Fuzzy相似矩阵方程X2=X与最优模糊等价矩阵的存在性

在文［１］基础上，对Ｆｕｚｚｙ 相似矩阵方程Ｘ２＝ Ｘ 的解的结构进行了进一步研究。首先提出了Ｆｕｚｚｙ 等价标准型的概念，为解的表达提供了工具； 第二，指出了相应标准分解过程的参数系的唯一性； 第三，在群作用观点下和平移等价类的意义下，讨论了解的类数计算公式； 第四，给出了解的分类表达式； 最后，证明了“失真”最小的模糊等价阵，即Ｆｕｚｚｙ 最优等价阵的存在性，为Ｆｕｚｚｙ 聚类提供了理论依据

Fuzzy Similar Matrix Equation X2=X and the Existence of Optimal Fuzzy Equivalence Matrix

The structure of the solution of fuzzy similar matrix equation X 2=X proposed by Li Hongxing is discussed in detail. First, the concept of fuzzy equivalent normal form is proposed in order to express the solutions. Second,the uniqueness of the parameter systems corresponding to standard resolution process is pointed out. Third, the formula about permutation equivalence classes and translation equivalence classes is given. Forth, the structure of the solution set is given. Finally, the existence of optimal fuzzy equivalence matrix is proved.