Notes on completely positive matrices

Abstract Let A be a n × n symmetric matrix and in the closure of inverse M -matrices. Then A can be factored as A = BB T 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 ).

[1]  John Hannah,et al.  Nonnegative factorization of completely positive matrices , 1983 .

[2]  M. Kaykobad.  On Nonnegative Factorization of Matrices , 1987 .

[3]  Charles R. Johnson,et al.  Notes on inverse M-matrices , 1987 .

[4]  M. Hall,et al.  Copositive and completely positive quadratic forms , 1963, Mathematical Proceedings of the Cambridge Philosophical Society.

[5]  P. H. Diananda.  On non-negative forms in real variables some or all of which are non-negative , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  Charles R. Johnson.  Inverse M-matrices☆ , 1982 .