The nonnegative matrix 
$V = \left( \begin{array}{cc}
1 & 1 \\
1 & 1 \end{array} \right)$
has nonnegative matrix factors $W = \left( \begin{array}{c} 1 \\ 1 \end{array} \right)$ and $H = \left( \begin{array}{cc} 1 & 1 \end{array} \right)$, i.e. $V=WH$. The identity matrix $\left( \begin{array}{cc}
1 & 0 \\
0 & 1 \end{array} \right)$, on the other hand, has no exact nonnegative factors of smaller dimensionality. 


In general, which nonnegative matrices have nonnegative factors, and which do not? 



Which nonnegative matrices have exact nonnegative matrix factors of smaller dimensionality?
Given the notation in the question, I think the actual concept that the OP is looking for is: [Nonnegative rank](http://en.wikipedia.org/wiki/Nonnegative_rank_%28linear_algebra%29) 


(Also, googling for "Nonnegative rank" will bring up several useful links).


Finally, this recent paper [on Nonnegative matrix factorization](http://www.ece.umn.edu/~nikos/06630130.pdf) covers what seems to be known about the desired (typically **low rank**) factorizations.


