Is the following Doubly Non-negative matrix Completely Positive:


$\frac{1}{6}\begin{bmatrix} 2 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\0 & 2 & 0 & 1 & 0 & 1 & 1 & 0 & 1 \\0 & 0 & 2 & 1 & 1 & 0 & 1 & 1 & 0 \\0 & 1 & 1 & 2 & 0 & 0 & 0 & 1 & 1 \\1 & 0 & 1 & 0 & 2 & 0 & 1 & 0 & 1 \\1 & 1 & 0 & 0 & 0 & 2 & 1 & 1 & 0 \\0 & 1 & 1 & 0 & 1 & 1 & 2 & 0 & 0\\1 & 0 & 1 & 1 & 0 & 1 & 0 & 2 & 0\\1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 2\end{bmatrix}$



Is the following DNN matrix CP?
Edit: I wanted the following matrix to be $W$. Robert Israel suggested
I call it $W^T$ instead. I defer to his years of experience and the fact
that it gives a better answer to the problem. End Edit.


For 6 times the given matrix, I nominate the following candidate for $W^T$


$$\begin{bmatrix}
1 & & & & 1 & & & & 1 \\\\
1 & & & & & 1 & & 1 & \\\\
 & 1 & & 1 & & & & & 1 \\\\
 & 1 & & & & 1 & 1 & & \\\\
 & & 1 & 1 & & & & 1 & \\\\
 & & 1 & & 1 & & 1 & & 
\end{bmatrix}
$$


Does this help?


Gerhard "Ask Me About Binary Matrices" Paseman, 2011.07.28


