Inequalities for the minimum eigenvalue of M-matrices

Abstract

Let A be a nonsingular M-matrix, and τ (A) denote its minimum eigenvalue. Shivakumar et al. [SIAM J. Matrix Anal. Appl., 17(2):298-312, 1996] presented some bounds of τ (A) when A is a weakly chained diagonally dominant M-matrix. The present paper establishes some new bounds of τ (A) for a general nonsingular M-matrix A. Numerical examples show that the results obtained are an improvement over some known results in certain cases.

New inequalities for the minimum eigenvalue ofM-matrices

“…In this article, we establish some new bounds of (A) for a general nonsingular M-matrix A. These new bounds improve the corresponding results in [6].…”

“…This example shows that the bound in Theorem 3.1 is sharp and is sharper than that in Lemma 2.6 of [6]. …”

“…We now give a comparison between the upper bound in Theorem 3.1 and the upper bound in Lemma 2.7 (Lemma 2.6 of [6]). …”

“…Estimating the bounds for (A) is an interesting subject in matrix theory and various refined bounds can be found in [5,6]. Shivakumar et al [5] …”

New bounds for the minimum eigenvalue of M-matrices

“…Subsequently, Tian et al [7] provided a lower bound for .A/ by using the spectral radius of the Jacobi iterative matrix…”