Swarte的引理5.4的证明及推广

The Proof and Generalization of Swart's Lemma 5.4

D. Crystal, H. Greenberg, A. Kolem, W. Morris, A. Raian, R. Rardin和 M. Trick指出:从我们对Swart的文章的研究,确信变量公式是正确的,但Swart对关键性引理5.4的证明是错误的。在这里,我们给出引理5.4的一个严格证明,证明引理是完全正确的,并进一步推广引理5.4的结果。 Swart引理5.4;给定了一个n×n双随机矩阵D,它的所有元素是非负整数,并且每一行和与列和都是正整数K,则D能分解成置换矩阵的线性组合。推论:给定一个n×n双随机矩阵D,它的所有元素是非负整数,并且每一行和与列和都正实数K,则D能分解成置换矩阵的线性组合。

D. Crystal, H. Greenberg, A. Kolem, W. Morris, A. Rajah, R. Rardin and M. Trick pointed out: We believe from our study of Swart's paper that the various formulations are essentially correct. Ubfortunately, it is wrong that Swart gets the proof of critical lemma 5.4. We give here a rigorous proof of lemma 5.4 to show that the Swart's lemma 5.4 is correct. The generalization of this lemma is also given. Swar's lemma 5.4: Given an×n doubly stochastic matrix D. Dcan be decomposed into a linear combination of permutation matrices. When all of its elements are nonnegative integers and the sum of the rows and / or columns is some positive integer K.. Corollary: Given an×n doubly stochastic matrix D. D can be decomposed into a linear combination of permutation matrices. When all of its elements are nonnegative real numbers for which the sum of the rows and / or columns is some positive real number K.