Complete positivity of matrices of special form Original Research Article

Given that B,C2,…,Ck are positive semidefinite (PSD) n-by-n real matrices and B is entrywise nonnegative, we characterize (fully for n=2 and partially for ngreater-or-equal, slanted3) when B may be written as B=∑i=2kBi so that Bi and Bi−Ci are PSD and Bi is entrywise nonnegative, i=2,…,k. These characterizations are used to give conditions under which an entrywise nonnegative, PSD matrix A with a special block form can be written as A=BBT, in which B is entrywise nonnegative.