Low rank approximation of a matrix Consider a $n\times m$ matrix $\mathbf{A}=\mathbf{U}\Sigma\mathbf{V}^T$. Also assume that all the entries of $\mathbf{A}$ are positive i-e $a_{ij}>0 \hspace{2mm} \forall \hspace{2mm} 1\leq i \leq n,1\leq j \leq m$.
My question is can we say some definite about the sign of entries of its rank-r approximation $\mathbf{A_r}$
\begin{equation}
\mathbf{A_r}=\mathbf{U_r}\Sigma_r\mathbf{V_r^T}
\end{equation}
where 


*

*$\mathbf{U_r}\rightarrow$ first r left singular vectors 

*$\Sigma_r\rightarrow$ diagonal matrix of first r-singular vaules 

*$\mathbf{V_r}\rightarrow$ first r right singular vectors
In other words, if we start from a non-negative matrix $\mathbf{A}$ then its SVD will also be non-negative.

Try e.g. the rank-$2$ approximation of 
$$ \pmatrix{1 & 0 & 1\cr 1 & 1 & 0\cr 0 & 0 & 1\cr} $$
EDIT: (for an example with nonnegative entries).  If you want strictly positive entries, change $0$ to  $\epsilon > 0$ for sufficiently small $\epsilon$: $\epsilon = 0.1$ will do.
