# Which nonnegative matrices have exact nonnegative matrix factors of smaller dimensionality?

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?

• What do you mean by "exact" factors? The identity is the product of two transposition matrices, so obviously you exclude these. – David Handelman Oct 18 '14 at 23:30
• Sorry to have to ask, but can you please point out to me which matrices multiply to get the identity matrix? The NMF solver I'm using (in R from CRAN) doesn't find factors for identity. – rhombidodecahedron Oct 18 '14 at 23:48
• Put the ones in the lower left and upper right. Square it. Voilà, the identity. – David Handelman Oct 19 '14 at 0:23
• Oh! I'm sorry, I meant to specify that the condition that the dimensionality of $W$ and $H$ have to be smaller than $V$. I'll edit the post. – rhombidodecahedron Oct 19 '14 at 0:38