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 t...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications 1996-04, Vol.17 (2), p.298-312
Main Authors: Shivakumar, P. N., Williams, Joseph J., Ye, Qiang, Marinov, Corneliu A.
Format: Article
Language:English
Subjects:
Applied mathematics
Eigenvectors
Engineering research
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Summary: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.
ISSN:0895-4798
1095-7162
DOI:10.1137/S0895479894276370