# 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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