Complete positivity of matrices of special form

Given that B, C-2,..., C-k are positive semidefinite (PSD) n-by-n real matr ices and B is entrywise nonnegative, we characterize (fully for n = 2 and p artially for n greater than or equal to 3) when B may be written as B = Sig ma (k)(i=2) B-i so that B-i and B-i - C-i are PSD and B-i is entrywise nonn egative, i = 2,...,k. These characterizations are used to give conditions u nder which an entrywise nonnegative, PSD matrix A with a special block form can be written as A = BBT, in which B is entrywise nonnegative. (C) 2001 E lsevier Science Inc. All rights reserved.