= Closedness of the completely positive cone
Use the <conic Carathéodory theorem> in the real space of <symmetric matrices>, of <dimension> $N=n(n+1)/2$. A convergent sequence $B_\ell$ in the <completely positive cone> has padded factorizations $B_\ell=\sum_{j=1}^N x_{\ell j}x_{\ell j}^T$ with nonnegative factors. Their total squared norms equal $\operatorname{tr}B_\ell$ and are uniformly bounded. A simultaneous convergent subsequence of the finite factor tuple gives $B=\sum_jx_jx_j^T$ with $x_j\geq0$. Thus the limit remains in the cone. The argument also bounds the required number of factors by $N$.
Back to article page
