cp-rank $A \leq \lim_{n \to \infty} \inf$ cp-rank $A_n$.

by user8795   Last Updated October 20, 2019 05:20 AM

Let $A$ be a $n \times n$ completely positive matrix and cp-rank is the minimal number of summands in a rank $1$ representation of $A$, $A = \sum_{i=1}^{k}b_ib_i^T, b_i \geq 0$, where $b_i \geq 0$ means that $b_i$ has entries $\geq 0$. So here cp-rank of $A$ is $k$.

Suppose that $A_n$ is completely positive for every $n$, and that $$A = \lim_{n \to \infty} A_n.$$

Then, cp-rank $A \leq \lim_{n \to \infty} \inf$ cp-rank $A_n$.

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