Summary: A note on "5 × 5 Completely positive matrices"
and Kurt Anstreicher
October 2009; revised April 2010
In their paper "5×5 Completely positive matrices," Berman and Xu [BX04] attempt
to characterize which 5 × 5 doubly nonnegative matrices are also completely positive.
Most of the analysis in [BX04] concerns a doubly nonnegative matrix A that has at least
one off-diagonal zero component. To handle the case where A is componentwise strictly
positive, Berman and Xu utilize an "edge-deletion" transformation of A that results
in a matrix A having an off-diagonal zero. Berman and Xu claim that A is completely
positive if and only if there is such an edge-deleted matrix A that is also completely
positive. We show that this claim is false. We also show that two conjectures made in
[BX04] regarding 5 × 5 completely positive matrices are both false.
Keywords: completely positive matrices, doubly nonnegative matrices, copositive matrices
MSC: 15A48, 15A21, 90C22
A real symmetric n×n matrix A is completely positive if there exists an entrywise nonnegative
n×r matrix B such that A = BBT
. We denote CPn as the cone of n×n completely positive