A completely positive matrix is a real 
square 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).
