A note on “5×5 Completely positive matrices”

In their paper “5×5 Completely positive matrices”, Berman and Xu (2004) [3], attempt to characterize which 5×5 doubly nonnegative matrices are also completely positive. Most of the analysis in [3], 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 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 that is also completely positive. We show that this claim is false. We also show that two conjectures made in [3] regarding 5×5 completely positive matrices are both false.