Journal of Applied Mathematics and Physics
Vol.06 No.05(2018), Article ID:84916,10 pages
10.4236/jamp.2018.65093
On Monotone Eigenvectors of a Max-T Fuzzy Matrix
Qing Wang, Nan Qin, Zixuan Yang, Lifen Sun, Liangjun Peng, Zhudeng Wang*
School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, China
Copyright © 2018 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: May 3, 2018; Accepted: May 27, 2018; Published: May 30, 2018
ABSTRACT
The eigenvectors of a fuzzy matrix correspond to steady states of a complex discrete-events system, characterized by the given transition matrix and fuzzy state vectors. The descriptions of the eigenspace for matrices in the max-Łukasiewicz algebra, max-min algebra, max-nilpotent-min algebra, max-product algebra and max-drast algebra have been presented in previous papers. In this paper, we investigate the monotone eigenvectors in a max-T algebra, list some particular properties of the monotone eigenvectors in max-Łukasiewicz algebra, max-min algebra, max-nilpotent-min algebra, max-product algebra and max-drast algebra, respectively, and illustrate the relations among eigenspaces in these algebras by some examples.
Keywords:
Fuzzy Matrix, Triangular Norm, Max-T Algebra, Eigenspace, Monotone Eigenvector
1. Introduction
The eigenproblem for a fuzzy matrix corresponds to finding a stable state (or all stable states) of the complex discrete-events system described by the given transition matrix and fuzzy state vectors. Therefore, the investigation of the eigenspace structure in fuzzy algebras is important for application. This problem has been solved in several types of so-called extremal algebras.
A max-T fuzzy algebra is defined over the interval and uses, instead of the conventional operations of addition and multiplication, the operations of maximum and one of the triangular norms, the so-called t-norm. These operations are extended in a natural way to the Cartesian products of vectors and matrices. The t-norms together with the t-conorms play a key role in fuzzy theory. These functions have applications in many areas, such as decision making, statistics, game theory, information and data fusion, probability theory, and risk management.
Although there exist various t-norms and families of t-norms (see, e.g., [1] ), let us mention the several main t-norms: the Łukasiewicz t-norm, the Gödel t-norm, the nilpotent minimum t-norm, the product t-norm, and the drastic t-norm.
The Łukasiewicz t-norm is computed as
The Gödel t-norm is the simplest t-norm and the conjunction is defined as the minimum of the entries, i.e.,
The nilpotent minimum t-norm is defined by
The definition of the product t-norm is
The drastic triangular t-norm is “the weakest norm” and the basic example of a non-divisible t-norm on any partially ordered set. The drastic triangular t-norm is defined as follows:
Recently, Gavalec et al. [2] [3] investigated the steady states of max-Łukasiewicz fuzzy systems and monotone interval eigenproblem in max-min algebra, Rashid et al. [4] discussed the eigenspace structure of a max-product fuzzy matrix and Gavalec et al. [5] studied the eigenspace structure of a max-drast fuzzy matrix. In this paper, based on these works, we further study eigenproblem. We investigate the eigenvectors in a max-T algebra, study monotone eigenvectors in max-nilpotent-min algebra, discuss the relation between the monotone eigenvectors in max-T algebra and max-drast algebra, and illustrate the relations among eigenspaces in these algebras by some examples.
2. Eigenvectors in a Max-T Algebra
Let T be one of the triangular norms used in fuzzy theory, let us denote the real unit interval by I. By the max-T algebra we understand the triple with the binary operations and on I. For given natural , we write . The set of all permutations on N will be denoted by . The notations and denote the set of all vectors and all square matrices of a given dimension over I, respectively. The operations and are extended to matrices and vectors in the standard way.
The eigenproblem for a given matrix in max-T algebra consists in finding an eigenvector for which holds true. The eigenspace of is denoted by
Theorem 2.1. Let be three triangular norms on I, and . If , and , then .
Proof. If and , then
When , we have that
i.e., . Thus, .
The theorem is proved.
The investigation of the eigenspace structure can be simplified by permuting any vector to a non-decreasing form.
For given permutations , we denote by the matrix with rows permuted by and columns permuted by , and we denote by the vector permuted by .
Theorem 2.2. (Gavalec [6] ). Let , and . Then if and only if .
We say a vector is increasing if holds for any and strictly increasing if whenever . The set of all increasing eigenvectors of a matrix A is denoted by and the set of all strictly increasing eigenvectors of a matrix A is denoted by . Similar notation and will be used without the condition .
Theorem 2.3. Let and . Then if and only if for every the following hold.
Proof. By definition, is equivalent with the condition
which is equivalent to for every and for some .
The theorem is proved.
Theorem 2.4. Let and . If , then
1) for all ,
2) .
Proof. If , then it follows from Theorem 2.3 that
When , , this is a contradiction. Thus, for every . Noting that is the largest triangular normon I, we see that
and hence .
The theorem is proved.
In particular, if with , then
3. Eigenvectors in Max-Łukasiewicz Algebra
The following theorem contains several logical consequences of the definition of Łukasiewicz triangular norm.
Theorem 3.1. (Rashid et al. [7] ). Let . Then
1) if and only if or ,
2) if and only if or ( and ),
3) if and only if ,
4) if and only if ,
5) if , then .
Combining Theorem 2.3 with Theorem 3.1, we have the following theorem.
Theorem 3.2. (Rashid et al. [7] ). Let and . Then if and only if for every the following hold:
1) for every and ,
2) if , then or for some ,
3) if , then for some .
The following theorem describes necessary conditions under which a given square matrix can have a strictly increasing eigenvector.
Theorem 3.3. (Rashid et al. [7] ). Let . If , then the following conditions are satisfied
1) for all and ,
2) .
The following theorem describes necessary and sufficient conditions under which a three-dimensional fuzzy matrix has a strictly increasing eigenvector.
Theorem 3.4. (Rashid et al. [7] ). Let . Then if and only if the following conditions are satisfied
1) , for all and ,
2) , or ,
3) .
Example 3.1. Let us consider the matrix
Matrix A satisfies conditions (1)-(3) in Theorem 3.4, hence and
4. Eigenvectors in Max-Min Algebra
In the case of the max-min (called also: bottleneck) algebras, the eigenproblem has been studied by many authors and interesting results describing the structure of the eigenspace have been found (see [3] [8] [9] [10] [11] [12] ). In particular, algorithms have been suggested for computing the maximal eigenvector of a given max-min matrix (see [13] ).
If the binary operation coincides with the minimum operation, then the strictly increasing eigenspace can be described as an interval of strictly increasing eigenvectors, where the bounds of the interval are defined as follows
If a maximum of an empty set should be computed in the above definition of , then we use the fact that by usual definition.
The following theorem has been proved in [6] .
Theorem 4.1. (Gavalec [6] ). Let and be a strictly increasing vector. Then if and only if , i.e.,
Hence, in view of Theorem 4.1, the structure of has been completely described for any .
5. Eigenvectors in Max-Nilpotent-Min Algebra
We know that the nilpotent minimum norm is left-continuous and the R-implication generated from is defined by
Moreover, it follows from Proposition 2.5.2 in [14] that and form an adjoint pair, i.e., they satisfy the following residual principle
and
If , then is equivalent with the two conditions:
1) for any , ,
2) there exist such that .
For ,
1) if , then ;
2) if , then if and only if
a) when , , ,
b) when , and
i.e., and hence ;
3) if , then and
6. Eigenvectors in Max-Product Algebra
For every vectors , define the quotient vector by
Then, if and only if fulfills the following inequalities
Noting that for any ,
Thus, it follows from Theorem 2.3 that
Theorem 6.1. (Rashid et al. [4] ). Suppose that and . Then if and only if for every the following two conditions hold.
1) for every ,
2) or for some .
When , and it follows from the proof of Theorem 2.4 that , i.e., .
Thus, the following theorem is a corollary of Theorem 2.4.
Theorem 6.2. (Rashid et al. [4] ). If and , then the following conditions satisfied
1) for all ,
2) .
This Theorem describes necessary conditions which a square matrix can have an increasing eigenvector.
7. Eigenvectors in Max-Drast Algebra
For , we have
for every .
Theorem 7.1. (Gavalec et al. [5] ). Let and . Then if and only if for every the following conditions hold
1) for every ,
2) if , then ,
3) for some , .
Moreover, the following theorem describes necessary and sufficient conditions which a square matrix possesses a strictly increasing eigenvector.
Theorem 7.2. (Gavalec et al. [5] ). Let . Then if and only if the following conditions are satisfied
1) for all ,
2) for all with ,
3) for all with ,
4) .
When and
i.e., . This shows that conditions (1) and (4) in Theorem 7.2 are also straightforward consequences of Theorem 2.4.
The next theorem characterizes all the eigenvectors of a given matrix. In other words, the theorem completely describes the eigenspace structure.
Theorem 7.3. (Gavalec et al. [5] ). Let , , and . Then if and only if the following conditions are satisfied
1) for all with ,
2) if , then for all with ,
3) if , then ,
4) if for some , then .
8. The Relations among These Eigenspaces
Now we discuss the relation between the monotone eigenvectors in max-T algebra and max-drast algebra.
Theorem 8.1. Let . If and , then
Proof. Assume that . For each , when , it follows from Theorem 2.4 that . If , then
Thus, .
When ,
i.e., there exist some such that
Therefore, by Proposition 3.3 in [5] .
The theorem is proved.
Finally, we illustrate the relations among eigenspaces in these algebras by two examples.
Example 8.1. Let
Then
Thus, , but . This illustrates that the condition is necessary in Theorem 6.2. Moreover, by a simple computation, we see that
i.e., .
This example shows that
even if .
Example 8.2. Let
Then the conditions (1)-(4) hold in Theorem 3.4 in [5] . Thus, . But,
i.e., and .
9. Conclusions and Further Works
The eigenproblem for a fuzzy matrix corresponds to finding a stable state of the complex discrete-events system described by the given transition matrix and fuzzy state vectors and the investigation of the eigenspace structure in fuzzy algebras is important for application. Gavalec et al. [2] [3] have investigated the steady states of max-Łukasiewicz fuzzy systems, Rashid et al. [4] and Gavalec et al. [5] have discussed the eigenspace structure of a max-product fuzzy matrix and a max-drast fuzzy matrix, respectively.
In this paper, we investigated the eigenvectors in a max-T algebra, discussed monotone eigenvectors in max-nilpotent-min algebra, and studied the relation between the monotone eigenvectors in max-T algebra and max-drast algebra.
In a forthcoming paper, we will further investigate monotone eigenvectors in max-nilpotent-min algebra and max-T algebra.
Acknowledgements
This work is funded by College Students Practice Innovation Training Program (201610324027Y).
Cite this paper
Wang, Q., Qin, N., Yang, Z.X., Sun, L.F., Peng, L.J. and Wang, Z.D. (2018) On Monotone Eigenvectors of a Max-T Fuzzy Matrix. Journal of Applied Mathematics and Physics, 6, 1076-1085. https://doi.org/10.4236/jamp.2018.65093
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