5 x 5 completely positive matrices

We study the problem of determining whether a given 5 x 5 matrix is completely positive, by investigating the number of negative entries of a Schur complement of A. We show that if this number is not 4, then A is completely positive, and present some sufficient conditions for A to be cp when this number is 4. We also show that the complete positivity of an elementwise positive doubly nonnegative 5 x 5 matrix is equivalent to the complete positivity of a related 5 x 5 doubly nonnegative matrix that is not elementwise positive. (C) 2004 Elsevier Inc. All rights reserved.