New methods for computing fuzzy eigenvalues and fuzzy eigenvectors of fuzzy matrices using nonlinear programming approach

In this paper, we propose a new method to obtain the eigenvalues and fuzzy triangular eigenvectors of a fuzzy triangular matrix A~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\tilde{A}} \right)$$\end{document}, where the elements of the fuzzy triangular matrix are given. For this purpose, we solve 1-cut of a fuzzy triangular matrix A~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\tilde{A}} \right)$$\end{document} to obtain 1-cut of eigenvalues and eigenvectors. Considering the interval system A~αX~α=λ~αX~α0≤α≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {\tilde{A}} \right]_{\alpha } \left[ {\tilde{X}} \right]_{\alpha } = \left[ {\tilde{\lambda }} \right]_{\alpha } \left[ {\tilde{X}} \right]_{\alpha } 0 \le \alpha \le 1$$\end{document} as α-cut of the fuzzy system A~X~=λ~X~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{A}\tilde{X} = \tilde{\lambda }\tilde{X}$$\end{document}, to determine the left and right width of eigenvalues λ~α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {\tilde{\lambda }} \right]_{\alpha }$$\end{document} and eigenvector elements X~α0≤α≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {\tilde{X}} \right]_{\alpha } 0 \le \alpha \le 1$$\end{document}, we make a system of linear and nonlinear equations and inequalities. And we propose nonlinear programming models to solve the system of linear and nonlinear equations and inequalities and to calculate λ~α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {\tilde{\lambda }} \right]_{\alpha }$$\end{document} and X~α0≤α≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {\tilde{X}} \right]_{\alpha } 0 \le \alpha \le 1$$\end{document}. Furthermore, we define three other new eigenvalues (namely, fuzzy escribed eigenvalue, fuzzy peripheral eigenvalue, and fuzzy approximate eigenvalue) for a fuzzy triangular matrix (Ã) that the fuzzy eigenvalue and fuzzy eigenvector cannot be obtained based on interval calculations. Therefore, the fuzzy escribed eigenvalue which is placed in a tolerable fuzzy triangular eigenvalue set, the fuzzy peripheral eigenvalue placed in a controllable fuzzy triangular eigenvalue set, and the fuzzy approximate eigenvalue placed in an approximate fuzzy triangular eigenvalue set is defined in this paper. Finally, numerical examples are presented to illustrate the proposed method.


triangular matrixÃ
À Á to obtain 1-cut of eigenvalues and eigenvectors. Considering the interval systemÃ Â Ã aX Â Ã a 1 k h i aX Â Ã a 0 a 1 as a-cut of the fuzzy systemÃX ¼kX, to determine the left and right width of eigenvaluesk h i a and eigenvector elementsX Â Ã a 0 a 1, we make a system of linear and nonlinear equations and inequalities. And we propose nonlinear programming models to solve the system of linear and nonlinear equations and inequalities and to calculatek h i a andX Â Ã a 0 a 1. Furthermore, we define three other new eigenvalues (namely, fuzzy escribed eigenvalue, fuzzy peripheral eigenvalue, and fuzzy approximate eigenvalue) for a fuzzy triangular matrix (A˜) that the fuzzy eigenvalue and fuzzy eigenvector cannot be obtained based on interval calculations. Therefore, the fuzzy escribed eigenvalue which is placed in a tolerable fuzzy triangular eigenvalue set, the fuzzy peripheral eigenvalue placed in a controllable fuzzy triangular eigenvalue set, and the fuzzy approximate eigenvalue placed in an approximate fuzzy triangular eigenvalue set is defined in this paper. Finally, numerical examples are presented to illustrate the proposed method.
Keywords Fuzzy number Á Fuzzy eigenvalues Á Fuzzy eigenvector Á Fuzzy triangular matrix 1 Introduction The system of linear equations plays a crucial role in various areas such as physics, statistics, operational research, engineering, and social sciences. One of the major applications of fuzzy number arithmetic is in solving linear systems whose parameters are all or partially represented by fuzzy numbers. Therefore, it is immensely important to develop a numerical procedure that would appropriately treat general fuzzy linear systems and solve them.
The system of linear equations A e X ¼ e b where the coefficient matrix A ¼ a ij À Á ; 1 i; j n is a crisp matrix and the elements, e x i ; e b i ; 1 i n of the vectors e X; e b are fuzzy numbers, called a fuzzy linear system (FLS). A general model for solving a FLS first was proposed by Friedman et al. (1998). Friedman et al. (2000) investigated a dual fuzzy linear system using nonnegative matrix theory. To continue Friedman et al.'s work, Allahviranloo (Zheng and Wang 2006) proposed various numerical methods to solve fuzzy linear systems. In addition, Abbasbandy et al. (2006) used the LU decomposition method for solving a fuzzy system of the linear equation when the coefficient matrix is symmetric positive definite. In  proposed a practical method to solve a FLS. In that method, they solved 1-cut of a fuzzy linear system. Then, by some fuzzification, they obtained the fuzzy vector solutions by symmetric spreads of each element of a fuzzy vector solution. Also, for more references, see (Wang and Zhend 2006;Zheng and Wang 2006). such that e a ij ; e b i ; e x j are fuzzy numbers, for all i; j ¼ 1; . . .; n; called a fully fuzzy linear system (FLS).
FFLSs have been studied by many authors. Buckley and Qu (1991) in their consecutive works suggested different solutions for FFLS.
To calculate fuzzy eigenvalues in a fuzzy matrix e A ¼ e a ij À Á nÂn , it is necessary to solve a system of nonlinear equations e A e X ¼ e k e X in which e k is a fuzzy number. This system is called a dual fully fuzzy system of equations that has been studied by Muzzioli and Reynaerts (2006). They introduced a solution algorithm with high complexity, especially for large ns. In general, it is difficult and complicated to find a fuzzy eigenvalue from a system of nonlinear equations e A e X ¼ e k e X because in addition to being nonlinear, these systems are complex in terms of calculations size with the consideration of a-cut theory. On the other hand, on the right side of these systems, the existence of two fuzzy unknown variables and applying calculations between them further complicates the examination of their solution.
The methods introduced by other researchers also have some deficiencies. At first in 1991, Buckley (1990) introduced a method for calculating the fuzzy eigenvalues in a positive fuzzy matrix e A that was based on a-cut theory and interval calculations. However, this method is not appropriate as it limits fuzzy eigenvalues and fuzzy eigenvectors. Then, Chiao (1998) studied generalized fuzzy eigenvalues in the form of e A e X ¼ e k e B e X. His method is similar to a-cut theory and Buckley's approach, having some restrictions in the performance and calculation of fuzzy eigenvalues.
In Buckley et al. (2002) introduced a method for calculating fuzzy eigenvalues using artificial neural nets, which has the same restrictions as the previous one. This method has some deficiencies based on a-cut theory and interval calculations given limiting fuzzy matrix, fuzzy eigenvalues and fuzzy eigenvectors. Dehghan et al. (2007) proposed some methods to solve FFLS such as the Cramer's rule, Gaussian elimination, LU decomposition (Doolittle algorithm), and linear programming (LP) for solving square and non-square fully fuzzy systems. However, their methods are not available for the nonnegative solution. Vroman et al. (2007) suggested a practical algorithm using parametric functions in which the variables were given by the fuzzy coefficients of the system. In addition, they showed that their algorithm is better than the method of Buckley and Qu.
In Theodorou et al. (2007) used a two-stage method with the analysis of the data of triangular fuzzy numbers in the form of a fuzzy matrix. Considering a-cuts of the fuzzy system e A e X ¼ e k e X and conducting interval calculations as well as limiting the sign of fuzzy eigenvalues and fuzzy eigenvectors along with assuming k MAX ¼ 1, it is timeconsuming and complex to perform. Tian (2010) extended the crisp eigenvectors of a crisp matrix to fuzzy eigenvalues. He studied the structure of fuzzy eigenspaces and relationships between crisp eigenspaces on a crisp matrix, upon which he examined the system of fuzzy linear equations and offered a solution for it.
In  suggested a method to solve FFLS. To this end, they solved 1-cut of an FFLS, then allocated some unknown symmetric spreads to each row of a 1-cut and then the symmetric spreads of the solution are computed by solving a 2n linear equations.
In Allahviranloo et al. (2013) presented a practical method for solving fully fuzzy linear systems. By considering the answer of 1-cut of the system e A e X ¼ e k e X, they allocated some unknown symmetric expansions for each cut of the system and then offered a system of linear equation for calculating fuzzy eigenvalues and fuzzy eigenvectors using interval algebra. In Allahviranloo and Hosseinzade. (2014) proposed a novel method to obtain the fuzzy trapezoidal solution for an FFLS. Their method is constructed based on solving two FILSs. In addition, they introduced two different models for those FFLSs that do not have a feasible solution. Also, for more references, see (Abbasi and Allahviranloo 2022;Akram et al. 2021;Chen and Guo 2021;Najariyan and Zhao 2020;Siahlooei and Fazeli 2018;Stanimirović and Micić 2022).
So, there are various methods for finding the fuzzy eigenvalues and eigenvectors of a fuzzy matrix. However, some of these existing methods suffer from a few shortcomings as below: 1) Some of them can be performed only when the signs of the fuzzy eigenvaluek and the fuzzy eigenvectorX obtained from the fuzzy systemÃX ¼kX are known. However, these signs cannot be predetermined. In that case, specific answers are found. Another problem is that if the sings ofX andk are regarded as being specific (positive or negative) in advance, the fuzzy systemÃX ¼kX may offer no answer. Refer to Allahviranloo and Hooshangian (2013), Chiao (1998), Theodorou et al. (2007), Tian (2010). 2) In some of these methods, a certain condition for the performance of the fuzzy matrixÃ is considered, for example the elements of the matrix must be fuzzy positive numbers (Theodorou et al. 2007). 3) Some of them define fuzzy eigenvalues and fuzzy eigenvectors based on a-cuts theory of the system ÃX ¼kX, where interval calculations are used. In these methods, fuzzy eigenvalues and eigenvectors are in fact obtained from solving the interval system However, if this system does not provide the interval answerk h i a andX Â Ã a , these methods cannot be performed. Refer to Allahviranloo and Hooshangian (2013), Chiao (1998).
Most of the methods have not considered this condition for calculating the interval eigenvalues k h i a 0 a 1 and the interval eigenvectorsX Â Ã a 0 a 1. Refer to Allahviranloo and Hooshangian (2013); Chiao (1998), Theodorou et al. (2007), Tian (2010).
It can generally be said that all methods proposed before have some shortages. Thus, here we intend to offer a method for calculating the fuzzy eigenvaluesk and fuzzy eigenvectorx from the fuzzy matrix obtained from the fuzzy systemÃX ¼kX, so that first for implementing the method there will be fewer limitations for the signsk andx and more general answers to the fuzzy eigenvalues and fuzzy eigenvector. Second given that the fuzzy eigenvalue and fuzzy eigenvector are gained from solving the interval is met. Solving the system of nonlinear equations such as AX ¼ kX has various applications in engineering science. Eigenvalue and eigenvector are among important and practical topics in linear algebra and matrix systems and variously used in computational fluid dynamics, the theory of elasticity, stability analysis, vibration analysis, atomic orbitals, data science, machine learning, data mining, face detection, diagonal matrix creation, the complexity analysis of matrix systems, and control systems [refer to Couillet and Liao (2022) (2019)], among others. Now given the ambiguity existing in the data of a system, if the matrix elements of A are fuzzy, the eigenvalue and eigenvector obtained from the matrix will be fuzzy, too. Hence, solving fuzzy linear systemsÃX ¼kX has always been the focus of attention among researchers.
One application for fuzzy eigenvalues is the problem of stable population situation. The objective here is to examine the stable population situation with the consideration of the maximum amount of migration to each city. If intercity migration occurs among n cities, T 1 ; T 2 ; . . .T n . One method to estimate the future migration is the investigation of migration statistics in the previous years.
Assuming every year with the probability of P ij , one resident from the city T i moves to the city T j . If f 1 ; f 2 ; . . .f n is the population of n cities and T 1 ; T 2 ; . . .T n is for a specific period of time like t; t þ 1 ½ . The relation for movement is F tþ1 ¼ AF t . The matrix A in which the sum of each column is equal to one is known as transition matrix in stochastic processes and Markov processes. In the stable situation, the largest eigenvalue of the transition matrix is k ¼ 1. That is, it is possible to find a vector like F t in which F tþ1 ¼ AF t ¼ kF t ¼ 1F t ¼ F t is true. Given the done calculations, if the probability of the movement of a resident from the city T i to the city T j is estimated as the fuzzy number P ij ¼ P ij ; P ij ; P ij À Á , then the transition system will in fact be asF tþ1 ¼ÃF t . In order for this fuzzy system to be stable, New methods for computing fuzzy eigenvalues and fuzzy eigenvectors of fuzzy matrices using… 4427 k ¼ k; k; k À Á as the largest eigenvalue of the transition fuzzy matrixÃ must be equal to one fuzzy. Thus, to examine the problem of the stable population situation using the fuzzy estimation of migration, it is necessary to investigate the fuzzy eigenvalues of the fuzzy matrix.
in this paper, we propose a method for obtaining eigenvalues and fuzzy triangular eigenvectors for a fuzzy triangular matrixÃ À Á using a nonlinear programming problem (NLP). Moreover, we define three other new eigenvalues namely, fuzzy escribed eigenvalue, fuzzy peripheral eigenvalue, and fuzzy approximate eigenvalue for a fuzzy triangular matrix.
The structure of this paper organized as follows: In Sect. 2, we introduce the notation, the definitions, and preliminary results, which will be used throughout. In Sect. 3, we design our new method to obtain eigenvalues and fuzzy triangular eigenvectors of a fuzzy triangular matrix. In Sect. 4, we define three new eigenvalues namely, fuzzy escribed eigenvalue, fuzzy peripheral eigenvalue, and fuzzy approximate eigenvalue for a fuzzy triangular matrix. Numerical examples are given in Sect. 5 to examine our method and conclusions drawn in Sect. 6.

Preliminaries
The basic definitions of a fuzzy number given in Goetschel and Voxman (1986), Zimmermann (1985) as follows: Definition 2.1 An interval number x ½ is defined as the set of real numbers such that We denote the set of all interval numbers by I.
, 1 i n are interval numbers, is called an interval number vector. In this case, we denote X ½ 2 I n . x; x ½ ø y; y Definition 2.4 The width of an interval number x ½ is defined as follows: (Allahviranloo et al. 2014) Definition 2.5 For arbitrary interval numbers x ½ ¼ x; x ½ and y ½ ¼ y; y h i , and arbitrary interval number vectors Definition 2.6 Ostad-Ali-Askari et al. 2017) For any two arbitrary interval number vectors X; Y 2 I n the metric, d : Obviously, I n is a complete metric space with the metric d.

Eigenvalues and fuzzy eigenvectors
The basic definition of fuzzy numbers given in Goetschel and Voxman (1986).
The set of all these fuzzy numbers is denoted by F.
g and the support and core ofÃ are defined by the sets Sũ ð Þ ¼ fx 2 Then, from i ð Þ À iv ð Þ it follows thatũ ½ r is a bounded closed interval for each 2 0; 1 ½ (Wu and Ming 1991). In this paper, we denote the r-cuts of fuzzy number u asũ ½ r ¼ u r ð Þ; u r ð Þ ½ ; for each r 2 0; 1 ½ .
; and k 2 R, r À cuts of the sumũ þṽ and the product k:ũ are defined based on interval arithmetic asũ Definition 3.2 Two fuzzy numbersũ andṽ are said to be equal, if and only ifũ ½ r ¼ṽ ½ r , i.e., u r ð Þ ¼ v r ð Þ and u r ð Þ ¼ v r ð Þ; for each r 2 0; 1 ½ .
Definition 3.3 A fuzzy triangular numberũ ¼ a 1 ; a 2 ; a 3 ð Þ ; defined as follows: where a 2 is the core, a 1 and a 3 are the left and right points of support.
We denote the set of fuzzy triangular numbers by F T . Clearly, for the fuzzy triangular numberã Definition 3.4 Two fuzzy triangular numbersÃ ¼ Definition 3.5. For arbitrary fuzzy triangular numbersÃ andB, addition, subtraction, and scalar multiplication are defined as follows: is called a fuzzy triangular number vector. In this case, we denoteX 2 F n T .
Consider the fuzzy square matrix ofÃ ¼ã ij Â Ã n i;j¼1 that a ij 2 F T .
Definition 3.7 The fuzzy triangular numberk 6 ¼0 is the fuzzy eigenvalue of the fuzzy triangular matrixÃ, if there is a fuzzy eigenvectorX 6 ¼0 such thatÃX ¼kX.
Otherwise, the n Â n linear system of equations is called a fully fuzzy nonlinear system (FFNLS) wherẽ a ij ;k andx i ; 1 i; j n; are fuzzy numbers. The matrix form of the system (3.4) is as follows: AX ¼kX: A method to solve the systemÃX ¼kX to calculate the fuzzy eigenvalue and the fuzzy eigenvector is the a À cut solution approach. In this approach, the systemÃX ¼kX is

and then based on interval calculations, the interval answer ofk
h i a andX Â Ã a is calculated for it. Based on this attitude, the following definitions are given below.
Definition 3.8 The n Â n linear system . . . ð3:5Þ is called the a-cut system of FFLS (3.4), where ½ã ij a ;x j Â Ã a andk h i a , 1 i; j n and a 2 0; 1 ½ , are a-cut of fuzzy numbersã ij ;x j andk; respectively. In addition, its matrix Definition 3.9 We define the following eigenvalue sets for FFLS (3.4) United fuzzy triangular eigenvalue set: New methods for computing fuzzy eigenvalues and fuzzy eigenvectors of fuzzy matrices using… 4429 Tolerable fuzzy triangular eigenvalue set: Controllable fuzzy triangular eigenvalue set: Approximate fuzzy triangular eigenvalue set: eigenvectors and interval number valued eigenvalues and ; 1 i n are 0 À cuts and 1 À cuts of fuzzy numbersx i andk i , respectively.
In definition 3.9. the set UTFES is an interval eigenvalue solution set ofk h i a for a ¼ 0 and a ¼ 1: It is clear that this set is empty if there is no interval solution for the system 3.1 Find eigenvalues and fuzzy eigenvectors of a fuzzy matrix using nonlinear programming models In this section, the reason for using nonlinear programming models to calculate the fuzzy eigenvalue and eigenvectors of a fuzzy matrix is explained. Also, the approach to using nonlinear programming models for controlling fuzzy eigenvalue and eigenvector based on the ambiguity of fuzzy matrix elements is investigated.
¼0 are considered as eigenvalues and fuzzy eigenvectors of a 0 a 1 are the a À cut of fuzzy numbers ofk andx. The

the values of the left and right boundaries ofk
h i a andx j Â Ã a , i.e., k a ; k a ; x ja ; x ja , j ¼ 1; . . .; n; based on the values of the left and right boundaries of the elements of the fuzzy matrix, in the same cut a can have different values considering different attitudes. Also, for each a À cut; the nonlinear equations of k a k a and x ja x ja ; j ¼ 1; . . .; n must also be satisfied. Therefore, in order to control the ambiguity of the fuzzy eigenvalue and the fuzzy eigenvector and simultaneously solve the nonlinear equations and linear and nonlinear inequalities, in this section, nonlinear programming models are used.
Hence, we consider the following system: AX ¼kX; ð3:6Þ Such that either a ij 0 ð Þ ! 0 or a ij 0 ð Þ 0; 1 i; j n: Now, to get a suitable solution of system (3.6), it is sufficient to solve the follows FILSs: . . .; n: ð3:9Þ . . .; n: ð3:10Þ Such that ð3:11Þ First, by solving the system (3.7), we obtain the eigenvalues k 1 ; . . .; k n and the corresponding eigenvectors X 1 ; . . .; X n for the crisp matrixÃ Â Ã 1 . Consequently, after obtaining the solution of the crisp system (3.7), we are going to determine the left and right widths for the eigenvalues k j ; 1 j n and the eigenvectors X j ; 1 j n, which are obtained from system (3.7).
Consider the eigenvalue k k ; k 2 1; . . .; n f g and corresponding eigenvector X k .
We assume that simultaneous zero cannot belong to Sk k and SX j À Á .
Therefore, we consider the following three case: Case A 0 6 2 SX j À Á ; 1 j n, Consider the follows partition.
Therefore, according to the sign of the values obtained x j ; k j from Eq. (3.7), Eq. (3.8) written as follows: ð3:15Þ In summary, due to be clear the sign of the intervals and using the interval multiplication definition, model (3.13) written as follows.
therefore, according to the sign of the values obtained x j ; k j from Eq. (3.7), Eq. (3.8) written as follows: That here: 19Þ New methods for computing fuzzy eigenvalues and fuzzy eigenvectors of fuzzy matrices using… 4431 Consequently, according to the definition (interval multiplication), model (3.17) written as follows: Case B 0 6 2 Sk k then k k ; k k ; k k [ 0 or k k ; k k ; k k \0; Consider the following partition.
Therefore, Eq. (3.8) is rewritten as follows: Solving models (3.16), (3.20) and (3.25) we obtain the values of k k ; k k and x j ; x j ; 1 j n; that is the suitable solution of the system (3.8). One of the important points is to control the width of the fuzzy eigenvaluek in the a-cut, i.e., k a À k a , and also to control the width of the elements of the fuzzy eigenvectorx j ; j ¼ 1; . . .; n ð Þin the a-cut, i.e., P n j¼1 x ja À x ja ; j ¼ 1; . . .; n.
Another point is the property of the fuzzy numberk and elements of the fuzzy eigenvectorx j in nested a-cuts.

Which must apply to the conditionsk
Therefore, in order to control the ambiguity of the fuzzy eigenvalue and the fuzzy eigenvector and simultaneously solve nonlinear equations and linear and nonlinear inequalities, the following nonlinear programming models have been used in (3.16), (3.20) and (3.25). Model (3.26) simultaneously by solving nonlinear equations and linear and nonlinear inequalities, models (3.16), (3.20) and (3.25), in its objective function it obtains the maximum width of the fuzzy eigenvaluek in the a-cut, i.e., k a À k a . Also, model (3.27) simultaneously by solving nonlinear equations and linear and nonlinear inequalities models (3.16), (3.20) and (3.25), in its objective function, obtains the maximum sum of the elements of the fuzzy eigenvector x j ; j ¼ 1; . . .; n ð Þin the a-cut, i.e., P n j¼1 x ja À x ja . The above NLPS (3.26) and (3.27) can be easily solved by using LINGO 18 software package. Is its optimum solution. Also, the fuzzy triangular number vectorX ¼x 1 ;x 2 ; . . .;x n ð Þ T , be constructed as follows: Then,X is a suitable solution for the system (3.6). Otherwise, if the NLPS (3.26) and (3.27) are unfeasible, the system (3.6) does not have any suitable solution.
(3.6)-(3.25) we can conclude that X ¼x 1 ;x 2 ; . . .;x n ð Þ T , wherẽ Is a suitable solution for system (3.6). Now, let us assume that the NLPS (3.26) and (3.27) are unfeasible. By contradiction assume that the system (3.6) has a suitable solutionX ¼x 1 ;x 2 ; . . .;x n ð Þ T , therefore we have: By considering the proposed method, for solving fuzzy triangular matrixÃ, in Sect. 3, we can rewrite: Then, with the approach of the previous section, nonlinear programming models are presented to calculate the fuzzy eigenvalue and the fuzzy eigenvector, in different acut Definition 4.1 The fuzzy triangular numberk called a fuzzy escribed eigenvalue ofÃ, if: We denote fuzzy escribed eigenvalue byk .
Definition 4.2 The fuzzy triangular numberk called a fuzzy peripheral eigenvalue ofÃ, if: We denote fuzzy escribed eigenvalue byk . In the following, we explain a method to obtaining k ;X andk ;X of the systemÃx ¼ e kx.
We first solve the 1-cut system of FFLS, i.e., New methods for computing fuzzy eigenvalues and fuzzy eigenvectors of fuzzy matrices using… 4433 We can construct the algebraic solution of the 1-cut system like the previous part technique as following: Then for finding 0-cut ofk ;X it is sufficient to solve the following NLP problem Now, we define: . . .; n; we have: ð4:32Þ Then by using (4.32) we can rewrite system (4.30) as follows: Min z s:t: Similarly, to obtaink ;X , it is sufficient to solve the following NLP problem Min z s:t:   (1)k 2 F T ;X 2 F n T ; AX ¼ kX We denote fuzzy approximate eigenvalue byk ' . We first solve the 1-cut system of FFLS, i.e., We can construct the algebraic solution of the 1-cut system like the previous part technique as following: Then for finding 0-cut ofk ' ;X ' it is sufficient to solve the following NLP problem Min e s:t: whereã ij; 1 i; j n; are fuzzy triangular numbers. First, we solve 1-cut system (5.36) which is crisp: So, the crisp solution (1-cut) obtained as follows: Now, we are going to get the three models mentioned in Sect. 3 for system (5.36).

ð5:56Þ
New methods for computing fuzzy eigenvalues and fuzzy eigenvectors of fuzzy matrices using… 4445 execution by using of the software package as LINGO 18 software package, then will see, that models are infeasible. Therefore, we obtain the fuzzy escribed eigenvalue, fuzzy peripheral eigenvalue, and fuzzy approximate eigenvalue of above, first, we solve the fuzzy triangular matrixÃ in a 1-cut position. So, we obtain: Clear the value of the objective function of each model is the distance betweenÃXandkX in 0-cut.
So here: Similarly, to obtain widths k 1 ; x 1 and k 3 ; x 3 , we behavior as above.
Therefore, the tables of values of T for proposed models assumed as follows: Therefore, the best solution in Table 1 is the escribed eigenvalue Tk TTFES ;X TTFES ¼ 0:000000 of model (3.25). Therefore, the best solution in Table 2 is the approximate eigenvalue Tk ATFES ;X ATFES ¼ 708:2773 of model (3.20). Therefore, the best solution in Table 3 is the approximate eigenvalue Tk ATFES ;X ATFES ¼ 943:4028 of model (3.20) (Table 4).

Conclusion
In this paper, we proposed some new approaches and definitions to find the eigenvalues and fuzzy triangular eigenvectors of a fuzzy matrixÃ. In these approaches, the solution of the interval systemÃ Â Ã aX Â Ã a 1 k h i aX Â Ã a 0 a 1 is considered as a-cut of the fuzzy systemÃX ¼kX and the eigenvaluesk h i a and elements of the eigenvectorX Â Ã a are obtained using models NLP. As a result of these new models, the exact solutions of United fuzzy triangular eigenvalue set and the approximate solutions of fuzzy described eigenvalue, fuzzy peripheral eigenvalue, and fuzzy approximate eigenvalue for fuzzy eigenvalue and fuzzy eigenvector are obtained in different a-cuts.The advantages of the method presented in this article are as follows: i) In the method presented, there are few restrictions in implementation for the sign of the fuzzy eigenvaluek and the fuzzy eigenvectorX, and better solutions are obtained in opinion diversity in the signk andX. ii) With the method presented, the interval system A Â Ã aX Â Ã a ¼k h i aX Â Ã a ; 0 a 1, definitely has an exact or approximate answer based on interval calculations. That the error of the approximate answer can be calculated. iii) With the method presented, it is possible to control the width of the eigenvaluek h i a and the eigenvec-torX Â Ã a in different a-cuts And fuzzy eigenvalue and fuzzy eigenvector with different widths calculated for a fuzzy matrix. iv) In the method of this article, the amount of . . .; n ð Þ , can be considered to calculate the interval eigenvalues ofk h i a 0 a 1 and the interval eigenvectors ofX Â Ã a 0 a 1.until title acuts be the fuzzy numberk and the fuzzy eigen-vectorX.
In general, the method of this article is not cost-effective for large matrices because the volume of calculations and the complexity of the models and their solution become problematic. Finally, the numerical examples show that the methods are effective and practical to obtain the eigenvalues and fuzzy eigenvectors of a fuzzy matrixÃ. Finally, it is suggested to the researchers that by developing the method presented in this article, models be with a simpler shape and less computational complexity with linear constraints.
Funding The authors have not disclosed any funding.
Data Availability Enquiries about data availability should be directed to the authors.

Declarations
Conflict of interest The authors have not disclosed any competing interests.