I believe the following statement should be true but somehow I don't see an argument:

For every integer $d>1$ there exists a constant $C=C(d)>1$ such that whenever $A$ is a $d \times d$ matrix with all entries integers $>0$, then

$$ \frac{1}{C} \|A\| \le \sigma(A) \le C \|A\|.$$

Here $\sigma(A)$ is the spectral radius of $A$ (which in this case is the Perron-Frobenius eigenvalue of $A$), and $\|A\|=\sup_{\|v\|=1} \|Av\|$ is the operator norm of $A$.

Does anyone see how to prove (or disprove) this statement?

Many thanks,

Ilya.

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