关于完全正矩阵的非负分解  

On Nonnegative Factorizations of Completely Positive Matrices

作　　者：徐常青[1] 李世航[1] 

机构地区：[1]安徽大学数学系

出　　处：《安徽大学学报（自然科学版）》1999年第1期6-10,共5页Journal of Anhui University(Natural Science Edition)

摘　　要：Ｎ阶矩阵Ａ称为完全正的，如果Ａ能分解成Ａ＝ｂ１ｂｔ１＋…＋ｂｍｂｔｍ，其中ｂｊ（ｊ＝１，２，…，ｍ）为ｎ维非负向量。满足此式的最小的正整数ｍ称为Ａ的分解指数。本文证明了一个秩≤２的非负半正定矩阵一定为完全正，并给出了一个秩为３的非负半正定矩阵为完全正的一个充分条件。An n×n matrix A is called completely positive if A can be factored as A=b 1b t 1+…+b mb t m. Here b j(j=1,2,…,m) are nonnegative vectors in R n .The least such integer m is called the factorization index (or CP rank) of A. It is shown that a doubly nonnegative matrix A with rank (A)≤2 is always completely positive. And a sufficient condition for a doubly nonnegative matrix with its rank equals 3 to be completely positive is presented here.

关 键 词：完全自矩阵 分解指数 半正定矩阵 非负分解 矩阵 

分 类 号：O151.21[理学—数学]