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I have a set of square matrices $A_i \in \mathbb{R}^{n \times n}$ for $i=1,\ldots,N$, such that $[A_i]_{jk} \ge 0$ for all $i$ and coordinates $j,k$.

If the largest eigenvalue of each $A_i$ is smaller than 1, is it going to be true also for $$\frac{1}{N} \sum_{i=1}^N A_i?$$

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    I think the answer is true, from triangle inequality. The spectral norm of each $A_i$ is smaller than 1, and therefore, the spectral norm of the sum is going less than $N$, dividing by $1/N$ yields the necessary result. Should have put more thought into it, thanks.2012-10-21

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