The Gershgorin bounds for the eigenvalues can be derived by multiplying on the left with the eigenvector with the maximum and minimum eigenvalues,
$$ \begin{align} \sum_i \psi^{\rm max}_i M_{ij} =& \lambda_{\rm max} \psi^{\rm max}_j \nonumber\\ \sum_i \psi^{\rm min}_i M_{ij} =& \lambda_{\rm min} \psi^{\rm min}_j \tag{12} \end{align} $$