Generalized fuzzy eigenvectors of real symmetric matrices 1

Abstract

In paper the generalized real eigenvalue and fuzzy eigenvector of a crisp real symmetric matrix with respect to another real symmetric matrix is studied. The original generalized fuzzy eigen problem is extended into a crisp generalized eigen problem of a real symmetric matrix with high orders using the arithmetic operation of LR fuzzy matrix and vector. Two cases are analysed: (a) the unknown eigenvalue λ is a non negative real number; (b) the unknown eigenvalue λ is a negative real number. Two computing models are established and an algorithm for finding the generalized fuzzy eigenvector of a real symmetric matrix is derived. Moreover, a sufficient condition for the existence of a strong generalized fuzzy eigenvector is given. Some numerical examples are shown to illustrated our proposed method.

Keywords

Fuzzy numbers fuzzy eigenvectors matrix computation fuzzy linear systems

1 Introduction

Many problems in physics, science and engineering are reduced to the problem of finding eigenvalues and eigenvectors of matrices that always inevitably involves the uncertainty of some parameters. Sometimes these uncertain elements are represented and computed by fuzzy numbers rather than crisp number. Therefore, the problem of fuzzy eigenvalues and eigenvectors has attracted the attention of many scholars in recent years. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [29], Dubois et al. [8] and Nahmias [22]. A different approach to fuzzy numbers and the structure of fuzzy number spaces was given by Puri and Ralescu [23], Goetschell et al. [10] and Wu Congxin et al. [26–27]. From Friedman et al. [9] discussed a class of semi fuzzy linear systems $A \tilde{x} = \tilde{b}$ by an embedding approach in 1998, some more general and complicated fuzzy linear systems such as dual fuzzy linear systems, general fuzzy linear systems, complex fuzzy linear systems, dual full fuzzy linear systems and general dual fuzzy linear systems were studied by many a scholar in past two decades see [1–5 , 14]. In recent years, new theories and approaches for fuzzy numbers and fuzzy linear systems has been emerging one after another [19–20 , 26]. For some linear systems related with fuzzy numbers, Guo al. made a few investigations [11–15 , 17–18].

Fuzzy eigenvalues were firstly studied by Buckley [6] on analyzing input-output of systems in 1998. Later, Chiao investigated generalized fuzzy eigenvalues in form $\tilde{A} \tilde{x} = \tilde{λ} \tilde{B} \tilde{x}$ using the same way as Buckley method [7]. After then, Thoedorou al. [24] studied fuzzy eigenvalues of fuzzy corresponding analysis and utilized a two-step method to obtain fuzzy eigenvectors based on triangular fuzzy numbers in. In 2010, Tian founded fuzzy eigenvectors of real matrix and given the structure of fuzzy eigenspaces and relationship between the real eigenvectors and fuzzy eigenvectors in [24]. In 2013, Allahviranloo al. [4] demonstrated how to obtained spreads lead to derive maximal and minimal eigenvalues as well as general fuzzy eigenvalues under different conditions.

Eigenvalue and eigenvector problems are often applied to both algorithms and physics. In terms of algorithm, eigenvalue analysis can reduce a coupled equation to a scalar equation. In physics, eigenvalue analysis provides insight into the properties of evolving systems determined by systems of linear equations. In practical problems, most matrices are symmetric. Symmetric matrix plays a very important role in theoretical research and practical application. In view of the universality of symmetric matrix application and LR fuzzy number is an extension of triangular fuzzy number, we investigate the generalized real eigenvalue and fuzzy eigenvector of a crisp real symmetric matrix in this paper. Two computing models are established and an algorithm for finding the fuzzy eigenvector of a real symmetric matrix is given. The structure of the paper is organized as follows:

In Section 2, some definitions and arithmetic operations on LR fuzzy numbers are recalled. In Section 3, The original generalized fuzzy eigen problem is extended into a simple eigen problem of a crisp real symmetric matrix with high orders and an algorithm for finding the generalized fuzzy eigenvector of a real symmetric matrix is proposed. Some illustrating numerical examples are given in Section 4. And in Section 5, the conclusion and further investigations are given.

2 Preliminaries

There are several definitions for the concept of fuzzy numbers (see [8 , 29]).

2.1 The LR fuzzy number

Definition 2.1. Let X be a nonempty set, A fuzzy set $\tilde{A}$ in X is characterized by a function $\begin{matrix} \begin{matrix} μ_{\tilde{A}} : X \to [0, 1], \end{matrix} \end{matrix}$ which associates each element x ∈ X with a real number in the closed interval [0, 1], where the value $μ_{\tilde{A}}$ represents the degree of membership of x in fuzzy set $\tilde{A}$ , the function $μ_{\tilde{A}} : X \to [0, 1]$ is called the membership function of $\tilde{A}$ . A fuzzy set $\tilde{A}$ is represented by the set of ordered pairs of element x and grade $μ_{\tilde{A}}$ which can be written as $\begin{matrix} \begin{matrix} \tilde{A} = {(x, μ_{\tilde{A}} (x)) | x \in X} . \end{matrix} \end{matrix}$

Definition 2.2. A fuzzy number is a fuzzy set like u : R → I = [0, 1] which satisfies:

(1) u is upper semi-continuous,

(2) u is fuzzy convex, i.e. u (λx + (1 - λ) y) ≥ min {u (x) , u (y)} for all x, y ∈ R, λ ∈ [0, 1],

(3) u is normal, i.e. there exists x₀ ∈ R such that u (x₀) =1,

(4) suppu = {x ∈ R ∣ u (x) >0} is the support of the u, and its closure cl(suppu) is compact.

Let E¹ be the set of all fuzzy numbers on R.

Definition 2.3. A fuzzy number $\tilde{m}$ is said to be a LR fuzzy number if $\begin{matrix} μ_{\tilde{m}} (x) = {\begin{matrix} L (\frac{m - x}{α}), & x \leq m, & α > 0, \\ R (\frac{x - m}{β}), & x \geq m, & β > 0, \end{matrix} \end{matrix}$ where m, α and β are called the mean value, left and right spreads of $\tilde{M}$ , respectively. The function L (·) , which is called left shape function satisfies:

(1) L (x) = L (- x) ,

(2) L (0) =1 and L (1) =0,

(3) L (x) is non increasing on [0, ∞) .

Similar, the right shape function satisfies:

(1) R (x) = R (- x) ,

(2) R (0) =1 and R (1) =0,

(3) R (x) is non decreasing on (- ∞ , 0].

Clearly, two LR fuzzy numbers $\tilde{m} = (m, α, β)_{LR}$ and $\tilde{n} = (n, γ, δ)_{LR}$ are said to be equal, if and only if m = n, α = γ and β = δ .

Definition 2.4. For arbitrary LR fuzzy numbers $\tilde{m} = (m, α, β)_{LR}$ and $\tilde{n} = (n, γ, δ)_{LR}$ , we have

(1) Addition $\begin{matrix} \tilde{m} + \tilde{n} = (m, α, β)_{LR} + (n, γ, δ)_{LR} \\ = (m + n, α + γ, β + δ)_{LR} . \end{matrix}$

(2) Subtraction $\begin{matrix} \tilde{m} - \tilde{n} = (m, α, β)_{LR} - (n, γ, δ)_{LR} \\ = (m - n, α - δ, β - γ)_{LR} . \end{matrix}$

(3) Scalar multiplication $\begin{matrix} λ \tilde{m} = λ (m, α, β)_{LR} ≅ {\begin{matrix} (λ m, λ α, λ β)_{LR}, λ \geq 0, \\ (λ m, - λ β, - λ α)_{RL}, λ < 0 . \end{matrix} \end{matrix}$

(4) multiplication

If $\tilde{m} > 0$ and $\tilde{n} > 0$ , then $\begin{matrix} \tilde{m} \times \tilde{n} ≅ (mn, m γ + n α, m δ + n β) . \end{matrix}$

If $\tilde{m} < 0$ and $\tilde{n} > 0$ , then $\begin{matrix} \tilde{m} \times \tilde{n} ≅ (mn, n α - m δ, n β - m γ) . \end{matrix}$

If $\tilde{m} < 0$ and $\tilde{n} < 0$ , then $\begin{matrix} \tilde{m} \times \tilde{n} ≅ (mn, - n β - m δ, - n α - m γ) . \end{matrix}$

2.2 Generalized fuzzy eigenvector of real matrix

Definition 2.5. If the mean value of a LR fuzzy number is 0 and the left and right spread values are α and β where 0 ≤ α, β < 1, this fuzzy number is called LR zero fuzzy number and denoted by $\tilde{0} = (0, α, β)$ .

A fuzzy vector $\tilde{x} = ({\tilde{x}}_{i}), i = 1, \dots, n$ is called a LR zero fuzzy vector, if each element ${\tilde{x}}_{i}$ of $\tilde{x}$ is a LR zero fuzzy number.

Definition 2.6. Let A, B to be a n × n real matrix. If the real number λ and the non zero fuzzy vector $\tilde{x}$ satisfies the following linear system

$A \tilde{x} = λ B \tilde{x},$ (1) i.e.,

$\begin{matrix} (\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{12} & \dots & a_{2 n} \\ hdotsfor 4 \\ a_{n 1} & a_{n 2} & \dots & a_{nn} \end{matrix}) (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \\ ⋮ \\ {\tilde{x}}_{n} \end{matrix}) \\ = λ (\begin{matrix} b_{11} & b_{12} & \dots & b_{1 n} \\ b_{21} & b_{12} & \dots & b_{2 n} \\ hdotsfor 4 \\ b_{n 1} & b_{n 2} & \dots & b_{nn} \end{matrix}) (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \\ ⋮ \\ {\tilde{x}}_{n} \end{matrix}), \end{matrix}$ (2) we call λ is a real eigenvalue of real matrix A with respect to matrix B, and $\tilde{x}$ is a fuzzy eigenvector of real matrix A with respect to matrix B corresponding to eigenvalue λ.

3 Finding the fuzzy eigenvectors

In this paper, we are devoted to the problem that how to compute the generalized fuzzy eigenvector of a real matrix. For simplification, we suppose matrices A and B are all crisp real symmetric matrices. Two cases that eigenvalue λ is a non negative and negative will be considered in this section.

3.1 Two extended models

Based on multiplication operations of LR fuzzy numbers proposed by Dubois et al. [8], we can obtain the following results.

(a) When λ is a non negative generalized eigenvalue of matrix A

Theorem 3.1. Let A and B to be crisp real symmetric matrices. When λ ≥ 0, the fuzzy generalized eigen problem Equation (1) can be extended into a crisp system of linear matrix equations as follows:

${\begin{matrix} Ax & = λ Bx, \\ (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) & = λ (\begin{matrix} B^{+} & - B^{-} \\ - B^{-} & B^{+} \end{matrix}) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) . \end{matrix}$ (3) where

$\tilde{x} = (x, x^{l}, x^{r}), A = A^{+} + A^{-}, B = B^{+} + B^{-} .$ (4) And the elements $a_{ij}^{+}$ of matrix A⁺ and $a_{ij}^{-}$ of matrix A^- are determined by this way: if $a_{ij} \geq 0, a_{ij}^{+} = a_{ij}$ else $a_{ij}^{+} = 0, 1 \leq i, j \leq n$ ; if $a_{ij} < 0, a_{ij}^{-} = a_{ij}$ else $a_{ij}^{-} = 0, 1 \leq i, j \leq n$ . The elements $b_{ij}^{+}$ of matrix B⁺ and $b_{ij}^{-}$ of matrix B^- are determined by the same as matrices A⁺ and A^-.

Proof. Let $A = A^{+} + A^{-}, B = B^{+} + B^{-}, \tilde{x} = (x, x^{l}, x^{r})$ . The elements $a_{ij}^{+}$ of matrix A⁺ and $a_{ij}^{-}$ of matrix A^- are determined by this way: if $a_{ij} \geq 0, a_{ij}^{+} = a_{ij}$ else $a_{ij}^{+} = 0, 1 \leq i, j \leq n$ ; if $a_{ij} < 0, a_{ij}^{-} = a_{ij}$ else $a_{ij}^{-} = 0, 1 \leq i, j \leq n$ . The elements $b_{ij}^{+}$ of matrix B⁺ and $b_{ij}^{-}$ of matrix B^- are determined by the same as matrices A⁺ and A^-.

Firstly

$\begin{matrix} A \tilde{x} & = (A^{+} + A^{-}) (x, x^{l}, x^{r}) \\ = (A^{+} x, A^{+} x^{l}, A^{+} x^{r}) + (A^{-} x, - A^{-} x^{r}, - A^{-} x^{l}) \\ = (A^{+} x + A^{-} x, A^{+} x^{l} - A^{-} x^{r}, A^{+} x^{r} - A^{-} x^{l}), \end{matrix}$ (5) On the other hand,

$\begin{matrix} B \tilde{x} & = (B^{+} + B^{-}) (x, x^{l}, x^{r}) \\ = (B^{+} x, B^{+} x^{l}, B^{+} x^{r}) + (B^{-} x, - B^{-} x^{r}, - B^{-} x^{l}) \\ = (B^{+} x + B^{-} x, B^{+} x^{l} - B^{-} x^{r}, B^{+} x^{r} - B^{-} x^{l}), \\ λ \tilde{x} & = (λ (B^{+} x + B^{-} x), λ (B^{+} x^{l} - B^{-} x^{r}), λ (B^{+} x^{r} - B^{-} x^{l})), \\ λ \geq 0 . \end{matrix}$ (6)

From $A \tilde{x} = λ B \tilde{x}$ , we have

${\begin{matrix} A^{+} x + A^{-} x = λ (B^{+} x + B^{-} x), \\ A^{+} x^{l} - A^{-} x^{r} = λ (B^{+} x^{l} - B^{-} x^{r}), \\ A^{+} x^{r} - A^{-} x^{l} = λ (B^{+} x^{r} - B^{-} x^{l}) . \end{matrix}$ (7)

By the matrix multiplication, the Equations (7) can be written as $\begin{matrix} {\begin{matrix} Ax & = λ Bx, \\ (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) & = λ (\begin{matrix} B^{+} & - B^{-} \\ - B^{-} & B^{+} \end{matrix}) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) . \end{matrix} \end{matrix}$ where $\begin{matrix} \tilde{x} = (x, x^{l}, x^{r}), A = A^{+} + A^{-}, B = B^{+} + B^{-} . \end{matrix}$

It is completed the proof.

(b) When λ is a negative generalized eigenvalue of matrix A

Corollary 3.1. Let A and B to be crisp real symmetric matrices. When λ < 0, the fuzzy generalized eigen problem Equation (1) can be extended into a crisp system of linear matrix equations as follows: $\begin{matrix} {\begin{matrix} Ax & = λ Bx, \\ (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) & = (- λ) (\begin{matrix} - B^{-} & B^{+} \\ B^{+} & - B^{-} \end{matrix}) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) . \end{matrix} \end{matrix}$ where

$\tilde{x} = (x, x^{l}, x^{r}), A = A^{+} + A^{-}, B = B^{+} + B^{-} .$ (8) And the elements $a_{ij}^{+}$ of matrix A⁺ and $a_{ij}^{-}$ of matrix A^- are determined by this way: if $a_{ij} \geq 0, a_{ij}^{+} = a_{ij}$ else $a_{ij}^{+} = 0, 1 \leq i, j \leq n$ ; if $a_{ij} < 0, a_{ij}^{-} = a_{ij}$ else $a_{ij}^{-} = 0, 1 \leq i, j \leq n$ . The elements $b_{ij}^{+}$ of matrix B⁺ and $b_{ij}^{-}$ of matrix B^- are determined by the same as matrices A⁺ and A^-.

Proof. The proof of Corollary 3.1. is similar to that of Theorem 3.1.

3.2 Solving the extended model

Now we consider the computation of the models Equations (3) and Equations (8).

At first, we compute all generalized eigenvalues and eigenvectors of real symmetric matrix A with respect to matrix B.

From Ax = λBx, i.e., we solve the roots to the equation respect to λ

$f (λ) = \det (λ B - A) = 0$ (9) and non zero solutions to the homogeneous systems of linear equations

$(λ B - A) x = o,$ (10) we can get all generalized eigenvalues and eigenvectors of real matrix A with respect to matrix B. Since matrices A and B are all crisp real symmetric matrices, we know generalized eigenvalues of matrix A with respect to matrix B are all real numbers and can be sorted by size. Let’s assume that all generalized eigenvalues of real matrix A are

$λ_{1} \geq λ_{2} \geq \dots \geq λ_{j - 1} \geq 0 \geq λ_{j + 1} \geq \dots \geq λ_{n},$ (11) and all eigenvectors of real matrix A are

$x_{1}, x_{2}, \dots, x_{n},$ (12) where x_i is a generalized eigenvector of real matrix A corresponding to eigenvalue λ_i.

Secondly, we solve the generalized eigenvectors of real matrix S with respect to matrix T.

For every λ_i, i = 1, 2, ⋯ , j, we solve non zero solutions to the homogeneous systems of linear equations

$(λ_{i} T - S) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) = 0,$ (13) where

$S = (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}), T = (\begin{matrix} B^{+} & - B^{-} \\ - B^{-} & B^{+} \end{matrix}) .$ (14)

For every λ_i, i = j + 1, ⋯ , n, we solve non zero solutions to the homogeneous systems of linear equations

$(λ_{i} I + S) (\begin{matrix} x^{r} \\ x^{l} \end{matrix}) = 0,$ (15) where

$S = (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}), T = (\begin{matrix} - B^{-} & B^{+} \\ B^{+} & - B^{-} \end{matrix}) .$ (16)

Finally, we get the solution of the models Equations (3) and Equations (8) is as follows:

$\begin{matrix} eigenvalues & : λ_{1}, λ_{2}, \dots, λ_{n}, \\ eigenvectors & : {\hat{x}}_{1}, {\hat{x}}_{2}, \dots, {\hat{x}}_{n}, \end{matrix}$ (17) where ${\hat{x}}_{i} = (x_{i}, x_{i}^{l}, x_{i}^{r}), i = 1, \dots, n$ .

Remark 3.1. When matrix B is invertible one, the generalized fuzzy eigenvectors of real matrix A with respect to matrix B is reduced to the ordinary fuzzy eigenvectors of real symmetric matrix $\bar{A}$ , i.e.,

$\bar{A} \tilde{x} = λ \tilde{x}, \bar{A} = B^{- 1} A .$ (18) In this case, all fuzzy eigenvectors of real symmetric matrix $\bar{A}$ are just generalized fuzzy eigenvectors of real matrix A with respect to matrix B.

Remark 3.2. When matrices A and B are all crisp real symmetric positive definite, the generalized eigenvalues of real matrix A with respect to matrix B are all positive $\begin{matrix} λ_{1} \geq λ_{2} \geq \dots \geq λ_{n} > 0, \end{matrix}$ and the generalized fuzzy eigenvectors $\begin{matrix} {\hat{x}}_{1}, {\hat{x}}_{2}, \dots, {\hat{x}}_{n} \end{matrix}$ of real matrix A with respect to matrix B are determined by the model Equations (3).

It is well known that the inverse matrix of a symmetric and negative definite matrix is also symmetric and negative definite. And the product of two symmetric negative definite matrices is a symmetric positive definite matrix. Thus we assert: when matrices A and B are all crisp real symmetric negative definite, the generalized eigenvalues and the generalized fuzzy eigenvectors of real matrix A with respect to matrix B are similar to the above result.

3.3 Fuzzy eigenvector

However, the solution vector $\hat{x}$ may still not be an appropriate LR fuzzy numbers vector except for x^l ≥ 0, x^r ≥ 0. Now we give the definition of LR fuzzy eigenvector to the problem Equation (1) as follows:

Definition 3.2. Let $\hat{x} = (x, x^{l}, x^{r})$ . If (x, x^l, x^r) is the solution that satisfied the Equations (3) or Equations (8), such that x^l × x^r ≥ 0, we call $\tilde{x} = \hat{x} = (x, | x^{l} |, | x^{r} |)$ is a strong generalized LR fuzzy eigenvector of fuzzy eigen problem Equation (1). Otherwise, the $\tilde{x} = (x, x^{l}, x^{r})$ is said to a weak generalized LR fuzzy eigenvector of fuzzy eigen problem Equation (1). In other words, the generalized LR fuzzy eigenvector of fuzzy eigen problem Equation (1) is given by the following formulas $\begin{matrix} \tilde{x} = \tilde{x_{i}}, \end{matrix}$ where

$\begin{matrix} {\tilde{x}}_{i} & = {\begin{matrix} (x_{i}, x_{i}^{l}, x_{i}^{r}), & x_{i}^{l} > 0, & x_{i}^{r} > 0, \\ (x_{i}, 0, max {- x_{i}^{l}, x_{i}^{r}}), & x_{i}^{l} < 0, & x_{i}^{r} > 0, \\ (x_{i}, max {x_{i}^{l}, - x_{i}^{r}}, 0,), & x_{i}^{l} > 0, & x_{i}^{r} < 0, \\ (- x_{i}, - x_{i}^{r}, - x_{i}^{l}), & x_{i}^{l} < 0, & x_{i}^{r} < 0 . \end{matrix} \\ i & = 1, \dots, n . \end{matrix}$ (19)

Remark 3.3. We know that if α ∈ R²ⁿ is a generalized eigenvectors of real symmetric matrix S ∈ R^2n×23n, the kα ∈ R²ⁿ, k ∈ R, k ≠ 0 is also the generalized eigenvectors of matrix S ∈ R^2n×2n. So the key point to make the solution vector be a strong generalized LR fuzzy eigenvector of fuzzy eigen problem Equation (1) is that x^lx^r ≥ 0.

Here we give a Algorithm for finding the generalized eigenvalues and fuzzy eigenvectors of real symmetric matrix A with respect to matrix B.

Algorithm 3.1. The steps for finding the fuzzy eigenvector of a real symmetric matrix is as follows:

Step 1. Decomposing the matrix A with A = A⁺ + A^- and matrix B with B = B⁺ + B^-.

Step 2. Computing all generalized eigenvalues and eigenvectors of real symmetric matrix A with respect to matrix B by the equation Ax = λBx, i.e., $\begin{matrix} \begin{matrix} eigenvalues : λ_{1}, λ_{2}, \dots, λ_{n}, \\ eigenvectors : x_{1}, x_{2}, \dots, x_{n} . \end{matrix} \end{matrix}$

Step 3. Solving the left and right spread values of fuzzy eigenvectors of real symmetric matrix A with respect to matrix B.

When λ_i ≥ 0, the fuzzy eigen problem Equation (1) can be computed by $\begin{matrix} (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) = λ_{i} (\begin{matrix} B^{+} & - B^{-} \\ - B^{-} & B^{+} \end{matrix}) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) . \end{matrix}$

When λ < 0, the fuzzy eigen problem Equation (1) can be computed by $\begin{matrix} (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) = - λ_{i} (\begin{matrix} - B^{-} & B^{+} \\ B^{+} & - B^{-} \end{matrix}) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) . \end{matrix}$

Step 4. Arranging generalized fuzzy eigenvectors of real symmetric matrix A with respect to matrix B, that is $\begin{matrix} \begin{matrix} eigenvalues : λ_{1}, λ_{2}, \dots, λ_{n}, \\ eigenvectors : {\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n}, \end{matrix} \end{matrix}$ where ${\tilde{x}}_{i} = (x_{i}, x_{i}^{l}, x_{i}^{r}), i = 1, \dots, n$ .

Step 5. Judging and giving strong generalized LR fuzzy eigenvector as $\begin{matrix} \tilde{x} = (x_{i}, x_{i}^{l}, x_{i}^{r}) \end{matrix}$ or weak generalized LR fuzzy eigenvector $\begin{matrix} {\tilde{x}}_{i} = {\begin{matrix} (x_{i}, x_{i}^{l}, x_{i}^{r}), x_{i}^{r} > 0, x_{i}^{l} > 0, \\ (x_{i}, 0, \max (- x_{i}^{r}, x_{i}^{l})), x_{i}^{r} < 0, x_{i}^{l} > 0, \\ (x_{i}, \max (x_{i}^{r}, - x_{i}^{l}), 0), x_{i}^{r} > 0, x_{i}^{l} < 0, \\ (x_{i}, - x_{i}^{r}, - x_{i}^{l}), x_{i}^{r} < 0, x_{i}^{l} < 0, \end{matrix} \end{matrix}$ $\begin{matrix} i = 1, 2, \dots, n, \end{matrix}$ by the Definition 3.2.

Definition 3.3. Matrix P ∈ R^2n×2n is called monotonous if the matrix P is invertible and its P^-1 is non negative.

Theorem 3.2.[16] Let Q ∈ R^2n×2n to be a non negative matrix and λ is a eigen value of matrix Q. Then there exists a non negative vector z ≥ o, z ≠ o that subject to Oz = λz.

Finally, we give a sufficient condition for the existence of a strong generalized fuzzy eigenvector.

Theorem 3.3. Suppose $T_{1} = (\begin{matrix} B^{+} & - B^{-} \\ - B^{-} & B^{+} \end{matrix}), T_{2} = (\begin{matrix} - B^{-} & B^{+} \\ B^{+} & - B^{-} \end{matrix})$ and $\begin{matrix} x^{lr} = (\begin{matrix} x^{r} \\ x^{l} \end{matrix}) . \end{matrix}$ If matrices T_i, i = 1, 2 are monotonous, the fuzzy eigen problem Equation (1) has a strong generalized LR fuzzy eigenvector.

Proof. Form the Equations (14) and (16), we know that the matrices S, T₁, T₂ are all non negative.

When the generalized eigen value λ_i ≥ 0 of the real symmetric matrix A with respect to matrix B, we are needed to consider the crisp generalized eigen problem of a real matrix

${Sx}^{lr} = λ_{i} T_{1} x^{lr} .$ (20) Since the matrix T₁ is monotonous, the above generalized eigen problem is equivalent to the following eigen problem

${ST}_{1}^{- 1} x^{lr} = λ_{i} x^{lr},$ (21) where ${ST}_{1}^{- 1}$ is a non negative matrix.

When the generalized eigen value λ_i < 0 of the real symmetric matrix A with respect to matrix B, we are needed to consider the crisp generalized eigen problem of a real matrix

${Sx}^{lr} = (- λ_{i}) T_{2} x^{lr} .$ (22) Since the matrix T₂ is monotonous, the above generalized eigen problem is equivalent to the following eigen problem

${ST}_{2}^{- 1} x^{lr} = (- λ_{i}) x^{lr},$ (23) where ${ST}_{2}^{- 1}$ is a non negative matrix.

According to the Theorem 3.2., non negative matrix ${ST}_{i}^{- 1}, i = 1, 2$ always exists a non negative vector x^lr ≥ o, x^lr ≠ o. It means x^l ≥ o, x^r ≥ o.

Thus the generalized eigen problem (1) always exists a strong LR fuzzy eigen vector $\tilde{x} = (x, x^{l}, x^{r})$ .

It is completed the proof.

Remark 3.4. This paper proposes a approach to find the generalized fuzzy eigenvectors of a real symmetric matrix by solving fuzzy linear systems. Although every eigenvalue of a real symmetric matrix is a real number, its generalized fuzzy eigenvector may not exist sometimes for each eigenvalue, because the solution of the corresponding fuzzy linear system may not exist sometimes [3].

4 Numerical examples

Example 4.1. Consider the following generalized eigen problem $\begin{matrix} A \tilde{x} = λ B \tilde{x}, \end{matrix}$ where $\begin{matrix} A = (\begin{matrix} 5 & 1 \\ 1 & 1 \end{matrix}), B = (\begin{matrix} 18 & 2 \\ 2 & 2 \end{matrix}) . \end{matrix}$

In this generalized eigen problem, the matrices A and B are all symmetric positive definite. Let $\begin{matrix} \begin{matrix} A & = A^{+} + A^{-} = (\begin{matrix} 5 & 1 \\ 1 & 1 \end{matrix}) + (\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}) . \\ B & = B^{+} + B^{-} = (\begin{matrix} 18 & 2 \\ 2 & 2 \end{matrix}) + (\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}) . \end{matrix} \end{matrix}$

From Ax = λBx, i.e., by solving the roots to the equation respect to λ $\begin{matrix} f (λ) = \det (λ B - A) = 0 \end{matrix}$ and non zero solutions to the homogeneous systems of linear equations $\begin{matrix} (λ B - A) x = O, \end{matrix}$ we can get $\begin{matrix} λ_{1} = 0.25, λ_{2} = 0.5, \end{matrix}$ $\begin{matrix} x_{1} = (\begin{matrix} 0.0000 \\ 1.0000 \end{matrix}), x_{2} = (\begin{matrix} 0.7071 \\ - 0.7071 \end{matrix}) \end{matrix}$ as all generalized eigenvalues and eigenvectors of A with respect to B.

For λ₁ = 0.25 > 0, we solve non zero solutions to the homogeneous systems of linear equation $\begin{matrix} (0.25 T - S) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) = O, \end{matrix}$ where $\begin{matrix} S = (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}), T = (\begin{matrix} B^{+} & - B^{-} \\ - B^{-} & A^{+} \end{matrix}) \end{matrix}$ and obtain $\begin{matrix} (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) = (\begin{matrix} 0.7071 \\ - 0.7071 \\ 0.0000 \\ 0.0000 \end{matrix}) . \end{matrix}$

By the same way, for λ₂ = 0.5 > 0, we solve non zero solutions to the homogeneous systems of linear equation $\begin{matrix} (0.5 T - S) (\begin{matrix} x^{r} \\ x^{l} \end{matrix}) = O \end{matrix}$ and get $\begin{matrix} (\begin{matrix} x^{r} \\ x^{l} \end{matrix}) = (\begin{matrix} 0.0000 \\ 1.0000 \\ 0.0000 \\ 0.0000 \end{matrix}) . \end{matrix}$

Thus we obtain all generalized real eigenvalues and fuzzy eigenvectors of real symmetric matrix A with respect to B are $\begin{matrix} λ_{1} = 0.25, λ_{2} = 0.5, \end{matrix}$ $\begin{matrix} \begin{matrix} {\tilde{x}}_{1} & = (\begin{matrix} (0.0000, 0.7071, 0.0000) \\ (1.0000, - 0.7071, 0.0000) \end{matrix}), \\ {\tilde{x}}_{2} & = (\begin{matrix} (0.7071, 0.0000, 0.0000) \\ (- 0.7071, 1.0000, 0.0000) \end{matrix}) . \end{matrix} \end{matrix}$

From above analysis,we know that the ${\tilde{x}}_{1}$ is a weak generalized LR fuzzy eigenvector of original real matrix A with respect to B corresponding to eigenvalue λ₁, the ${\tilde{x}}_{2}$ is a strong generalized LR fuzzy eigenvector of original real matrix A with respect to B corresponding to eigenvalue λ₂.

Example 4.2. Consider the following generalized eigen problem $\begin{matrix} A \tilde{x} = λ B \tilde{x}, \end{matrix}$ where $\begin{matrix} (\begin{matrix} 1 & - 2 & 2 \\ - 2 & - 2 & 4 \\ 2 & 4 & - 2 \end{matrix}), B = (\begin{matrix} 2 & 2 & - 2 \\ 2 & 5 & - 4 \\ - 2 & - 4 & 5 \end{matrix}) . \end{matrix}$

In this generalized eigen problem, the matrices A is symmetric and B is symmetric positive definite. Let $\begin{matrix} A = A^{+} + A^{-} = (\begin{matrix} 1 & 0 & 2 \\ 0 & 0 & 4 \\ 2 & 4 & 0 \end{matrix}) + (\begin{matrix} 0 & - 2 & 0 \\ - 2 & - 2 & 0 \\ 0 & 0 & - 2 \end{matrix}), \end{matrix}$ $\begin{matrix} B = B^{+} + B^{-} = (\begin{matrix} 2 & 2 & 0 \\ 2 & 5 & 0 \\ 0 & 0 & 5 \end{matrix}) + (\begin{matrix} 0 & 0 & - 2 \\ 0 & 0 & - 4 \\ - 2 & - 4 & 0 \end{matrix}) . \end{matrix}$

From Ax = λBx, i.e., by solving the roots to the equation respect to λ $\begin{matrix} f (λ) = \det (λ B - A) = 0 \end{matrix}$ and non zero solutions to the homogeneous systems of linear equations $\begin{matrix} (λ B - A) x = O, \end{matrix}$ we can get $\begin{matrix} λ_{1, 2} = 2, λ_{3} = 0.7, \end{matrix}$ $\begin{matrix} x_{1} = (\begin{matrix} - 0.9428 \\ 0.2357 \\ - 0.2357 \end{matrix}), x_{2} = (\begin{matrix} 0.1912 \\ 0.6446 \\ 0.7402 \end{matrix}), \end{matrix}$ $\begin{matrix} x_{3} = (\begin{matrix} - 0.3333 \\ - 0.6667 \\ 0.6667 \end{matrix}) . \end{matrix}$ as all generalized eigenvalues and eigenvectors of A with respect to B.

For λ_1,2 = 2 >0, we solve non zero solutions to the homogeneous systems of linear equation $\begin{matrix} (2 T - S) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) = O, \end{matrix}$ where $\begin{matrix} S = (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}), T = (\begin{matrix} B^{+} & - B^{-} \\ - B^{-} & A^{+} \end{matrix}) \end{matrix}$ and obtain $\begin{matrix} (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) = (\begin{matrix} - 0.6667 & - 0.0180 \\ 0.1667 & - 0.4953 \\ - 0.1667 & - 0.5043 \\ 0.6667 & 0.0180 \\ - 0.1667 & 0.4953 \\ 0.1667 & 0.5043 \end{matrix}) . \end{matrix}$

For λ₂ = -0.7 < 0, we solve non zero solutions to the homogeneous systems of linear equation $\begin{matrix} (- 0.7 T - S) (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) = O, \end{matrix}$ where $\begin{matrix} S = (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}), T = (\begin{matrix} - B^{-} & B^{+} \\ B^{+} & - B^{-} \end{matrix}) \end{matrix}$ and get $\begin{matrix} (\begin{matrix} x^{l} \\ x^{r} \end{matrix}) = (\begin{matrix} - 0.2357 \\ - 0.4714 \\ 0.4714 \\ 0.2357 \\ 0.4714 \\ - 0.4714 \end{matrix}) . \end{matrix}$

Thus we obtain all generalized real eigenvalues and fuzzy eigenvectors of real symmetric matrix A with respect to B are $\begin{matrix} λ_{1, 2} = 2, λ_{3} = - 0.7, \end{matrix}$ $\begin{matrix} {\tilde{x}}_{1} = (\begin{matrix} (- 0.9428, - 0.6667, 0.6667) \\ (0.2357, 0.1667, - 0.1667) \\ (- 0.2357, - 0.1667, 0.1667) \end{matrix}), \end{matrix}$ $\begin{matrix} x_{2} = (\begin{matrix} (0.1912, - 0.0180, 0.0180) \\ (0.6446, - 0.4953, 0.4953) \\ (0.7402, - 0.5043, 0.5043) \end{matrix}), \end{matrix}$ $\begin{matrix} x_{3} = (\begin{matrix} (- 0.3333, - 0.2357, 0.2357) \\ (- 0.6667, - 0.4714, 0.4714) \\ (0.6667, 0.4714, - 0.4714) \end{matrix}) . \end{matrix}$

From above analysis, we know that the ${\tilde{x}}_{1}, {\tilde{x}}_{2}$ are weak generalized LR fuzzy eigenvector of original real matrix A with respect to B corresponding to eigenvalue λ_1,2, the ${\tilde{x}}_{3}$ is a weak generalized LR fuzzy eigenvector of original real matrix A with respect to B corresponding to eigenvalue λ₃.

5 Conclusion

In this work we investigated the generalized fuzzy eigenvectors of a crisp real symmetric matrix. According to the unknown real eigenvalue λ is non negative or not, we established two computing models and proposed a algorithm for finding the generalized real eigenvalues and fuzzy eigenvector of a real symmetric matrix with respect to another real symmetric matrix. Some numerical examples are given to illustrated the method we put up. We will investigate the fuzzy eigenvalues and fuzzy eigenvectors of LR fuzzy matrix in the next step. Our results enrich the fuzzy linear systems theory.

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