Solving complex LR fuzzy matrix equation Z ˜ C = W ˜ 1

Abstract

In paper a general complex fuzzy matrix equation $\tilde{Z} C = \tilde{W}$ in which C is a n × p complex matrix and $\tilde{W}$ is a m × p complex LR fuzzy numbers one, is investigated. A numerical procedure for calculating the solution is designed in detail and a sufficient condition for the existence of strong complex LR fuzzy solution is derived. Some examples are given to illustrate the proposed method.

Keywords

Complex fuzzy numbers matrix analysis complex fuzzy matrix systems fuzzy approximate solutions

1 Introduction

There are lots of imprecisions in the real-world engineering systems. Fuzzy mathematics is one of powerful tools for modeling and dealing them. In many linear systems, some of the system parameters are vague or imprecise, and fuzzy number is a better expression than crisp one for handling the problem, and hence solving a linear system involved with fuzzy numbers is becoming more important. The fuzzy numbers and their arithmetic operations were firstly introduced and investigated by Zadeh [32], Dubois and Prade [15] and Nahmias [25]. A different approach to fuzzy numbers and the structure of fuzzy number spaces was given by Puri and Ralescu [27], Goetschell and Voxman [17] and Wu and Ma Ming [30, 31].

Since Friedman et al. [16] proposed a general model for solving an n × n fuzzy linear systems $A \tilde{x} = \tilde{b}$ by an embedding approach in 1998, many a work has been done about some fuzzy linear systems such as dual fuzzy linear systems (DFLS), general fuzzy linear systems (GFLS), dual full fuzzy systems linear systems (DFFLS) and general dual fuzzy linear systems (GDFLS) see [1 –4, 7, 13, 14, 24, 29, 33]. For a system of linear equations involved with complex fuzzy numbers which was introduced firstly by J.J. Buckley [12] in 1989, few researchers have developed methods to investigate it in the past decades. In 2010, M.A. Jahanigh [23] studied a n × n fuzzy complex linear systems. Solution of fuzzy complex linear system of linear equations was described and applied to circuit analysis problem by Rahgooy et al. [28]. In 2014, Behera and Chakraverty [8] discussed the fuzzy complex system of linear equations by the embedding method, and they redefine the complex fuzzy number [9] in 2016. Recently, Guo et al. [21] proposed a computing method for solving LR complex linear systems.

Although fuzzy matrix equation has a wide use in control theory and control engineering, few work has been done in the past. In 2009, Allahviranloo et al. [6] firstly discussed the fuzzy linear matrix equations (FLME) of the form $A \tilde{X} B = \tilde{C}$ . By means of the parametric form of the fuzzy number, they derived a necessary and sufficient conditions for the existence condition of fuzzy solutions and designed a numerical procedure for calculating its fuzzy solutions. In 2011, Gong et al. [18] studied least squares solutions of the inconsistent fuzzy matrix equation $A \tilde{X} = \tilde{B}$ by the same way. Recently, Guo et al. [22] investigated a class of fuzzy matrix equation $\tilde{X} A = \tilde{B}$ by the matrix method.

In general, the uncertain elements of fuzzy linear systems were denoted by the parametric form of fuzzy numbers and the systems were extended into a crisp function linear systems. Thus it may lead to two defects. The one is that the extended linear equations always contains parameter r, 0 ≤ r ≤ 1, which makes their computation inconvenient in some sense. The other is that sometimes the weak fuzzy solution of fuzzy linear systems does not exist [5]. To overcome above two defects and handle the full fuzzy linear systems (FFLS), D. Dubois and H. Prade [15] introduced the LR fuzzy number. In 2006, Dehgham et al. [14] discussed computational methods for fully fuzzy linear systems $\tilde{A} \tilde{x} = \tilde{b}$ whose coefficient matrix and the right-hand side vector are LR fuzzy numbers. In the past ten years, some researchers paid more attention to LR fuzzy linear systems. In 2013, Guo et al. [20] proposed a computing method for the fuzzy Sylvester matrix equations $A \tilde{X} + \tilde{X} B = \tilde{C}$ with LR fuzzy numbers. In 2014, Gong and Guo et al. [19] studied the general LR dual fuzzy linear matrix systems $A \tilde{X} + \tilde{B} = C \tilde{X} + \tilde{D}$ . In this paper we introduce the complex LR fuzzy matrix equation $\tilde{Z} C = \tilde{W}$ and propose a general method to deal with.

2 Preliminaries

There are several definitions for the concept of fuzzy numbers (see [15, 25, 32, 34]).

2.1 The fuzzy number

Definition 2.1. A fuzzy number is a fuzzy set like u : R → I = [0, 1] which satisfies:

u is upper semicontinuous,

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

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

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 2.2. A fuzzy number u in parametric form is a pair $(\underline{u}, \bar{u})$ of functions $\underline{u} (r)$ , $\bar{u} (r)$ , 0 ≤ r ≤ 1, which satisfies the requirements:

$\underline{u} (r)$ is a bounded monotonic increasing left continuous function,

$\bar{u} (r)$ is a bounded monotonic decreasing left continuous function,

$\underline{u} (r) \leq \bar{u} (r)$ , 0 ≤ r ≤ 1.

Definition 2.3. A fuzzy number $\tilde{M}$ is said to be a LR fuzzy number if $μ_{\tilde{M}} (x) = {\begin{matrix} L (\frac{m - x}{α}), & x \leq m, & α > 0, \\ R (\frac{x - m}{β}), & x \geq m, & β > 0, \end{matrix}$ 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:

L (x) = L (- x),

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

L (x) is non increasing on [0, ∞).

The definition of a right shape function R (·) is similar to that of L (·).

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

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

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

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

Definition 2.5. An arbitrary LR complex fuzzy number should be represented as $\tilde{x} = \tilde{p} + i \tilde{q}$ , where $\tilde{p} = (p, p^{l}, p^{r})$ and $\tilde{q} = (q, q^{l}, q^{r})$ . In this case, $\tilde{x}$ can be written as $\tilde{x} = (p, p^{l}, p^{r}) + i (q, q^{l}, q^{r})$ .

Definition 2.6. For any two arbitrary complex fuzzy numbers $\tilde{x} = \tilde{p} + i \tilde{q}$ and $\tilde{y} = \tilde{u} + i \tilde{v}$ where $\tilde{p}, \tilde{q}, \tilde{u}, \tilde{v}$ are fuzzy numbers, their arithmetic is as follows:

$\tilde{x} + \tilde{y} = (\tilde{p} + \tilde{u}) + i (\tilde{q} + \tilde{v})$ ,

$k \tilde{x} = k \tilde{p} + ik \tilde{q}, k \in R,$

$\tilde{x} \times \tilde{y} = (\tilde{p} \times \tilde{u} - \tilde{q} \times \tilde{v}) + i (\tilde{p} \times \tilde{v} + \tilde{q} \times \tilde{u})$ .

2.2 Fuzzy matrix equations

Definition 2.7. The matrix system

$\begin{matrix} (\begin{matrix} {\tilde{z}}_{11} & {\tilde{z}}_{12} & \dots & {\tilde{z}}_{1 n} \\ {\tilde{z}}_{21} & {\tilde{z}}_{22} & \dots & {\tilde{z}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{z}}_{m 1} & {\tilde{z}}_{m 2} & \dots & {\tilde{z}}_{mn} \end{matrix}) (\begin{matrix} c_{11} & c_{12} & \dots & c_{1 p} \\ c_{21} & c_{22} & \dots & c_{2 p} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ c_{n 1} & c_{n 2} & \dots & c_{np} \end{matrix}) \\ = (\begin{matrix} {\tilde{w}}_{11} & {\tilde{w}}_{12} & \dots & {\tilde{w}}_{1 p} \\ {\tilde{w}}_{21} & {\tilde{w}}_{22} & \dots & {\tilde{w}}_{2 p} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{w}}_{m 1} & {\tilde{w}}_{m 2} & \dots & {\tilde{w}}_{mp} \end{matrix}), \end{matrix}$ (1) where c_ij, 1 ≤ i ≤ n, 1 ≤ j ≤ p are crisp complex numbers and ${\tilde{w}}_{ij}, 1 \leq i \leq m, 1 \leq j \leq p$ are complex LR fuzzy numbers, is called a complex LR fuzzy matrix equations (CLRFME).

Using matrix notation, we have $\tilde{Z} C = \tilde{W} .$ (2)

A fuzzy numbers matrix $\begin{matrix} \tilde{Z} = ({\tilde{z}}_{ij}) & = & (p, p_{ij}^{l}, p_{ij}^{r}) + i (q, q_{ij}^{l}, q_{ij}^{r}), \\ 1 \leq i \leq m, 1 \leq j \leq n \end{matrix}$ is called a solution of the complex fuzzy matrix equation (2.1) if $\tilde{Z}$ satisfies $\tilde{Z} C = \tilde{W} .$

3 Solving complex fuzzy matrix equation

Definition 3.1. A matrix $\tilde{A} = ({\tilde{a}}_{ij})$ is called a LR fuzzy matrix, if each element ${\tilde{a}}_{ij}$ of $\tilde{A}$ is a LR fuzzy number. For example, we represent m × n LR fuzzy matrix $\tilde{A} = ({\tilde{a}}_{ij})$ , that ${\tilde{a}}_{ij} = (a_{ij}, α_{ij}, β_{ij})_{LR}$ with new notation $\tilde{A} = (A, M, N)$ , where A = (a_ij), M = (α_ij) and N = (β_ij) are three m × n crisp matrices.

Definition 3.2. An arbitrary complex LR fuzzy matrix should be represented as $\tilde{Z} = \tilde{P} + i \tilde{Q}$ , where $\tilde{P} = (P, P^{l}, P^{r})$ and $\tilde{Q} = (Q, Q^{l}, Q^{r})$ are two LR fuzzy numbers matrices. In this case, $\tilde{Z}$ can be written as $\tilde{Z} = (P, P^{l}, P^{r}) + i (Q, Q^{l}, Q^{r}) .$

3.1 Equivalent fuzzy matrix system

Theorem 3.1. The n × p complex fuzzy matrix system (2.2) is equivalent to a 2n × 2p order fuzzy matrix equation $\tilde{X} A = \tilde{B},$ (3)where $A = (\begin{matrix} G & H \\ - H & G \end{matrix}), \tilde{X} = (\begin{matrix} \tilde{P} & \tilde{Q} \end{matrix}), \tilde{B} = (\begin{matrix} \tilde{U} & \tilde{V} \end{matrix}) .$ (4)

Proof. We denote C = G + iH, G, H ∈ R^n×p and $\tilde{W} = \tilde{U} + i \tilde{V}$ where $\tilde{U}$ and $\tilde{V}$ are fuzzy number matrices. We also suppose the unknown matrix $\tilde{Z} = \tilde{P} + i \tilde{Q}$ where $\tilde{P}$ and $\tilde{Q}$ are two unknown fuzzy number matrices.

Since $\tilde{Z} C = \tilde{W}$ , we have $(\tilde{P} + i \tilde{Q}) (G + iH) = \tilde{U} + i \tilde{V}$ i.e., $(\tilde{P} G - \tilde{Q} H) + i (\tilde{Q} G + \tilde{P} H) = \tilde{U} + i \tilde{V} .$

Comparing with the coefficient of i, we have ${\begin{matrix} \tilde{P} G - \tilde{Q} H = \tilde{U}, \\ \tilde{Q} G + \tilde{P} H = \tilde{V} . \end{matrix}$ i.e. $(\begin{matrix} \tilde{P} & \tilde{Q} \end{matrix}) (\begin{matrix} G & H \\ - H & G \end{matrix}) = (\begin{matrix} \tilde{U} & \tilde{V} \end{matrix}),$ (5) it admits a high order fuzzy matrix system.

Expressing (3.3) in matrix form, we have $\tilde{X} A = \tilde{B} .$

3.2 Solving fuzzy matrix system

In order to solve the complex fuzzy linear system (2.2), we need to solve the fuzzy matrix Equations (3.1).

Definition 3.3. For two LR fuzzy numbers matrices $\tilde{A} = (A, A^{l}, A^{r})$ and $\tilde{B} = (B, B^{l}, B^{r})$ , we have

Addition $\begin{matrix} \tilde{A} + \tilde{B} & = & (A, A^{l}, A^{r}) + (B, B^{l}, B^{r}) \\ = & (A + B, A^{l} + B^{l}, A^{r} + B^{r}) . \end{matrix}$

Subtraction $\begin{matrix} \tilde{A} - \tilde{B} & = & (A, A^{l}, A^{r}) - (B, B^{l}, B^{r}) \\ = & (A + B, A^{l} - B^{r}, A^{r} - B^{l}) . \end{matrix}$

Scalar multiplication

$\begin{matrix} λ \tilde{A} & = & λ (A, A^{l}, A^{r}) \\ ≅ & {\begin{matrix} (λ A, λ A^{l}, λ A^{r}), λ \geq 0, \\ (λ A, - λ A^{r}, - λ A^{l}), λ < 0 . \end{matrix} \end{matrix}$ (6)

Theorem 3.2. The fuzzy linear system $\tilde{X} A = \tilde{B}$ can be extended into the following system of linear matrix equations ${\begin{matrix} X (A^{+} + A^{-}) = B, \\ (\begin{matrix} X^{l}, X^{r} \end{matrix}) (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) = (\begin{matrix} B^{l}, B^{r} \end{matrix}) \end{matrix}$ (7) where $\tilde{X} = (X, X^{l}, X^{r}) .$ And the elements $a_{ij}^{+}$ of matrix A⁺ and $a_{ij}^{-}$ of matrix A^- are determined by the following way:

if $a_{ij} \geq 0, a_{ij}^{+} = a_{ij}$ else $a_{ij}^{+} = 0, 1 \leq i \leq n, 1 \leq j \leq p$ ; if $a_{ij} < 0, a_{ij}^{-} = a_{ij}$ else $a_{ij}^{-} = 0, 1 \leq i \leq n, 1 \leq j \leq p$ .

Proof. We denote the right fuzzy matrix $\tilde{B}$ with $\tilde{B} = (B, B^{l}, B^{r}) = (b_{ij}, b_{ij}^{l}, b_{ij}^{r})_{m \times p}$ and the unknown fuzzy matrix $\tilde{X}$ by $\tilde{X} = (X, X^{l}, X^{r}) = (x_{ij}, x_{ij}^{l}, x_{ij}^{r})_{m \times n}$ . We also suppose A = A⁺ + A^- in which the elements $a_{ij}^{+}$ of matrix A⁺ and $a_{ij}^{-}$ of matrix A^- are determined by the following way:

if $a_{ij} \geq 0, a_{ij}^{+} = a_{ij}$ else $a_{ij}^{+} = 0, 1 \leq i \leq n, 1 \leq j \leq p$ ; if $a_{ij} < 0, a_{ij}^{-} = a_{ij}$ else $a_{ij}^{-} = 0, 1 \leq i \leq n, 1 \leq j \leq p$ .

For fuzzy matrix equation $\tilde{X} A = \tilde{B}$ , we can express it as

$(X, X^{l}, X^{r}) (A^{+} + A^{-}) = (B, B^{l}, B^{r}) .$ (8)

Since

${\tilde{x}}_{ij} k = {\begin{matrix} ({kx}_{ij}, {kx}_{ij}^{l}, {kx}_{ij}^{r}), & k \geq 0, \\ ({kx}_{ij}, - {kx}_{ij}^{r}, - {kx}_{ij}^{l}), & k < 0 \end{matrix},$ we have

$\tilde{X} A = {\begin{matrix} (XA, X^{l} A, X^{r} A), & A \geq 0, \\ (XA, - X^{r}, - X^{l} A), & A < 0 \end{matrix}$

So the Equations (3.2) be rewritten is

$\begin{matrix} (X, X^{l}, X^{r}) A^{+} + (X, X^{l}, X^{r}) A^{-} \\ = ({XA}^{+}, X^{l} A^{+}, X^{r} A^{+}) \\ + ({XA}^{-}, - X^{r} A^{-}, - X^{l} A^{-}) \end{matrix}$

$\begin{matrix} ({XA}^{+} + {XA}^{-}, X^{l} A^{+} - X^{r} A^{-}, X^{r} A^{+} - X^{l} A^{-}) \\ = (B, B^{l}, B^{r}) . \end{matrix}$

Thus we get ${\begin{matrix} {XA}^{+} + {XA}^{-} = B, \\ X^{l} A^{+} - X^{r} A^{-} = B^{l}, \\ X^{r} A^{+} - X^{l} A^{-} = B^{r} . \end{matrix}$ (9) or ${\begin{matrix} X (A^{+} + A^{-}) = B, \\ (\begin{matrix} X^{l}, X^{r} \end{matrix}) (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) = (\begin{matrix} B^{l}, B^{r} \end{matrix}) . \end{matrix}$

Theorem 3.3. The matrix S is nonsingular if and only if the matrices A⁺ + A^- and A⁺ - A^- are both nonsingular. If S is nonsingular, then $S^{- 1} = \frac{1}{2} (\begin{matrix} (A^{+} - A^{-})^{- 1} + (A^{+} + A^{-})^{- 1} & (A^{+} - A^{-})^{- 1} - (A^{+} + A^{-})^{- 1} \\ (A^{+} - A^{-})^{- 1} - (A^{+} + A^{-})^{- 1} & (A^{+} - A^{-})^{- 1} + (A^{+} + A^{-})^{- 1} \end{matrix})$ where (A⁺ + A^-) ^-1, (A⁺ - A^-) ^-1 are inverse matrices of matrices A⁺ + A^- and A⁺ - A^-, respectively. If matrices A and S are nonsingular simultaneously, the solution of model Equation (3.4) is ${\begin{matrix} X = {BA}^{- 1}, \\ (\begin{matrix} X^{l}, X^{r} \end{matrix}) = (\begin{matrix} B^{l}, B^{r} \end{matrix}) {(\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix})}^{- 1} . \end{matrix}$ (10)

Theorem 3.4. Let S belong to R^n×p and C belong to R^m×p. Then the minimal solution X^* of the matrix equation XS = C is expressed by $X^{*} = {CS}^{†} .$ (11)

Proof. The proof is straight forward.

To solve complex fuzzy matrix Equations (2.2), we need to consider the fuzzy matrix systems (3.1). In order to solve the Equations (3.1), we need to consider the matrix system (3.4). By Theorem 3.4., we obtain the minimal solution of the function system (3.4) as ${\begin{matrix} X = {BA}^{†}, \\ (\begin{matrix} X^{l}, X^{r} \end{matrix}) = (\begin{matrix} B^{l}, B^{r} \end{matrix}) {(\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix})}^{†}, \end{matrix}$ (12) where S^† is the Moore-Penrose generalized inverse of matrix S.

From the Equations (3.1), we know

$\begin{matrix} X = (PQ), X^{l} = (P^{l} Q^{l}), X^{r} = (P^{r} Q^{r}), \\ B = (UV), B^{l} = (U^{l} V^{l}), B^{r} = (U^{r} V^{r}) . \end{matrix}$

Thus we get the minimal solution of fuzzy matrix Equation (3.1) is

${\begin{matrix} (PQ) = (UV) A^{†}, \\ (\begin{matrix} (P^{l} Q^{l}), (P^{r} Q^{r}) \end{matrix}) \\ = (\begin{matrix} (U^{l} V^{l}), (U^{r} V^{r}) \end{matrix}) {(\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix})}^{†}, \end{matrix}$ (13)

Since complex fuzzy matrix Equation (2.2) is equivalent to the fuzzy matrix Equation (3.1), we obtain the solution of the original fuzzy system (2.2) is as follows:

$\tilde{Z} = (P, P^{l}, P^{r}) + i (Q, Q^{l}, Q^{r}) .$ (14)

It seems that we have obtained the complex fuzzy solution as the above expression (3.12). However, the solution matrices $\tilde{P} = (P, P^{l}, P^{r})$ and $\tilde{Q} = (Q, Q^{l}, Q^{r})$ may still not be appropriate LR fuzzy numbers ones except for

$X^{l} = (P^{l} Q^{l}) \geq O, X^{r} = (P^{r} Q^{r}) \geq O .$

So we give the definition of complex LR fuzzy solution to the Equation (2.2) as follows:

Definition 3.4. Let

$\tilde{X} = (X, X^{l}, X^{r}) .$

If (X, X^l, X^r) is the minimal solution of Equations (3.5) such that X^l ≥ O, X^r ≥ O, we call $\tilde{X} = (X, X^{l}, X^{r})$ is the strong LR fuzzy minimal solution of fuzzy matrix Equation (3.1), i.e., we call (3.12) is the strong complex LR fuzzy minimal solution of complex fuzzy matrix Equation (2.2). Otherwise, the (3.12) is said to a weak complex LR fuzzy minimal solution of the Equations (2.2) given by

$\tilde{X} = ({\tilde{x}}_{ij})_{m \times n},$ where ${\tilde{x}}_{ij} = {\begin{matrix} (x_{ij}, x_{ij}^{l}, x_{ij}^{r}), & x_{ij}^{l} > 0, & x_{ij}^{r} > 0, \\ (x_{ij}, 0, max {- x_{ij}^{l}, x_{ij}^{r}}), & x_{ij}^{l} < 0, & x_{ij}^{r} > 0, \\ (x_{ij}, max {x_{ij}^{l}, - x_{ij}^{r}}, 0,), & x_{ij}^{l} > 0, & x_{ij}^{r} < 0, \\ (x_{ij}, - x_{ij}^{l}, - x_{ij}^{r}), & x_{ij}^{l} < 0, & x_{ij}^{r} < 0, \\ i = 1, \dots, m, j = 1, \dots, n . \end{matrix}$ (15) ∥

3.3 A sufficient condition for complex fuzzy solution

To illustrate the expression (3.10) to be a fuzzy solution matrix, we now discuss the generalized inverses of non negative matrix $S = (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix})$ in a special structure.

Lemma 3.1. ([11]) Let $S = (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) .$

Then the matrix

$S^{†} = \frac{1}{2} (\begin{matrix} (A^{+} - A^{-})^{†} + (A^{+} + A^{-})^{†} & (A^{+} - A^{-})^{†} - (A^{+} + A^{-})^{†} \\ (A^{+} - A^{-})^{†} - (A^{+} + A^{-})^{†} & (A^{+} - A^{-})^{†} + (A^{+} + A^{-})^{†} \end{matrix})$ (16)is the Moore-Penrose inverse of the matrix S, where (A⁺ + A^-) ^†, (A⁺ - A^-) ^† are Moore-Penrose inverses of matrices A⁺ + A^- and A⁺ - A^-, respectively.

The key point to make the solution matrix be a strong LR fuzzy solution is that $\tilde{X} = (X, X^{l}, X^{r})$ is LR fuzzy matrix, i.e., $\tilde{P} = (P, P^{l}, P^{r})$ and $\tilde{Q} = (Q, Q^{l}, Q^{r})$ are both LR fuzzy numbers matrices. By the the following analysis, we know that it is equivalent to the condition S^† ≥ O.

Theorem 3.5. if $\begin{matrix} (A^{+} - A^{-})^{†} + (A^{+} + A^{-})^{†}) \geq O, \\ (A^{+} - A^{-})^{†} - (A^{+} + A^{-})^{†}) \geq O, \end{matrix}$ the fuzzy matrix Equation (2.2) has a strong complex LR fuzzy minimal solution as follows: $\tilde{Z} = (P, P^{l}, P^{r}) + i (Q, Q^{l}, Q^{r})$ where ${\begin{matrix} (PQ) = (UV) A^{†}, \\ (\begin{matrix} (P^{l} Q^{l}), (P^{r} Q^{r}) \end{matrix}) \\ = (\begin{matrix} (U^{l} V^{l}), (U^{r} V^{r}) \end{matrix}) {(\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix})}^{†}, \end{matrix}$ (17)

Proof. Since (U^lV^l) and (U^rV^r) are the left and right spreads fuzzy matrix $\tilde{UV}$ , we have that (U^lV^l) ≥ O and (U^rV^r) ≥ O. It means $(\begin{matrix} [(U^{l} V^{l}), (U^{r} V^{r})] \end{matrix})$ is a non negative matrix.

Let $S^{†} = (\begin{matrix} E & F \\ F & E \end{matrix}) = \frac{1}{2} (\begin{matrix} (A^{+} - A^{-})^{†} + (A^{+} + A^{-})^{†} & (A^{+} - A^{-})^{†} - (A^{+} + A^{-})^{†} \\ (A^{+} - A^{-})^{†} - (A^{+} + A^{-})^{†} & (A^{+} - A^{-})^{†} + (A^{+} + A^{-})^{†} \end{matrix}) .$

We know the condition that S^† ≥ 0 is equivalent to E ≥ 0 and F ≥ 0.

Now that E ≥ O and F ≥ O, the product of two non negative matrices $\begin{matrix} (\begin{matrix} [(U^{l} V^{l}), (U^{r} V^{r})] \end{matrix}) (\begin{matrix} E & F \\ F & E \end{matrix}) \\ = (\begin{matrix} (U^{l} V^{l}) E + (U^{r} V^{r}) F, (U^{l} V^{l}) F + (U^{r} V^{r}) E \end{matrix}) \\ \geq O \end{matrix}$ is non negative in nature. i.e. (P^lQ^l) ≥ O and (P^rQ^r) ≥ O.

The following Theorems give some results for such S^-1 and S^† to be nonnegative. As usual, (.) ^⊤ denotes the transpose of a matrix (.).

Theorem 3.6. [26] The inverse S^-1 of a nonnegative matrix S is nonnegative if and only if S is a generalized permutation matrix.

Theorem 3.7. [10] Let S be an 2n × 2m nonnegative matrix with rank r. Then the following assertions are equivalent:

S^† ≥ 0.

There exists a permutation matrix P, such that PS has the form $PS = (\begin{matrix} T_{1} \\ T_{2} \\ ⋮ \\ T_{r} \\ O \end{matrix}),$ where each T_i has rank 1 and the rows of T_i are orthogonal to the rows of T_j, whenever i ≠ j, the zero matrix may be absent.

$S^{†} = (\begin{matrix} {GC}^{⊤} & {GD}^{⊤} \\ {GD}^{⊤} & {GC}^{⊤} \end{matrix})$ for some positive diagonal matrix G. In this case, $\begin{matrix} (C + D)^{†} = G (C + D)^{⊤}, (C - D)^{†} \\ = G (C - D)^{⊤} . \end{matrix}$

4 Numerical examples

Example 4.1. Consider the following complex fuzzy matrix system $(\begin{matrix} {\tilde{z}}_{11} & {\tilde{z}}_{12} \end{matrix}) (\begin{matrix} 1 \\ 2 \end{matrix}) = (\begin{matrix} (5, 1, 2) + i (3, 2, 1)_{LR} \end{matrix}) .$

Denoting C = G + iH where $H = O, \tilde{W} = \tilde{U} + i \tilde{V}$ and $\tilde{Z} = \tilde{P} + i \tilde{Q}$ , we have the equivalent fuzzy matrix equation $\tilde{X} A = \tilde{B}$ as follows $[(\begin{matrix} {\tilde{p}}_{11} & {\tilde{p}}_{12} \end{matrix}), (\begin{matrix} {\tilde{q}}_{11} & {\tilde{q}}_{12} \end{matrix})] A = [(\begin{matrix} {\tilde{u}}_{11} \end{matrix}), (\begin{matrix} {\tilde{v}}_{11} \end{matrix})],$ where $A = (\begin{matrix} G & H \\ - H & G \end{matrix}) = (\begin{matrix} 1 & 0 \\ 2 & 0 \\ 0 & 1 \\ 0 & 2 \end{matrix}) .$

By Theorem 3.2., the above fuzzy matrix equation is extended into the following function linear system ${\begin{matrix} (PQ) A = (UV), \\ (\begin{matrix} (P^{l} Q^{l}), (P^{r} Q^{r}) \end{matrix}) (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) \\ = (\begin{matrix} (U^{l} V^{l}), (U^{r} V^{r} \end{matrix}), \end{matrix}$ in which $A^{+} = (\begin{matrix} 1 & 0 \\ 2 & 0 \\ 0 & 1 \\ 0 & 2 \end{matrix}), A^{-} = (\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}) .$

From the Equations (3.10), we obtain the minimal solution of fuzzy matrix Equation (3.5) is $\begin{matrix} (PQ) = (UV) A^{†} \\ = (\begin{matrix} 1.0000 & 2.0000 & 0.6000 & 1.2000 \end{matrix}), \\ (P^{l} Q^{l}, P^{r} Q^{r}) = (U^{l} V^{l}, U^{r} V^{r}) {(\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix})}^{†} \\ = (\begin{matrix} 1 & 2 & 2 & 1 \end{matrix}) {(\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix})}^{†} \\ = (\begin{matrix} 0.2000 & 0.4000 & 0.4000 & 0.8000 & 0.4000 & 0.8000 & 0.2000 & 0.4000 \end{matrix}) . \end{matrix}$

It means $\begin{matrix} \tilde{P} = {(\begin{matrix} (1.0000, 0.2000, 0.4000) & (2.0000, 0.4000, 0.8000) \end{matrix})}_{LR}, \\ \tilde{Q} = {(\begin{matrix} (0.6000, 0.4000, 0.2000) & (1.2000, 0.8000, 0.4000) \end{matrix})}_{LR} . \end{matrix}$

Since $\tilde{P}$ and $\tilde{Q}$ are appropriate LR fuzzy numbers matrices, the solution of the complex fuzzy matrix system we obtained is $\begin{matrix} \tilde{Z} = \tilde{P} + i \tilde{Q} \\ = {(\begin{matrix} (1.0000, 0.2000, 0.4000) & (2.0000, 0.4000, 0.8000) \end{matrix})}_{LR} \\ + i {(\begin{matrix} (0.6000, 0.4000, 0.2000) & (1.2000, 0.8000, 0.4000) \end{matrix})}_{LR}, \end{matrix}$ it admits a strong complex LR fuzzy solution by Definition 3.4.

Example 4.2. Consider the following fuzzy matrix system $\begin{matrix} (\begin{matrix} {\tilde{z}}_{11} & {\tilde{z}}_{12} \\ {\tilde{z}}_{21} & {\tilde{z}}_{22} \end{matrix}) (\begin{matrix} 4 - i & 2 + i & 1 - 2 i \\ 1 + i & - 1 + i & 3 - i \end{matrix}) \\ = (\begin{matrix} (3, 2, 1) + i (2, 0, 1) & (1, 1, 1) + i (3, 2, 3) & (4, 3, 1) + i (- 1, 1, 2) \\ (2, 1, 2) + i (4, 3, 2) & (1, 0, 2) + i (1, 2, 0) & (3, 2, 1) + i (1, 1, 2) \end{matrix}) . \end{matrix}$

Denoting $C = G + iH, \tilde{W} = \tilde{U} + i \tilde{V}$ and $\tilde{Z} = \tilde{P} + i \tilde{Q}$ , we have the equivalent fuzzy matrix equation $\tilde{X} A = \tilde{B}$ as follows $[(\begin{matrix} {\tilde{p}}_{11} & {\tilde{p}}_{12} \\ {\tilde{p}}_{21} & {\tilde{p}}_{22} \end{matrix}), (\begin{matrix} {\tilde{q}}_{11} & {\tilde{q}}_{12} \\ {\tilde{q}}_{21} & {\tilde{q}}_{22} \end{matrix})] A = [(\begin{matrix} {\tilde{u}}_{11} & {\tilde{u}}_{12} & {\tilde{u}}_{13} \\ {\tilde{u}}_{21} & {\tilde{u}}_{22} & {\tilde{u}}_{23} \end{matrix}), (\begin{matrix} {\tilde{v}}_{11} & {\tilde{v}}_{12} & {\tilde{v}}_{13} \\ {\tilde{v}}_{21} & {\tilde{v}}_{22} & {\tilde{v}}_{23} \end{matrix})],$ where $A = (\begin{matrix} 4 & 2 & 1 & - 1 & 1 & - 2 \\ 1 & - 1 & 3 & 1 & 1 & - 1 \\ 1 & - 1 & 2 & 4 & 2 & 1 \\ - 1 & - 1 & 1 & 1 & - 1 & 3 \end{matrix}) .$

By the same way, we obtain the minimal solution of fuzzy matrix Equation (3.5) is $\begin{matrix} (PQ) = (UV) A^{†} = (\begin{matrix} 3 & 1 & 4 & 2 & 3 & - 1 \\ 2 & 1 & 3 & 4 & 1 & 1 \end{matrix}) {(\begin{matrix} 4 & 2 & 1 & - 1 & 1 & - 2 \\ 1 & - 1 & 3 & 1 & 1 & - 1 \\ 1 & - 1 & 2 & 4 & 2 & 1 \\ - 1 & - 1 & 1 & 1 & - 1 & 3 \end{matrix})}^{†} \\ = (\begin{matrix} 0.6375 & 0.5875 & 0.5625 & 0.0250 \\ 0.5750 & - 0.1000 & 0.8750 & 0.5250 \end{matrix}), \end{matrix}$

$\begin{matrix} (P^{l} Q^{l}, P^{r} Q^{r}) = (U^{l} V^{l}, U^{r} V^{r}) {(\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix})}^{†} \\ = (\begin{matrix} 0.3012 & 0.9923 & - 0.2738 & 0.4548 & 0.0262 & 0.1673 & 0.3012 & 0.0548 \\ - 0.1613 & 0.3861 & 0.6012 & - 0.2014 & 0.5387 & - 0.3264 & 0.3762 & 0.5611 \end{matrix}) . \end{matrix}$

It means $\begin{matrix} \tilde{P} & = & {(\begin{matrix} (0.6375, 0.3012, 0.0262) & (0.5875, 0.9923, 0.1673) \\ (0.5750, 0.0000, 0.5387) & (- 0.1000, 0.3861, 0.0000) \end{matrix})}_{LR}, \\ \tilde{Q} & = & {(\begin{matrix} (0.5625, 0.0000, 0.3012) & (0.0250, 0.4548, 0.0548) \\ (0.8750, 0.6012, 0.3762) & (0.5250, 0.0000, 0.5611) \end{matrix})}_{LR} . \end{matrix}$

Since $\tilde{P}$ and $\tilde{Q}$ are not appropriate LR fuzzy numbers matrices, we obtained the solution of the complex fuzzy matrix system is $\begin{matrix} \tilde{Z} = \tilde{P} + i \tilde{Q} \\ = {(\begin{matrix} (0.6375, 0.3012, 0.0262) & (0.5875, 0.9923, 0.1673) \\ (0.5750, 0.0000, 0.5387) & (- 0.1000, 0.3861, 0.0000) \end{matrix})}_{LR} \\ + i {(\begin{matrix} (0.5625, 0.0000, 0.3012) & (0.0250, 0.4548, 0.0548) \\ (0.8750, 0.6012, 0.3762) & (0.5250, 0.0000, 0.5611) \end{matrix})}_{LR}, \end{matrix}$ it admits a weak complex fuzzy solution by Definition 3.4.

5 Conclusion

In this work we introduced the complex LR fuzzy matrix equation $\tilde{Z} C = \tilde{W}$ and proposed a general model for solving it. We first converted the complex fuzzy matrix equation to a equivalent high order LR fuzzy matrix system. Then we extended the LR fuzzy matrix equation into a crisp system of matrix equations. The fuzzy minimal solution of the fuzzy matrix equation was derived from solving the crisp linear matrix system. Moreover, the existence condition of the strong complex fuzzy minimal solution was discussed. Numerical examples showed that our method is feasible to solve this type of complex fuzzy matrix equations.

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