  The primary aim of this paper is to establish bounds on the joint spectral
radius for a finite set of nonnegative matrices based on their diagonal
elements. The efficacy of this approach is evaluated in comparison to existing
and related results in the field. In particular, let $\Sigma$ be any finite set
of $D\times D$ nonnegative matrices with the largest value $U$ and the smallest
value $V$ over all positive entries. For each $i=1,\dots,D$, let $m_i$ be any
number so that there exist $A_1,\dots,A_{m_i}\in\Sigma$ satisfying $(A_1\dots
A_{m_i})_{i,i} > 0$, or let $m_i=1$ if there are no such matrices. We prove
that the joint spectral radius $\rho(\Sigma)$ is bounded by \[
  \max_i \sqrt[m_i]{\max_{A_1,\dots,A_{m_i}\in\Sigma} (A_1\dots A_{m_i})_{i,i}}
\le \rho(\Sigma) \le \max_i \sqrt[m_i]{\left(\frac{UD}{V}\right)^{3D^2}
\max_{A_1,\dots,A_{m_i}\in\Sigma} (A_1\dots A_{m_i})_{i,i}}. \]
