Inequalities for the Minimum Eigenvalue of Doubly Strictly Diagonally DominantM-Matrices

Abstract

LetAbe a doubly strictly diagonally dominantM-matrix. Inequalities on upper and lower bounds for the entries of the inverse ofAare given. And some new inequalities on the lower bound for the minimal eigenvalue ofAand the corresponding eigenvector are presented to establish an upper bound for theL1-norm of the solutionx(t)for the linear differential systemdx/dt=-Ax(t),x(0)=x0>0.

Lower bounds of the minimum eigenvalue for $M$-matrices

“…Next, we use only the entries of A to give the lower bounds of τ (A). Numerical results obtained from Theorem 4.1 of [3], Corollary 3.4 of [4], Corollary 4.4 of [5], Corollary 1 of [7], Theorem 14 of [8], and Theorem 5, i.e., inequalities (1), ( 3), ( 5), ( 8) and (12) are given in Table 2 for the total number of iterations T = 10. In fact, τ (A) = 1.0987.…”

“…Estimating the bounds for the minimum eigenvalue of M -matrices is an interesting subject in matrix theory, it has important applications in many practical problems (see [3][4][5][6][7][8][9][10][11][12]) and various refined bounds can be found in [3][4][5][6][7][8]. Hence, it is necessary to estimate the bounds for τ (A).…”

A fast and stable test to check if a weakly diagonally dominant matrix is a nonsingular M-matrix

“…We list a few other interesting recent results concerning w.c.d.d. matrices and M-matrices here: [20,12,25,24,22,13,11,26].…”