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?