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We give an effective bound of the joint spectral radius $\rho(\Sigma)$ for afinite set $\Sigma$ of nonnegative matrices: For every $n$,  \[  \sqrt[n]{\left(\frac{V}{UD}\right)^{D} \max_C \max_{i,j\in C}\max_{A_1,\dots,A_n\in\Sigma}(A_1\dots A_n)_{i,j}} \le \rho(\Sigma)  \le \sqrt[n]{D \max_C \max_{i,j\in C} \max_{A_1,\dots,A_n\in\Sigma}(A_1\dotsA_n)_{i,j}},  \] where $D\times D$ is the dimension of the matrices, $U,V$ are respectivelythe largest entry and the smallest entry over all the positive entries of thematrices in $\Sigma$, and $C$ is taken over all strongly connected componentsin the dependency graph. The dependency graph is a directed graph where thevertices are the dimensions and there is an edge from $i$ to $j$ if and only if$A_{i,j}\ne 0$ for some matrix $A\in\Sigma$.  Furthermore, a bound on the norm is also given: If $\rho(\Sigma)>0$ thenthere exist a nonnegative integer $r$ and two positive numbers $\alpha,\beta$so that for every $n$,  \[  \alpha n^r{\rho(\Sigma)}^n \le \max_{A_1,\dots,A_n\in\Sigma} \|A_1\dots A_n\|\le \beta n^r{\rho(\Sigma)}^n.  \]  Corollaries of the approach include a simple proof for the joint spectraltheorem for finite sets of nonnegative matrices and the convergence rate ofsome sequences. The method in use is mostly based on Fekete's lemma, for bothsubmultiplicative and supermultiplicative sequences.

On the joint spectral radius of nonnegative matrices