Similarity relations,eigenvalues and eigenvectors of bipolar fuzzy matrix

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 n _ ( x ) , μ p ¯ ( x ) ) } where μ n _ : X → [ - 1 , 0 ] and μ p ¯ : X → [ 0 , 1 ] are two mappings.

The positive membership degree μ 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 n _ ( x ) denotes the satisfaction degree of x to some implicit counter-property of ℬ _F .

If μ p ¯ ( x ) ≠ 0 and μ n _ ( x ) = 0, it is the situation that x is regarded as having only positive satisfaction for ℬ _F . If μ p ¯ ( x ) = 0 and μ n _ ( x ) ≠ 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 p ¯ ( x ) ≠ 0 and μ n _ ( x ) ≠ 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 ∈ 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 x + y = ( - x n , x p ) + ( - y n , y p ) = ( - max { x n , y n } , max { x p , y p } ) = ( - { x n ∨ y n } , { x p ∨ y p } ) .

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).

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 ) ≥ B n ( x ) ∧ B n ( y ) and B p ( ax + by ) ≥ B p ( x ) ∧ 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 x + y = [ x 1 + y 1 , x 2 + y 2 , x 3 + y 3 , … , x m + y m ] ∈ V m and for any a ∈ ℬ _F ax = [ ax 1 , ax 2 , ax 3 , … , ax m ] ∈ 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 ≥ M R 2,

i.e. r n ( x , z ) ≥ max y ∈ X { min { r n ( x , y ) , r n ( y , z ) } } and r p ( x , z ) ≥ min y ∈ 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:

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

P ^T is a reflexive BFM,

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 _m Q 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 ^T P 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 ≥ P ≥ I m .(1) Also, P is transitive, then P 2 ≤ 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 = [ ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ] ≱ I 2 . Hence P is not reflexive. But P 2 = P ⊙ P = [ ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ] ⊙ [ ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ] = [ ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ] = P .

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 ( 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 ≤ PQ . 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} p ij ≥ max k { min ( p ik , p kj ) } for all i , j , i . e . p ii ≥ max k { min ( p ik , p ki ) } for i = j and each k ≥ min ( p ij , p ji ) for k = j for each i This gives p _ii  ≥ p _ij [Since p _ij  = p _ji ] . □

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 = [ ∑ k = 1 m a 1 k x k 1 ∑ k = 1 m a 2 k x k 2 ⋮ ∑ k = 1 m a mk x km ] = [ o b o b ⋮ a ii ⋮ o b ] = a ii [ o b o b ⋮ i b ⋮ o b ] [Since the ith entry ∑ k = 1 m a ik x ki = a i 1 · o b + a i 2 · o b + ⋯ + a ii · i b + ⋯ + a in · o b = a ii.] Therefore, AX = a _ii X.

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 = [ ( - 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 ) ] and X = [(0, 0)   (-1, 1)   (0, 0)]  ^T . Then AX = [ ( - 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 ) ] ⊙ [ ( 0 , 0 ) ( - 1 , 1 ) ( 0 , 0 ) ] = [ ( 0 , 0 ) ( - 0.4 , 0.3 ) ( 0 , 0 ) ] = ( - 0.4 , 0.3 ) [ ( 0 , 0 ) ( - 1 , 1 ) ( 0 , 0 ) ] = ( - 0.4 , 0.3 ) X . 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 = [ ( - 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 ) ] and X = ((0, 0)   (-1, 1)   (0, 0)). Then XA = ( ( 0 , 0 ) ( - 1 , 1 ) ( 0 , 0 ) ) ⊙ [ ( - 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 ) ]

= ( ( 0 , 0 ) ( - 0.6 , 0.4 ) ( 0 , 0 ) ) = ( - 0.6 , 0.4 ) ( ( 0 , 0 ) ( - 1 , 1 ) ( 0 , 0 ) ) . 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, ∑ j = 1 m a ij = λ. Also X = [i _b , i _b , i _b , …, i _b ]  ^T  ∈ V ^m . Then AX = [ ∑ j = 1 m a 1 j i b ∑ j = 1 m a 2 j i b ⋮ ∑ j = 1 m a mj i b ] = [ ∑ j = 1 m a 1 j ∑ j = 1 m a 2 j ⋮ ∑ j = 1 m a mj ] = [ λ λ ⋮ λ ] = λ [ i b i b ⋮ i b ] = λ X This shows, λ is an eigenvalue of A corresponding to the column eigenvector X. □

Example 5. Let A = [ ( - 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 ) ] and X = [(-1, 1)   (-1, 1)   (-1, 1)]  ^T . Then AX = [ ( - 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 ) ]

⊙ [ ( - 1 , 1 ) ( - 1 , 1 ) ( - 1 , 1 ) ] = [ ( - 0.8 , 0.9 ) ( - 0.8 , 0.9 ) ( - 0.8 , 0.9 ) ] = ( - 0.8 , 0.9 ) [ ( - 1 , 1 ) ( - 1 , 1 ) ( - 1 , 1 ) ] .

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 = [ ( - 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 ) ] and X = ((-1, 1)   (-1, 1)   (-1, 1)) . Then XA = ( ( - 1 , 1 ) ( - 1 , 1 ) ( - 1 , 1 ) ) ⊙ [ ( - 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 ) ] = ( ( - 0.9 , 0.8 ) ( - 0.9 , 0.8 ) ( - 0.9 , 0.8 ) ) = ( - 0.9 , 0.8 ) ( ( - 1 , 1 ) ( - 1 , 1 ) ( - 1 , 1 ) ) . 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 ≥ ∑ j ≠ i , j = 1 m a ij. A is called column diagonally dominant if a ii ≥ ∑ i ≠ 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 ∑ j = 1 m a ij = a ii = c and ∑ i = 1 m a ij = a jj = c. Also X = α [i _b , i _b , i _b , …, i _b ]  ^T  ∈ V ^m . Then AX = [ α ∑ j = 1 m a 1 j i b α ∑ j = 1 m a 2 j i b ⋮ α ∑ j = 1 m a mj i b ] = [ α ∑ j = 1 m a 1 j α ∑ j = 1 m a 2 j ⋮ α ∑ j = 1 m a mj ] = [ α c α c ⋮ α c ] = c α [ i b i b ⋮ i b ] = cX . 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 = [ ( - 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 ) ] and X = ((-0.9, 0.8)   (-0.9, 0.8)   (-0.9, 0.8)). Then XA = ( ( - 0.9 , 0.8 ) ( - 0.9 , 0.8 ) ( - 0.9 , 0.8 ) ) ⊙ [ ( - 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 ) ] = ( ( - 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 ) ) . 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 ∑ j = 1 m a kj = ( - ∑ j = 1 m a kjn , ∑ 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 AX = [ α ∑ j = 1 m a 1 j i b α ∑ j = 1 m a 2 j i b ⋮ α ∑ j = 1 m a mj i b ] = [ α ∑ j = 1 m a 1 j α ∑ j = 1 m a 2 j ⋮ α ∑ j = 1 m a mj ] = [ α λ α λ ⋮ α λ ] = λ α [ i b i b ⋮ i b ] = λ X .

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

Example 8. Let A = [ ( - 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 ) ] and X = [(-1, 1)   (-1, 1)   (-1, 1)]  ^T . Then AX = [ ( - 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 ) ] ⊙ [ ( - 1 , 1 ) ( - 1 , 1 ) ( - 1 , 1 ) ] = [ ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ( - 0.8 , 0.7 ) ] = ( - 0.8 , 0.7 ) [ ( - 1 , 1 ) ( - 1 , 1 ) ( - 1 , 1 ) ] . 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 = [ ( - 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 ) ] and X = (-0.9, 0.9) ((-1, 1)   (-1, 1)   (-1, 1)). Therefore, XA = ( - 0.9 , 0.9 ) ( ( - 1 , 1 ) ( - 1 , 1 ) ( - 1 , 1 ) ) ⊙ [ ( - 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 ) ] = ( - 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 ) ) . 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 ∑ j = 1 m a 1 j = ∑ j = 1 m a 2 j = ∑ j = 1 m a 3 j = ⋯ = ∑ 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 ∑ i = 1 m a i 1 = ∑ i = 1 m a i 2 = ∑ i = 1 m a i 3 = ⋯ = ∑ 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 _b X. 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 _b X. We assume that x _i  ≠ o _b for i ∈ {1, 2, 3, …, m}. Then AX = o _b X implies that ∑ k = 1 m a jk x k = o b for each j ∈ {1, 2, 3, …, m}. This implies a _jk x _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 _b X = 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 ^m X = A^m-1AX = A^m-1X = A^m-2X = ⋯ = A²X = AX = X,

i.e. X = A ^m X ≤ A ^m e ≤ B ^m e.

[Since X ≤ e and A ≤ B.]

Since X is non-zero hence B ^m e is non-zero.

Now, if Y = B ^m e, then BY = B^m+1e = B ^m e = Y = i _b Y. 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.