An $n$ by $n$ matrix $A$ is completely positive if $A$ has the factorization $A = V^T V$ for some $k$ by $n$ non-negative matrix $V$. $k$ can any integer. As a matter of fact, the space of completely positive matrix forms a convex cone inside the space of positive definite matrix. It's closed in addition, and thus $A +B$ is completely positive for any $A$, $B$ completely positive.
Now given two completely positive matrix $A$, $B$. Let $V^T V = A$, $W^T W = B$ denote their factorization so that both $V, W$ are $k$ by $n$ non-negative matrix. How can we find the factorization of $A + B = V^T V + W^T W$?
A naive solution is consider $C = \begin{bmatrix} V \\W \end{bmatrix}$. Then $C^T C=\begin{bmatrix} V^T &W^T \end{bmatrix} \begin{bmatrix} V \\W \end{bmatrix} = V^TV+W^TW$ is trivially the solution. However, I'm wondering if there is anyway to maintain $k$ small. It would be awesome if $C$ can be the size as $V$ and $W$. For now I'm thinking about finding an orthonormal set for the orthogonal complement of the image of $C$ and extend that to an orthogonal transformation $O$ in $\mathbb{R}^{2k}$ andchoose $OC$ as the desired factorization. But I'm not sure of its detail. Any suggestion will be much appreciated.