1 Introduction
A fuzzy matrix is a matrix over the max-min fuzzy algebra F =[0,1] with operations defined as a+b = max{a,b} and ab = min{a,b} for all a,bF and the standard order ≥ of real numbers over F. A matrix AFmn is said to be regular if there exists X Fmn such that AXA = A. X is called a generalized inverse of A and is denoted by A . In [1], Thomason has studied the convergence of powers of a fuzzy matrix. In [2], Kim and Roush have developed a theory for fuzzy matrices analogous to that for Boolean matrices [3]. A finite fuzzy relational equation can be expressed in the form of a fuzzy matrix equation as x.A = b for some fuzzy coefficient matrix A. If A is regular, then x.A=b is consistent and bX. is a solution for some g-inverse X of A [4]. For more details on fuzzy matrices one may refer to [5, 6]. Recently, the concept of the interval valued fuzzy matrix (IVFM) as a generalization of fuzzy matrix has been introduced and developed by Shyamal and Pal [7]. In earlier work, we have studied the regularity of IVFM [8] and analogous to that for complex matrices [9].
In this paper, we discuss the g-inverses of interval valued fuzzy matrices (IVFM) as a generalization of the g-inverses of regular fuzzy matrices studied in [2, 6], and as an extension of the regularity of the IVFM discussed in [8]. In section 2, we present the basic definition, notation of the IVFM and required results of g-inverses of regular fuzzy matrices. In Section 3, the existence and construction of g-inverses, {1, 2} inverses, {1, 3} inverses and {1, 4} inverses of interval-valued fuzzy matrices are determined in terms of the row and column spaces of IVFM.
Copyright © 2013 Published by ITB Journal Publisher, ISSN: 2337-5760, DOI: 10.5614/j.math.fund.sci.2013.45.1.7
2 Preliminaries
In this section, some basic definitions and results needed are given. Let IVFM denote the set of all interval-valued fuzzy matrices, that is, fuzzy matrices whose entries are all subintervals of the interval [0, 1].
Definition 2.1. For a pair of fuzzy matrices \(E = (e_{ij})\) and \(F = (f_{ij})\) in \(\mathcal{F}_{mn}\) such that \(E \leq F\), the interval valued fuzzy matrix \([E, F] = ([e_{ij}, f_{ij}])\), is the matrix, whose \(ij^{th}\) entry is the interval with lower limit \(e_{ij}\) and upper limit \(f_{ij}\).
In particular for E = F, IVFM [E,E] reduces to the fuzzy matrix \(E \in \mathcal{F}_{mn}\).
For \(A=(a_{ij})=([a_{ijL}\,,\,a_{ijU}])\in (IVFM)_{mxn}\), let us define \(A_L=(a_{ijL})\) and \(A_U=(a_{ijU})\). Clearly, the fuzzy matrices \(A_L\) and \(A_U\) belong to \(\boldsymbol{\mathcal{F}}_{mn}\) such that \(A_L\leq A_U\). Therefore, by Definition (2.1), A can be written as
\[A = [A_L, A_U] \tag{1}\] where A<sub>L</sub> and A<sub>U</sub> are called the lower and upper limits of A respectively.
Here we shall follow the basic operation on IVFM as given in [8].
For \(A=(a_{ij})=([a_{ijL},a_{ijU}])\) and \(B=(b_{ij})=([b_{ijL},b_{ijU}])\) of order mxn, their sum, denoted as A+B, is defined as
\[A+B = (a_{ij}+b_{ij}) = [(a_{ijL}+b_{ijL}), (a_{ijU}+b_{ijU})]\] (2)
For \(A = (a_{ij})_{mxn}\) and \(B = (b_{ij})_{nxp}\) their product, denoted as AB, is defined as
AB = \[(C_{ij}) = [\sum_{k=1}^{n} a_{ik} b_{kj}]\] \(i = 1, 2, ... m \text{ and } j = 1, 2, ... ... p\)
= \([\sum_{k=1}^{n} (a_{ikL}, b_{kiL}), \sum_{k=1}^{n} (a_{ikU}, b_{kiU})]\)
If \(A = [A_L, A_U]\) and \(B = [B_L, B_U]\) then \(A + B = [A_L + B_L, A_U + B_U]\)
\[AB = [A_L B_L, A_U B_U] \tag{3}\]
\(A \ge B\) if and only if \(a_{ijL} \ge b_{ijL}\) and
\[a_{ijU} \ge b_{ijU}\] if and only if A+B =A (4)
In particular if \(a_{ijL} = a_{ijU}\) and \(b_{ijL} = b_{ijU}\) then by Eq. (3) reduces to the standard max. min. composition of fuzzy matrices [2, 6].
For \(A \in (IVFM)_{mn}\), \(A^T\), \(\mathcal{R}(A)\), \(\mathcal{C}(A)\), \(A^T\), \(A\{1\}\) denotes the transpose, row space, column space, g-inverses and set of all g-inverses of A, respectively.
Lemma 2.2. (Lemma 2 [5]) For A, \(B \in \mathcal{F}_{mn}\), if A is regular, then (i) \(\Re(B) \subseteq \Re(A) \Leftrightarrow B = BA^-A\) for each \(A^- \in A\{1\}\)
(ii) \(\mathcal{C}(B) \subseteq \mathcal{C}(A) \Leftrightarrow B = AA^{-}B\) for each \(A^{-} \in A\{1\}\).
Lemma 2.3. If \(A \in \mathcal{F}_{mn}\) with \(\mathbb{R}(A) = \mathbb{R}(A^TA)\), then \(A^TA\) is regular fuzzy matrix if and only if A is a regular fuzzy matrix. If \(A \in \mathcal{F}_{mn}\) with \(\mathcal{C}(A) = \mathcal{C}(AA^T)\), then \(AA^T\) is a regular fuzzy matrix if and only if A is a regular fuzzy matrix.
In the following, we will make use of the following results proved in our earlier work [8]. For the sake of completeness we will provide the proof.
Lemma 2.4. (Theorem 3.3 [8])
Let \(A = [A_L, A_U] \in (IVFM)_{mn}\)
Then the following holds:
- (i) A is regular IVFM \(\Leftrightarrow\) \(A_L\) and \(A_U \in \mathcal{F}_{mn}\) are regular
- (ii) \(\Re(A) = [\Re(A_L), \Re(A_U)]\) and \(\mathcal{C}(A) = [\mathcal{C}(A_L), \mathcal{C}(A_U)]\).
Proof.
(i) Since \(A \in (IVFM)_{mn}\), any vector \(x \in R(A)\) is of the form x = y.A for some \(y \in (IVFM)_{1n}\), that is, x is an interval valued vector with n components.
Let us compute \(x \in R(A)\) as follows:
x is a linear combination of the rows of A \[\Rightarrow\] x = \(\sum_{i=1}^{\infty} \alpha_i\). Ai* where A<sub>i*</sub> is the i<sup>th</sup> row of A. Equating the j<sup>th</sup> component on both sides yields
\[x_j = \sum_{i=1}^{m} \alpha_i. \ a_{ij}.\]
Since, \(a_{ij} = [a_{ijL}, a_{ijU}]\)
\[\begin{split} x_j &= \sum_{i=1}^{m} \alpha_i. \; [a_{ijL}, \, a_{ijU}] \\ &= \sum_{i=1}^{m} [\alpha_i \, a_{ijL}, \, \alpha_i \, a_{ijU}] \\ &= \left( \sum_{i=1}^{m} \left( \alpha_i. \, a_{ijL} \right) \;, \; \sum_{i=1}^{m} \left( \alpha_i. \, a_{ijU} \right) \\ &= [x_{jL}, \, x_{jU}]. \end{split}\]
\(x_{jL}\) is the \(j^{th}\) component of \(x_L \in R(A_L)\) and \(x_{jU}\) is the \(j^{th}\) component of \(x_U \in R(A_U)\). Hence \(x = [x_L, x_U]\). Therefore, \(R(A) = [R(A_L), R(A_U)]\)
(ii) For \(A = [A_L, A_U]\), the transpose of A is \(A^T = [A_L^T, A_U^T]\). By using (i) we get, \(C(A) = R(A^T) = [R(A_L^T), R(A_U^T)] = [C(A_L), C(A_U)]\).
Lemma 2.5. (Theorem 3.7 [8])
For A and \(B \in (IVFM)_{mn}\)
- (i) \(\Re(B) \subseteq \Re(A) \Leftrightarrow B = XA \text{ for some } X \in (IVFM)_m\)
- (ii) \(\mathcal{C}(B) \subset \mathcal{C}(A) \Leftrightarrow B = AY \text{ for some } Y \in (IVFM)_n\)
Proof.
(i) Let \(A = [A_L, A_U]\) and \(B = [B_L, B_U]\). Since, B = XA, for some \(X \in (IVFM)\), put \(X = [X_L, X_U]\). Then, by Equation (3), \(B_L = X_L \ A_L\) and \(B_U = X_U \ A_U\). Hence, by (Lemma (2.2)), \(\Re (B_L) \subseteq \Re (A_L)\) and \(\Re (B_U) \subseteq \Re (A_U)\)
By Lemma (2.4)(ii), \(\Re\) (B) = \([R(B_L), \Re\) (B<sub>U</sub>)] \(\subseteq\) \([\Re\) (A<sub>L</sub>), \(\Re\) (A<sub>U</sub>)] = \(\Re\) (A). Thus \(\Re\) (B) \(\subseteq\) \(\Re\) (A). Conversely, \(\Re\) (B) \(\subseteq\) \(\Re\) (A).
\[\begin{array}{l} \Rightarrow \mathcal{R}\left(B_{L}\right) \subseteq \mathcal{R}\left(A_{L}\right) \text{ and } \mathcal{R}\left(B_{U}\right) \subseteq \mathcal{R}\left(A_{U}\right) & \text{ (By Lemma (2.4) (ii))} \\ \Rightarrow B_{L} = YA_{L} \text{ and } B_{U} = ZA_{U} & \text{ (By Lemma (2.2))} \end{array}\] \[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\]
(ii) This can be proved along the same lines as that of (i) and hence omitted.
3 g- Inverses of Interval Valued Fuzzy Matrices
In this section, we will discuss the g-inverses of an IVFM and their relations in terms of the row and column spaces of the matrix as a generalization of the results available in the literature on fuzzy matrices [2, 6] as a development of our earlier work [8] on regular IVFMs and analogous to that for complex matrices [9].
Definition 3.1. For \(A \in (IVFM)_{mn}\) if there exists \(X \in (IVFM)_{nm}\) such that
- (1) AXA = A
- \((2) \quad XAX = X\)
- \((3) \quad (AX)^T = (AX)\)
- (4) \((XA)^T = (XA)\), then X is called a g-inverse of A.
X is said to be a \(\lambda\)- inverse of A and \(X \in A\{\lambda\}\) if X satisfies \(\lambda\) equation where \(\lambda\) is a subset of \(\{1, 2, 3, 4\}\). A \(\{\lambda\}\) denotes the set of all \(\lambda\)- inverses of A. In particular if \(\lambda = \{1, 2, 3, 4\}\) then X unique and is called the Moore Penrose inverse of A, denoted as \(A^{\dagger}\).
Remark 3.2. From Definition (3.1) of \(\lambda\)-inverses for \(A \in (IVFM)\), by applying Eq. (3) for \(A = [A_L, A_U]\) and \(X = [X_L, X_U]\) it can be verified that the existence and construction of \(\{\lambda\}\)-inverses of \(A \in (IVFM)_{mn}\) reduces to that of the \(\{\lambda\}\)-inverses of \(A_L, A_U \in F_{mn}\).
Theorem 3.3. Let \(A \in (IVFM)_{mn}\) and \(X \in A\{1\}\), then \(X \in A\{2\}\) if and only if \(\mathbb{R}(AX) = \mathbb{R}(X)\)
Proof.
Since \[A = [A_L, A_U]\] and \(X = [X_L, X_U]\) \[X \in A\{2\} \Rightarrow XAX = X, \text{ then by Eq. (3),}\] \[\Rightarrow X_L A_L X_L = X_L \text{ and } X_U A_U X_U = X_U; X_L \in A_L\{2\} \text{ and } X_U \in A_U\{2\}\] \[\Rightarrow A_L \in X_L\{1\} \text{ and } A_U \in X_U\{1\}\] \[\Rightarrow R(X_L) = R(A_L X_L) \text{ and } R(X_U) = R(A_U X_U)\] \[\Rightarrow R(AX) = R(X). \qquad (By Lemma (2.4))\]
Conversely,
Let \[\Re\] (AX) = \(\Re\) (X), then by Lemma (2.4), \(\Re\) (X) \(\subseteq\) \(\Re\) (AX) implies X = YAX for some Y \(\in\) (IVFM)m. X(AX) = (YAX)(AX) \[XAX = Y(AXA)X\]\[= YAX \qquad (By Definition (3.1))\]\[= X\]
Thus \(X \in A\{2\}\).
Remark 3.4. In the above Theorem (3.3), the condition XA{1} is essential. This is illustrated in the following example.
Example 3.5.
Let A = \[\begin{bmatrix} [0,1] & [1,1] \\ [1,1] & [0,0] \end{bmatrix}\], \(X = \begin{bmatrix} [1,1] & [0,1] \\ [0,0] & [0,1] \end{bmatrix}\)
Then by representation (1) we have, 0 1 1 1 AL = 1 0 , AU = 1 0
\[X_L = \left( \begin{array}{ccc} 1 & & 0 \\ \\ 0 & & 0 \end{array} \right) \quad \text{and} \quad X_U = \left( \begin{array}{ccc} 1 & & 1 \\ \\ 0 & & 1 \end{array} \right),\]
0 0 1 1 ALXLAL = 0 1 AL implies XLAL{1}and AUXUAU = 1 1 AU implies XUAU{1}
\[A_L X_L = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \quad \text{and} \quad A_U X_{U} = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}\]
But \[X_L A_L X_L = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \neq X_L\]. and \(X_U A_U X_U = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \neq X_U\).
Hence \(X_L \notin A_L\{2\}\) and \(X_U \notin A_U\{2\}\). Then by Eq. (3) we have, \(AXA \neq A\), therefore \(X \notin A\{1\}\). Here \(\Re(X_L) = \Re(A_LX_L)\) and \(\Re(X_U) = \Re(A_UX_U)\). Therefore by Lemma (2.4), \(\Re(X) = \Re(AX)\), but \(XAX \neq X\). Hence \(X \notin A\{2\}\).
Theorem 3.6. For \(A \in (IVFM)_{mn}\), A has a {1, 3} inverse if and only if \(A^TA\) is a regular IVFM and \(\Re(A^TA) = \Re(A)\).
Proof. Since A is regular, Lemma (2.4), \(A_L\) and \(A_U\) are regular. Let A has a \(\{1, 3\}\) inverse X (say) then by Eq. (3), \(A_L\) has a \(\{1, 3\}\) inverse \(X_L\) and \(A_U\) has a \(\{1, 3\}\) inverse \(X_U\).
Then \[A_L X_L A_L = A_L\] and \((A_L X_L)^T = A_L X_L\) \[A_L^T (A_L X_L A_L) = A_L^T A_L\] \[(A_L^T A_L X_L) A_L = A_L^T A_L\] \[\Re (A_L^T A_L) \subset \Re (A_L) \qquad (By Lemma (2.2))\]
Similarly, \(\Re(A_U^T A_U) \subseteq \Re(A_U)\)
Therefore by Equation (3) we have, \(\Re(A^TA) \subseteq \Re(A)\)
Also \[(A_L X_L)^T A_L = A_L X_L A_L\]
\[\Rightarrow X_L^T A_L^T A_L = A_L\]
\[\Rightarrow X_L^T (A_L^T A_L) = A_L\]
\[\Re (A_L) \subseteq \Re (A_L^T A_L)\] (By Lemma (2.2))
Similarly, \(\Re\) \((A_U) \subseteq \Re\) \((A_U^T A_U)\). By Equation (3) we have, \(\Re\) \((A) \subseteq \Re\) \((A^T A)\). Thus, \(\Re\) \((A) = \Re\) \((A^T A)\). Since \(X \in A\{1\}\), \(\Re\) \((A) = \Re\) (XA). Hence, \(\Re\) \((A^T A) = \Re\) \((A) = \Re\) (XA). Since \(\Re\) \((A^T A) \supseteq \Re\) (XA) (By Lemma (2.5)),
\[YA^{T}A = XA\] let \(Y = [Y_{L}, Y_{U}]\) then, \(A_{L}^{T}A_{L} (Y_{L}A_{L}^{T}A_{L}) = A_{L}^{T}A_{L} (X_{L}A_{L})\) \[(A_{L}^{T}A_{L})Y_{L} (A_{L}^{T}A_{L}) = A_{L}^{T} (A_{L}X_{L}A_{L})\] \[= A_{L}^{T}A_{L}\]
Similarly, \(A_U^T A_U (Y_U A_U^T A_U) = A_U^T A_U\). By (3) we have, \(A^T A (Y A^T A) = A^T A\)Thus \(A^T A\) is a regular interval valued fuzzy matrix. Conversely, let \(A^T A\) be a regular interval-valued fuzzy matrix and \(\Re (A) = \Re (A^T A)\). By Lemma (2.3),
A is a regular IVFM. Let us take \(Y = (A^T)^-A^T \in (IVFM)\). We claim that \(Y \in A \{1, 3\}\).
\(\mathbb{R}\) (A) = \(\mathbb{R}\) (A<sup>T</sup>A) and A<sup>T</sup>A is regular, by Lemma (2.3) A = A(A<sup>T</sup>A)<sup>-</sup>A<sup>T</sup>A) = AYA, Y \in A{\{1\}} and since \(\mathbb{R}\) (A) = \(\mathbb{R}\) (A<sup>T</sup>A), A = XA<sup>T</sup>A, by Lemma (2.4), A<sub>L</sub> = X<sub>L</sub>A<sub>L</sub><sup>T</sup>A<sub>L</sub> and A<sub>U</sub> = X<sub>U</sub>A<sub>U</sub><sup>T</sup>A<sub>U</sub>. Let Y = [Y<sub>L</sub>, Y<sub>U</sub>].
Then, \[A_{L}Y_{L} = X_{L}A_{L}^{T}A_{L} (A_{L}^{T}A_{L})^{-}A_{L}^{T}\]
\(= X_{L}A_{L}^{T}A_{L} (A_{L}^{T}A_{L})^{-}A_{L}^{T}A_{L}X_{L}^{T}\)
\(= X_{L} (A_{L}^{T}A_{L})(A_{L}^{T}A_{L})^{-} (A_{L}^{T}A_{L})X_{L}^{T}\)
\(= X_{L} (A_{L}^{T}A_{L}X_{L}^{T})\)
\(= X_{L} A_{L}^{T}\)
Similarly, \(A_U Y_U = X_U A_U^T\). Then by Eq. (3) we have, \(AY = XA^T\)
\[(A_L Y_L)^T = (X_L A_L^T)^T\] \[= A_L X_L^T\] \[= X_L A_L^T A_L X_L^T\] \[= X_L A_L^T = A_L Y_L\]
Similarly, \((A_U Y_U)^T = X_U A_U^T = A_U Y_U\). Then by Equation (3) we have, \((AY)^T = AY\), \(Y \in A\{3\}\). Since \(\mathbb{R}(A) = \mathbb{R}(A^T A)\) by Lemma (2.4) and regularity of \(A^T A\) we get
\(A = A(A^{T}A)^{-1}(A^{T}A) = AYA, Y \in A\{1\}.\) Thus A has a {1, 3} inverse.
Theorem 3.7. For \(A \in (IVFM)_{mn}\), A has \(\{1, 4\}\) inverse if and only if \(AA^{T}\) is regular and \(\mathcal{C}(AA^{T}) = \mathcal{C}(A)\).
Proof. This can be proved in the same manner as that of Theorem (3.6).
Corollary 3.8. Let \(A \in (IVFM)_{mn}\) be a regular IVFM with \(A^TA\) is a regular IVFM and \(\Re(A^TA) = \Re(A)\), then \(Y = (A^TA)^TA^T \in A\{1, 2, 3\}\).
Proof. \(Y \in A\{1, 3\}\) follows from Theorem (3.6), it is enough verify \(Y = [Y_L, Y_U] \in A\{2\}\) that is, \(Y_L A_L Y_{L} = Y_L\) and \(Y_U A_U Y_U = Y_U\).
\[Y_{L}A_{L}Y_{L} = Y_{L} (X_{L}^{T}A_{L}^{T}A_{L}) (A_{L}^{T}A_{L})^{-} A_{L}^{T}\]
\[\begin{split} &= Y_{L}X_{L}^{T} (A_{L}^{T}A_{L}) (A_{L}^{T}A_{L})^{-} (A_{L}^{T}A_{L}X_{L}) \\ &= Y_{L}X_{L}^{T} (A_{L}^{T}A_{L}) (A_{L}^{T}A_{L})^{-} (A_{L}^{T}A_{L})X_{L} \\ &= Y_{L}X_{L}^{T}A_{L}^{T}A_{L}X_{L} \\ &= Y_{L}A_{L}X_{L} \\ &= [(A_{L}^{T}A_{L})^{-}A_{L}^{T}]A_{L}X_{L} \\ &= (A_{L}^{T}A_{L})^{-} (A_{L}^{T}A_{L}X_{L}) \\ &= (A_{L}^{T}A_{L})^{-}A_{L}^{T} \\ &= Y_{L} \end{split}\]
Similarly, \(Y_UA_UY_U=Y_U\). Then by Eq. (3), YAY=Y.
Thus \(Y \in A\{1,2,3\}\).
Theorem 3.9. Let \(A \in (IVFM)_{mn}\) be a regular IVFM with \(AA^T\) is a regular IVFM and \(\Re(A^T) = \Re(AA^T)\) then \(Z = A^T(AA^T)^- \in A\{1, 2, 4\}\).
Proof. Similar to the proof of Theorem (3.7) and Corollary (3.8) hence omitted.
4 Conclusion
The main results of the present paper are the generalization of the results on ginverses of regular fuzzy matrices found in [2, 6] and the extension of our earlier work on regular IVFMs [8].