A completely positive matrix is a realsquare matrix
that can be factorized as

where
stands for the transpose of and is any (not necessarily square) matrix with nonnegative elements. The least possible
number of columns () of is called the factorization index (or the cp-rank) of .
The study of complete positivity originated in inequality theory and quadratic forms
(Diananda 1962, Hall and Newman 1963).

There are two basic problems concerning complete positivity.

1. When is a given real matrix completely positive?

2. How can the cp-rank of be calculated?

These two fundamental problems still remains open. The applications of completely positive matrices can be found in block designs (Hall
and Newman 1963) and economic modelling (Gray and Wilson 1980).

Ando, T. Completely Positive Matrices. Lecture Notes. Sapporo, Japan, 1991.Berman, A. "Complete Positivity." Linear
Algebra Appl.107, 57-63, 1988.Berman, A. "Completely
Positive Graphs." In Combinatorial
and Graph-Theoretical Problems in Linear Algebra: Papers from the IMA Workshop held
in Minneapolis, Minnesota, November 11-15, 1991 (Ed. R. A. Brualdi,
S. Friedland, and V. Klee). New York: Springer-Verlag, pp. 229-233,
1991.Berman, A. and Shaked-Monderer, N. Completely
Positive Matrices. Singapore: World Scientific, 2003.Diananda,
P. H. "On Nonnegative Forms in Real Variables Some or All of Which Are
Nonnegative." Proc. Cambridge Philos. Soc.58, 17-25, 1962.Gray,
L. J. and Wilson, D. G. "Nonnegative Factorization of Positive Semidefinite
Nonnegative Matrices." Linear Algebra Appl. Appl.31, 119-127,
1980.Hall, M. Jr. and Newman, M. "Copositive and Completely Positive
Quadratic Forms." Proc. Cambridge Philos. Soc.59, 329-339, 1963.