摘要
A real matrix A of order n is called doubly nonnegative (denoting A∈DPn) if it is non- negative entrywise and positive semidefinite as well. A is called completely positive (denoting A∈CPn) if there exist k nonnegative column vectors b1,b2, …,bk∈Rn for some nonnegative integer k such that A=b1b′1+…+bkb′k. The smallest such number k is called the factorization index of A and is denoted by φ(A). This paper gives an effective criterion for any doubly nonnegative matrix A of order 5 whose associated graph is isomorphic neither to K5(the complete graph) nor to K5-e (a subgraph of K5 obtained by cutting off an edge from it) to be completely positive.
A real matrix A of order n is called doubly nonnegative (denoting A∈DPn) if it is non- negative entrywise and positive semidefinite as well. A is called completely positive (denoting A∈CPn) if there exist k nonnegative column vectors b1,b2, …,bk∈Rn for some nonnegative integer k such that A=b1b′1+…+bkb′k. The smallest such number k is called the factorization index of A and is denoted by φ(A). This paper gives an effective criterion for any doubly nonnegative matrix A of order 5 whose associated graph is isomorphic neither to K5(the complete graph) nor to K5-e (a subgraph of K5 obtained by cutting off an edge from it) to be completely positive.
基金
the fund of Anhui Education Committee.
