Inequalities for the Minimum Eigenvalue of Doubly Strictly Diagonally Dominant M-Matrices

Abstract

Let $A$ be a doubly strictly diagonally dominant $M$-matrix. Inequalities on upper and lower bounds for the entries of the inverse of $A$ are given. And some new inequalities on the lower bound for the minimal eigenvalue of $A$ and the corresponding eigenvector are presented to establish an upper bound for the ${\mathcal{L}}_1$-norm of the solution $x(t)$ for the linear differential system $dx/dt=-Ax(t)$, $x(0)=x^0>0$.