广义模糊矩阵若干问题的研究
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摘要
广义模糊矩阵是模糊数学的重要组成部分，它能广泛应用于不同的领域，如切换电路的设计、自动机理论、图论、信息系统、聚类分析、复杂系统的建模、动态编程及决策理论，故学者们视广义模糊矩阵是最为重要的研究课题之一。本学位论文研究了广义模糊矩阵中几个常见且重要的问题。全文总共有七章，主要研究工作分五个部分。
第一部分讨论了一般广义模糊矩阵幂序列性质。首先对一般广义模糊矩阵的幂的收敛性进行了探讨，其中一些矩阵幂的收敛指数比以往研究的收敛指数要低，起到了降阶的作用。接着利用伴随矩阵研究了广义模糊矩阵幂序列性质，其中所得的一些成果是对以往相应成果的推广。最后引进特殊算子后，研究了广义模糊矩阵幂的组合性质，从而将以往相关的重要结论推广到了斜代数上。
第二部分研究了广义模糊矩阵的传递性。传递矩阵是广义模糊矩阵中的一类重要的典型阵。传递矩阵代表了传递关系，而传递关系在聚类分析、信息检索、优化等领域都有很重要的应用。广义模糊矩阵的传递闭包也能广泛应用于交通网络、模糊控制、经济管理、模糊聚类分析、计算机科学技术及稳定性。故广义模糊矩阵的传递性引起了很多学者的重视。第二部分首先在分配格上，定义了几类特殊的传递阵，并研究了其性质，进而丰富了典型传递阵的研究内容。接着讨论了广义传递阵幂的收敛性，其中一些主要结论是对以往相关结论的完善和改进，并将有关成果推广到了斜代数及路代数上。然后在分配格及斜代数上完善了广义模糊矩阵的传递闭包的性质及传递闭包的算法，并将有关成果推广到了路代数上。最后将广义模糊传递阵的构造及规则形状问题中所得的一些重要结论推广到了斜代数及路代数上。
第三部分研究了广义模糊矩阵的幂零性。幂零矩阵是广义模糊矩阵中的另一类重要的典型阵。幂零矩阵可用来表示非循环图，而非循环图能用来代表一致性，且更能表示优先关系，故对广义模糊矩阵中的幂零阵的研究是很有价值的。第三部分一方面在路代数上研究了矩阵的幂零性，并完善了对幂零指数的研究，其中所得的一些结论是对以往相应成果的推广。另外，通过对以往矩阵幂零性的归纳总结，定义了一类特殊幂零阵――圈可控阵，并研究了圈可控阵的性质，其中包括用图论的知识去研究圈可控阵的表达式，所取得的一些成果改进了关于幂零阵表达式的已有结论。
第四部分研究了广义模糊矩阵的行列式及可逆性问题。首先将广义模糊矩阵行列式的相关重要成果推广到路代数上。最后完善了路代数上带有周期的矩阵的可逆性问题的研究。
第五部分研究了广义模糊矩阵的特征问题。在完全完备分配格上，利用拟基刻画了特征值的特征向量子格，从而完善了特征问题在格上的研究。
Generalized fuzzy matrix is an important part in the theory of fuzzy mathematics anduseful in diverse areas such as design of switching circuits, automata theory, graph theory,information systems, clustering, complex system modeling, dynamical programming anddecision theory. Consequently, generalized fuzzy matrix is one of the most focused prob-lems in fuzzy mathematics. This doctoral dissertation studies several problems which areusually important in generalized fuzzy matrices. This paper consists of five parts withseven chapters:
Part1is to discuss the properties of powers of commonly generalized fuzzy matrices.Firstly, the convergence for powers of commonly generalized fuzzy matrices is discussed.The convergent index of some studied matrices is smaller than previous considered index.Therefore these matrices here play the role of reducing order. Next some properties ofpowers of generalized fuzzy matrices are established through adjoint matrix, and someresults obtained generalize and develop the corresponding results in previous literatures.At last, by introducing special operator, some properties of compositions of generalizedfuzzy matrices are given, thus some corresponding important results are extended to in-clines.
Part2is devoted to investigating the transitivity of generalized fuzzy matrices. Tran-sitive matrix is an important type of generalized matrices which represent transitive re-lation. Transitive relation plays an important role in clustering, information retrieval,preference, and so on. The transitive closure of a generalized matrix is also widely usedfor traffic network, fuzzy control, economic management, fuzzy clustering analysis, com-puter technology and reliability. Therefore, the transitivity problems of generalized matri-ces have been discussed by many authors. In part two, several special kinds of transitivematrices are firstly defined over distributive lattices, and some properties of these transi-tive matrices are obtained. These types of transitive matrices aim at enrich research ontransitivity of lattice matrices. Then, the convergence for powers of generalized transitivematrices is considered. Some results contained here perfect and improve the correspond-ing results in previous literatures. And, furthermore, these obtained results are generalizedto inclines and path algebras. Next, some properties and the algorithm of the transitive closure of generalized fuzzy matrices are perfected over distributive lattices and inclines.Also, some results established are extended to path algebras. Finally, some properties ofcompositions and the issue of the canonical form of generalized transitive matrices areextended to inclines and path algebras.
Part3is to study the nilpotency for powers of generalized fuzzy matrices. Nilpotentmatrices are another important typical matrices of generalized matrices which representacyclic graphs. Acyclic graphs are used to represent consistent systems and are importantin the representation of precedence relations. Then, the nilpotent matrix of generalizedfuzzy matrices is valuable. In part3, the nilpotency for powers of matrices is concerned,and the nilpotent index is completed over path algebras. Some results established heregeneralize the corresponding results in the previous literatures. In addition, the authorpresent the definition of the cycle controllable matrix, which is an interesting inductionand generalization of studied nilpotent matrices. Furthermore, the paper gives some prop-erties for cycle controllable matrices, which include the expressions of cycle controllablematrices in terms of graph-theory. The results proposed here improve some known resultsabout the expressions of nilpotent matrices.
Part4is about the studies of the determinant and invertibility of generalized fuzzymatrices. In this paper, the author firstly extend some important results on the determi-nant of generalized fuzzy matrices to the matrices over a path algebra. In the end, thisdissertation perfects the invertibility of a matrix with a period over a path algebra.
Part5is to study the eigen problem of generalized fuzzy matrices. In completeand completely distributive lattices, the aim of the present paper is to characterize theeigenspace of each eigenvalue by using the simulated basis, so as to improve the correla-tive theory of eigen problem.
第一部分讨论了一般广义模糊矩阵幂序列性质。首先对一般广义模糊矩阵的幂的收敛性进行了探讨，其中一些矩阵幂的收敛指数比以往研究的收敛指数要低，起到了降阶的作用。接着利用伴随矩阵研究了广义模糊矩阵幂序列性质，其中所得的一些成果是对以往相应成果的推广。最后引进特殊算子后，研究了广义模糊矩阵幂的组合性质，从而将以往相关的重要结论推广到了斜代数上。
第二部分研究了广义模糊矩阵的传递性。传递矩阵是广义模糊矩阵中的一类重要的典型阵。传递矩阵代表了传递关系，而传递关系在聚类分析、信息检索、优化等领域都有很重要的应用。广义模糊矩阵的传递闭包也能广泛应用于交通网络、模糊控制、经济管理、模糊聚类分析、计算机科学技术及稳定性。故广义模糊矩阵的传递性引起了很多学者的重视。第二部分首先在分配格上，定义了几类特殊的传递阵，并研究了其性质，进而丰富了典型传递阵的研究内容。接着讨论了广义传递阵幂的收敛性，其中一些主要结论是对以往相关结论的完善和改进，并将有关成果推广到了斜代数及路代数上。然后在分配格及斜代数上完善了广义模糊矩阵的传递闭包的性质及传递闭包的算法，并将有关成果推广到了路代数上。最后将广义模糊传递阵的构造及规则形状问题中所得的一些重要结论推广到了斜代数及路代数上。
第三部分研究了广义模糊矩阵的幂零性。幂零矩阵是广义模糊矩阵中的另一类重要的典型阵。幂零矩阵可用来表示非循环图，而非循环图能用来代表一致性，且更能表示优先关系，故对广义模糊矩阵中的幂零阵的研究是很有价值的。第三部分一方面在路代数上研究了矩阵的幂零性，并完善了对幂零指数的研究，其中所得的一些结论是对以往相应成果的推广。另外，通过对以往矩阵幂零性的归纳总结，定义了一类特殊幂零阵――圈可控阵，并研究了圈可控阵的性质，其中包括用图论的知识去研究圈可控阵的表达式，所取得的一些成果改进了关于幂零阵表达式的已有结论。
第四部分研究了广义模糊矩阵的行列式及可逆性问题。首先将广义模糊矩阵行列式的相关重要成果推广到路代数上。最后完善了路代数上带有周期的矩阵的可逆性问题的研究。
第五部分研究了广义模糊矩阵的特征问题。在完全完备分配格上，利用拟基刻画了特征值的特征向量子格，从而完善了特征问题在格上的研究。
Generalized fuzzy matrix is an important part in the theory of fuzzy mathematics anduseful in diverse areas such as design of switching circuits, automata theory, graph theory,information systems, clustering, complex system modeling, dynamical programming anddecision theory. Consequently, generalized fuzzy matrix is one of the most focused prob-lems in fuzzy mathematics. This doctoral dissertation studies several problems which areusually important in generalized fuzzy matrices. This paper consists of five parts withseven chapters:
Part1is to discuss the properties of powers of commonly generalized fuzzy matrices.Firstly, the convergence for powers of commonly generalized fuzzy matrices is discussed.The convergent index of some studied matrices is smaller than previous considered index.Therefore these matrices here play the role of reducing order. Next some properties ofpowers of generalized fuzzy matrices are established through adjoint matrix, and someresults obtained generalize and develop the corresponding results in previous literatures.At last, by introducing special operator, some properties of compositions of generalizedfuzzy matrices are given, thus some corresponding important results are extended to in-clines.
Part2is devoted to investigating the transitivity of generalized fuzzy matrices. Tran-sitive matrix is an important type of generalized matrices which represent transitive re-lation. Transitive relation plays an important role in clustering, information retrieval,preference, and so on. The transitive closure of a generalized matrix is also widely usedfor traffic network, fuzzy control, economic management, fuzzy clustering analysis, com-puter technology and reliability. Therefore, the transitivity problems of generalized matri-ces have been discussed by many authors. In part two, several special kinds of transitivematrices are firstly defined over distributive lattices, and some properties of these transi-tive matrices are obtained. These types of transitive matrices aim at enrich research ontransitivity of lattice matrices. Then, the convergence for powers of generalized transitivematrices is considered. Some results contained here perfect and improve the correspond-ing results in previous literatures. And, furthermore, these obtained results are generalizedto inclines and path algebras. Next, some properties and the algorithm of the transitive closure of generalized fuzzy matrices are perfected over distributive lattices and inclines.Also, some results established are extended to path algebras. Finally, some properties ofcompositions and the issue of the canonical form of generalized transitive matrices areextended to inclines and path algebras.
Part3is to study the nilpotency for powers of generalized fuzzy matrices. Nilpotentmatrices are another important typical matrices of generalized matrices which representacyclic graphs. Acyclic graphs are used to represent consistent systems and are importantin the representation of precedence relations. Then, the nilpotent matrix of generalizedfuzzy matrices is valuable. In part3, the nilpotency for powers of matrices is concerned,and the nilpotent index is completed over path algebras. Some results established heregeneralize the corresponding results in the previous literatures. In addition, the authorpresent the definition of the cycle controllable matrix, which is an interesting inductionand generalization of studied nilpotent matrices. Furthermore, the paper gives some prop-erties for cycle controllable matrices, which include the expressions of cycle controllablematrices in terms of graph-theory. The results proposed here improve some known resultsabout the expressions of nilpotent matrices.
Part4is about the studies of the determinant and invertibility of generalized fuzzymatrices. In this paper, the author firstly extend some important results on the determi-nant of generalized fuzzy matrices to the matrices over a path algebra. In the end, thisdissertation perfects the invertibility of a matrix with a period over a path algebra.
Part5is to study the eigen problem of generalized fuzzy matrices. In completeand completely distributive lattices, the aim of the present paper is to characterize theeigenspace of each eigenvalue by using the simulated basis, so as to improve the correla-tive theory of eigen problem.
引文
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