Abstract
In this paper, gives the definition of complex fuzzy matrix, and study its convergence problems based on the fuzzy matrix theory, which included the Convergence in norm and the Convergence in Power, some important conclusions are obtained, to build and to improve the solid foundation for complex fuzzy matrix theory.
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1 Introduction
The fuzzy matrix is an important part of fuzzy mathematics, which plays an important role in the fuzziness expression of two dimensional relationship, which has important applications in the circuit design the exchange of information, cluster analysis, etc., but, because the diversity of the research fields and the target, to solve complicated system problems will require higher dimensions, in this paper, based on the fuzzy matrix theory, gives the definition of complex fuzzy matrix, and study its convergence problems of complex fuzzy matrix, to establish complex fuzzy matrix theory and to laid a solid foundation for the further research.
2 Complex Fuzzy Sets and Complex Fuzzy Number
Definition 1
([1]) Let R be the real field, C is complex field. For \(\forall X,Y \in F\left( R \right) \), \(Z = X + iY\), called the complex fuzzy sets, referred to as the complex fuzzy sets.
Definition 2
([1]) Suppose \(\mathop {{Z_1}}\limits _{} = \mathop {{X_1}}\limits _{} + \mathop {i{Y_1}}\limits _{},\mathop {{Z_2}}\limits _{} = \mathop {{X_2}}\limits _{} + \mathop {i{Y_2}}\limits _{} \in {C^F}(C)\), the Operation of intersection and union be defined as: \(\forall z = x + iy \in C\)
Definition 3
Suppose \(\{ \mathop {{Z_\gamma }}\limits _{} = \mathop {{X_\gamma }}\limits _{} +\, \mathop {{iY_\gamma }}\limits _{},\gamma \in \varGamma \} \subseteq {C^F}(C)\), the operations of infinite intersection and infinite unit of the complex fuzzy sets are defined as: \(\forall z = x + iy \in C\),
\(\mathop {{Z_1}}\limits _{} \cap \mathop {{Z_2}}\limits _{} \), \(\mathop {{Z_1}}\limits _{} \cup \mathop {{Z_2}}\limits _{} \), \(\mathop \cap \limits _{\gamma \in \varGamma } \mathop {{Z_\gamma }}\limits _{}, \mathop \cup \limits _{\gamma \in \varGamma } \mathop {{Z_\gamma }}\limits _{} \in {C^F}(C).\)
Definition 4
The regular convex complex fuzzy sets in the complex field, \(\mathop Z = \mathop X +\,\mathop iY \) is called a complex fuzzy numbers.
3 Fuzzy Matrix
Definition 5
([2]) All the elements of a matrix are in the closed interval \(\left[ {0,1} \right] \), which called the fuzzy matrix.
Union, intersection, complement operation, the corresponding element of the two fuzzy matrix: take big, take small, take up, which obtain a new element matrix called their union, intersection, complement operation.
Concrete concepts and properties of fuzzy matrix, please refer to the Ref. [2].
4 Complex Fuzzy Matrix
Definition 6
Suppose X is a non empty sets of real numbers, called
is complex fuzzy sets on X, \(A\left( x \right) \), \(B\left( x \right) \) are real fuzzy number, \(A\left( x \right) \,+\,iB\left( x \right) \) is complex fuzzy number, where \(A\left( x \right) :X \rightarrow \left[ {0,1} \right] \), \(B\left( x \right) :X \rightarrow \left[ {0,1} \right] \), which expressed the real parts and the imaginary parts of C separately, \(0 \le A\left( x \right) + B\left( x \right) \le ~1.\)
Definition 7
Suppose \(C = {\left( {{A_{ij}}\left( x \right) + i{B_{ij}}\left( x \right) } \right) _{n \times m}}\) is matrix, all of the \({A_{ij}}\left( x \right) + i{B_{ij}}\left( x \right) \) is complex fuzzy number for \(i,j\left( {1 \le i \le n,1 \le j \le n} \right) \), then called C is complex fuzzy matrix, note as \(CFM\left( {n,m} \right) \).
Definition 8
Suppose
are complex fuzzy matrix, the operation of the A and B is defined as:
Definition 9
Suppose
are complex fuzzy matrix, the Product operation of the A and B is defined as:
5 Complex Fuzzy Matrix Norm and Convergence in Norm
Definition 10
([3]) Let \(F = R\) or C, V is a linear space over F. If the real vector function \(\left\| * \right\| \) on V satisfies the following properties:
-
1.
For arbitrary \(x \in V\), \(\left\| x \right\| \ge 0\), and \(\left\| x \right\| = 0 \Leftrightarrow x = 0\),
-
2.
For arbitrary \(k \in F\), \(x \in V\) get \(\left\| {kx} \right\| = \left\| k \right\| \left\| x \right\| \),
-
3.
For arbitrary \(x,y \in V\), get \(\left\| {x + y} \right\| \le \left\| x \right\| + \left\| y \right\| \),
then \(\left\| x \right\| \) is called vector norm of X in V.
Definition 11
Suppose \(\left\| * \right\| \) is non negative real function on \({F^{n \times n}}\), if
then called \(\left\| * \right\| \) is \(CFM\left( {n,m} \right) \).
Definition 12
([4]) Suppose \(\left( {V,\left\| * \right\| } \right) \) is a n-dimensional normed linear space, \({x_1},{x_2}\), \(\ldots , {x_k}, \ldots \) is a vector sequence of V, \(\alpha \) is a fixed vector V, if
then called vector sequence \({x_1}, {x_2}, \ldots , {x_k}, \ldots \) convergence in norm, A is the limit of a sequence, note as:
vector sequence does not converge called divergence.
Definition 13
Suppose \(\left( {V,\left\| * \right\| } \right) \) is a n-dimensional normed linear space, \({{c_1}, {c_2}, \ldots , {c_k}, \ldots }\) is a complex fuzzy matrix sequence of V, \(c\left( k \right) \) constitutes a complex fuzzy matrix function, \(\alpha = \alpha R + i\alpha I\) is a fixed complex fuzzy matrix of V, if
then called complex fuzzy matrix sequence \({c_1}\), \({c_2}\), \( \ldots \), \({c_k}\), \(\ldots \) convergence in norm, \(\alpha = \alpha R + i\alpha I\) is the limit of a sequence, note to:\(\mathop {\mathrm{{lim}}}\limits _{k \rightarrow \infty } x\left( k \right) = \alpha \) or \({x_k} \rightarrow \alpha . \)
Definition 14
Suppose \(\left( {V,\left\| * \right\| } \right) \) is a n-dimensional normed linear space \(C\left( k \right) \), \(\left( {n = 1,2, \ldots } \right) \), \(c:C\mathrm{{FM}}\left( {n,m} \right) \rightarrow C\mathrm{{FM}}\left( {n,m} \right) \), then
-
1.
\(\left\{ {c\left( k \right) } \right\} \) almost everywhere convergence c, in V, if there is \(E \in V\), \(c\left( E \right) = 0\), makes \(\left\{ {c\left( k \right) } \right\} \) pointiest convergence on c in \(V - E\), note as \(c\left( k \right) \mathop \rightarrow \limits ^{a.e} c\) in V,
-
2.
\(\left\{ {c\left( k \right) } \right\} \) almost uniform convergence c in V, if there is \(E \in V\), for any \(\varepsilon > 0\), \(\left\| {c\left( E \right) } \right\| < \varepsilon \), makes \(\left\{ {c\left( k \right) } \right\} \) pointiest uniform convergence on c in \(V - E\), note as \(c\left( k \right) \mathop \rightarrow \limits ^{a.u} c\) in V,
-
3.
\(\left\{ {c\left( k \right) } \right\} \) pseudo almost everywhere convergence c in V, if there is \(E \in V\), \( c \left( {V - E} \right) = c \left( E \right) \), makes \(\left\{ {c\left( k \right) } \right\} \) pointiest convergence on c in \(V - E\), note as \(c\left( k \right) \mathop \rightarrow \limits ^{p.a.e} c\) in V,
-
4.
\(\left\{ {c\left( k \right) } \right\} \) pseudo almost uniform convergence c in V, if there is \(\left\{ {{E_k}} \right\} \subset V\), \(\mathop {\mathrm{{lim}}}\limits _{n \rightarrow \infty } c\left( {V - {E_k}} \right) = c\left( A \right) \) makes \(\left\{ {c\left( k \right) } \right\} \) pointiest uniform convergence on c in \(V - E\), note as \(c\left( k \right) \mathop \rightarrow \limits ^{p.a.u} c\) in V.
Theorem 1
A necessary and sufficient condition of null additives of c is for any \(A \in V\), \(c \in C\mathrm{{FM}}\left( {n,m} \right) \), \(c\left( k \right) \in C\mathrm{{FM}}\left( {n,m} \right) \) and \(\left\| {CMF\left( {n,m} \right) } \right\| < \infty \) has
Proof: Necessity:
by consistency theorem has
Sufficiency: Suppose for any \(B \in V\) has \(c\left( B \right) = 0,\) let \(x \in V - B,\) \(c\left( x \right) = 0\), so obviously there is \(c\left( k \right) \mathop \rightarrow \limits _V^{a.e} 0\), exist \({P_k}\), when \(k \rightarrow \infty \) has \(c\left( {V - {P_k}} \right) \rightarrow C\left( V \right) \), and \(\left\{ {c\left( k \right) } \right\} \) uniform convergence 0 on \(V - {p_k}\), \(n = 1,2, \ldots \), at this time there will be \(V - B \supseteq V - {P_k}\), \(k = 1, 2, \ldots \), \(c\left( {V - B} \right) \ge c\left( {V - {P_k}} \right) \rightarrow c\left( V \right) \), therefore it is \(c\left( {V - B} \right) = c \left( V \right) \).
6 The Power of Complex Fuzzy Matrix and Its Power Convergence
Definition 15
Suppose \(A \in CMF\left( {n,n} \right) \) power K of A is defined as \({A^k}\), among them \({A^1} = A\), \({A^k} = {A^{k - 1}}A\).
Theorem 2
Suppose \(A \in CFM\left( {n,n} \right) \), there are must be a positive integer P and K, making \(\forall k \ge K\) has \({A^{k + p}} = {A^k}\).
Proof: Let \(\forall k \ge 1\), \(A = \left( {{A_{ij}}\left( x \right) + i{B_{ij}}\left( x \right) } \right) \),
because the \( \wedge , \vee \) is closed, so the elements number of \(\left\{ {{A^k},k \ge 1} \right\} \) in not more than \({\left( {{n^{4n}}} \right) ^n}\) there must be a positive integer P and K, make \({A^{k + p}} = {A^k}\), so for any \(k > K\) has
Theorem 3
Suppose \(A \in CFM\left( {n,n} \right) \), which is power sequence monotone increasing of A, hence
that is, power sequences of A is Convergence.
Proof: Because power sequence of A monotone increasing, then \({A^n} \le {A^{n + 1}}\), and from the \({A^{n + 1}} \le A \vee {A^2} \vee \cdots \vee {A^n} = {A^n}\), so \({A^n} = {A^{n + 1}}\).
7 Conclusion
In this paper, by studied complex fuzzy matrix convergence problems based on the fuzzy matrix theory, which included the Convergence in norm and the Convergence in Power, which is important work in fuzzy complex analysis, to build and to lay a solid foundation for our future research on complex fuzzy matrix theory.
References
Ma, S.: The Theory of Fuzzy Complex Analysis Foundation. Science Press, Beijing (2010)
Fan, Z.: Theory and Application of Fuzzy Matrix. Science Press, Beijing (2006)
Yan, L., Gao, Y.: The Principle of Fuzzy Mathematics and Applications, 3rd edn., pp. 90–94. South China University of Technology Press, Guangzhou (2001)
Zhang, Y.: Matrix Theory and Application. Science Press, Beijing (2011)
Acknowledgments
This work is supported by International Science and Technology Cooperation Program of China (2012DFA11270).
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© 2016 Springer International Publishing Switzerland
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Zhao, Zq., Ma, Sq. (2016). Complex Fuzzy Matrix and Its Convergence Problem Research. In: Cao, BY., Liu, ZL., Zhong, YB., Mi, HH. (eds) Fuzzy Systems & Operations Research and Management. Advances in Intelligent Systems and Computing, vol 367. Springer, Cham. https://doi.org/10.1007/978-3-319-19105-8_15
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DOI: https://doi.org/10.1007/978-3-319-19105-8_15
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