Notes on completely positive matrices

Let A be a n × n symmetric matrix and in the closure of inverse M-matrices. Then A can be factored as A = BBT for some nonnegative lower triangular n × n matrix B, and cp-rank A n. If A is a positive semidefinite (0, 1) matrix, then A is completely positive and cp-rank A = rank A; if A is a nonnegative symmetric H-matrix, then A is completely positive and cp-rank A n(n + 1)/2 - N - (n - μ), where μ is the number of connected components of the graph G(A).