Fuzzy向量组的相关性与Fuzzy矩阵的秩

THE DEPENDENCE OF A SYSTEM OF FUZZY VECTORS AND THE RANK OF A FUZZY MATRIX

本文在[1]、[2]的基础上,首先建立了半环I=([0,1),a+b=max{a,b}ab=min{a,b})上的Fuzzy向量组相关性的若干命题,进而证明了Fuzzy向量组S以至它生成的子空间的任一生成集的极大独立组基数的唯一性。得到向量组S是有限生成子空间W的基的充要条件。从而,将半环I上的Fuzzy矩阵行(列)空间的基及其行(列)秩的计算转化为求该矩阵的行(列)向量组的极大独立组及其基数的问题。最后,利用Brouwer格上α算子的性质,得到判别Fuzzy矩阵的行(列)向量组相关性的充要条件及若干简易的充分条件。使Fuzzy矩阵秩的计算问题得到完满解决。

In this paper. the author, on the basis of [1]、[2], has firstly set up a number of propositions for the dependence of a system of fuzzy vectors over semi-ring I=([0,1], a+b=max{a,b}, ab=min{a,b}). The auther has further proved the unioueness of the cardinal number of the max imum independent group in a system of a fuzzy vector S up to any spanning set generated sub-space . Anecessary and sufficient condition that the system of vectors is the basis of finitely generated sud-space is obtained. Thereby, finding the basis of row (column) space of a given fuzzy matrix over semi-ring I and its row (column) rank has been turne into a problem as finding maximum independent group of the row (column) vectors of this matrix and its cardinal numder.Finally, by using the properties of operators a over Brouwer lattice, we obtained the necessary and sufficient conditions as well as some simple but sufficient conditions in discriminating the dependence for a system of row (column) vectors of a given matrix, providing a perfect solution in determing the rank of a fuzzy matrix.