On Two-Sided Bounds Related to Weakly Diagonally Dominant M-Matrices with Application to Digital Circuit Dynamics

Let $A$ be a real weakly diagonally dominant $M$-matrix. We establish upper and lower bounds for the minimal eigenvalue of $A$, for its corresponding eigenvector, and for the entries of the inverse of $A$. Our results are applied to find meaningful two-sided bounds for both the $\ell_{1}$-norm and the weighted Perron-norm of the solution $x(t)$ to the linear differential system $\dot{x}=-Ax$, $x(0)=x_{0}>0$. These systems occur in a number of applications, including compartmental analysis and RC electrical circuits. A detailed analysis of a model for the transient behaviour of digital circuits is given to illustrate the theory.