pal

, Soft matrices, Journal of Uncertain Systems7(4) (2013), 254–264.

33.

Mondal

and Pal

, Similarity relations, invertibility and eigenvalues of intuitoinistic fuzzy matrix, Fuzzy Inf Eng5(4) (2013), 431–443.

34.

Mondal

and Pal

, Intuitionistic fuzzy incline matrix and determinant, Annals of Fuzzy Mathematics and Informatics8(1) (2014), 19–32.

35.

Mondal

and Pal

, Rank of interval-valued fuzzy matrices, Afr Mat. DOI: 10.1007/s13370-015-0325-8

36.

Mondal

and Pal

, Bipolar fuzzy matrices, communicated.

37.

Mondal

and Pal

, Bipolar fuzzy vector space and rank of bipolar fuzzy matrix, communicated.

38.

Mondal

and Pal

, Solution of assignment problem by using the concept of soft matrix, communicated.

39.

Pal

, Intuitionistic fuzzy determinant, V.U.J. Physical Sciences7 (2001), 87–93.

40.

Pal

, Khan

S.K.

and Shyamal

A.K.

, Intuitionistic fuzzy matrices, Notes on Intuitionistic Fuzzy Sets8(2) (2002), 51–62.

41.

Pal

, Fuzzy matrices with fuzzy rows and columns, to appear in, Journal of Intelligent and Fuzzy Systems.

42.

Pal

, Interval-valued fuzzy matrices with interval-valued rows and columns, Communicated.

43.

Pradhan

and Pal

, Intuitionistic fuzzy linear transformations, Annals of Pure and Applied Mathematics1(1) (2012), 57–68.

44.

Pradhan

and Pal

, Generalized inverse of block intuitionistic fuzzy matrices, International Journal of Applications of Fuzzy Sets and Artificial Intelligence3 (2013), 23–38.

45.

Pradhan

and Pal

, Convergence of maxgeneralized meanmingeneralized mean powers of intuitionistic fuzzy matrices, The Journal of Fuzzy Mathematics22(2) (2013), 477–492.

46.

Shyamal

A.K.

and Pal

, Distances between intuitionistic fuzzy matrices, V.U.J. Physical Sciences8 (2002), 81–91.

47.

Shyamal

A.K.

and Pal

, Two new operations on fuzzy matrices, Journal of Applied Mathematics and Computing15(1-2) (2004), 91–107.

48.

Shyamal

A.K.

and Pal

, Distance between intuitionistic fuzzy matrices and its applications, Natural and Physical Sciences19(1) (2005), 39–58.

49.

Shyamal

A.K.

and Pal

, Interval-valued fuzzy matrices, The Journal of Fuzzy Mathematics14(3) (2006), 583–604.

50.

Theodorou

, Drossos

and Alevizos

, Correspondence analysis with fuzzy data: The fuzzy eigenvalue problem, Fussy Sets and Systems158 (2007), 704–721.

51.

Thomason

M.G.

, Convergence of powers of a fuzzy matrix, J Math Anal Appl57 (1977), 476–480.

52.

Wang

Y.M.

and Chin

K.S.

, An eigenvector method for generating normalized interval and fuzzy weights, Applied Mathematics and Computation181 (2006), 1257–1275.

53.

Xin

L.J.

, Convergence of power of a controllable fuzzy matrices, Fuzzy Sets and System45 (1992), 313–319.

54.

Zhang

, Bipolar fuzzy sets and relations: A computational framework for cognitive modeling and multiagent decision analysis, IEEE (1994), 305–309.

55.

Zhang

, Bipolar Fuzzy Sets, in: Proceedings of IEEE World Congress on Computational Science–Fuzz-IEEE, Anchorage, AK, 1998, pp. 835–840.

56.

Zhang

, Fuzzy equilibrium relations and bipolar fuzzy clustering, in: Proceedings of 18th Internat Conference of the North American Fuzzy Information Processing Society–NAFIPS, New York, 1999, pp. 361–365.

57.

Zhang

, Bipolar logic and bipolar fuzzy partial ordering for clustering and coordination, in: Procedings of 6th International Conferance on Information Science, Duke University, NC, 2002, pp. 85–88.

58.

Zhang

, Bipolar logic and bipolar fuzzy logic, in: Proceedings of Internat Conference of the North American Fuzzy Information Processing Society (NAFIPS), New Orleans, LA, 2002, pp. 286–291.

59.

Zhang

and Zhang

, Yin Yan bipolar logic and bipolar fuzzy logic, Information Sciences165 (2004), 265–287.