On Two-Sided Bounds Related to Weakly Diagonally Dominant M -Matrices with Application to Digital Circuit Dynamics
P. N. Shivakumar, J. J. Williams, Qiang John Ye, Corneliu A. Marinov · SIAM Journal on Matrix Analysis and Applications · 1996
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.