双随机情形下的完全正矩阵

一个实方阵A称为双非负矩阵 ,若A为元素非负的半正定矩阵 ;A称为完全正的 ,若有 (不必方的 )n×m的非负矩阵B ,满足A=BB′.B的最小可能的列数m称为矩阵A的分解指数 .已知任何一个不可约双非负矩阵都具有双随机型 .因此一个双非负矩阵的完全正性等价于其对应的双随机矩阵的完全正性 .本文研究双随机矩阵的完全正 ,并给出了几类特殊的双随机矩阵为完全正的充要条件 .

Doubly Stochastical Matrices and Complete Positivity

A real square matrix A is called doubly nonnegative, if A is entrywise nonnegative and positive semidefinite as well; A is called completely positive, if there exists an (not necessarily square) n×m entrywise nonnegative matrix B , such that A = BB＇ . The least possible number m of columns of B is called the factorization index of A . It is known that every irreducible doubly nonnegative matrix has a doubly stochastic pattern. Therefore the complete positivity of a doubly nonnegative matrix can be reduced to the case for a doubly stochastic、positive semidefinite matrix. The paper concerns the complete positivity of doubly stochastic matrices. Also necessary and sufficient conditions for some special types of doubly stochastic matrices to be completely positive are given here.