New inequalities for the minimum eigenvalue ofM-matrices

Abstract

Some new inequalities for the minimum eigenvalue of M-matrices are established. These inequalities improve the results in [G. Tian and T. Huang, Inequalities for the minimum eigenvalue of M-matrices, Electr.

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

“…Hence inequalities (1), ( 3), ( 5), ( 8), ( 12) and (15) can not be used to estimate the lower bounds of τ (A). Numerical results obtained from Theorem 3.1 of [4], Theorem 4.1 of [5], Theorem 4 of [6], Theorem 3 of [7] and Theorem 2, i.e., inequalities (2), ( 4), ( 6), ( 7) and (10) are given in Table 1 for the total number of iterations T = 10. In fact, τ (A) = 0.8873.…”

“…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).…”

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

“…Estimating the bounds of the minimum eigenvalue ( ) of an -matrix and its corresponding eigenvector is an interesting subject in matrix theory and has important applications in many practical problems; see [4,[6][7][8]. In particular, these bounds are used to estimate upper bounds of the L 1 -norm of the solution ( ) for the following system of ordinary differential equations:…”

A new characterization of symmetric $H^+$-tensors and $M$-tensors

“…In particular, in Section 4, we consider the problem of computing the minimum H-eigenvalue of M -tensors (which generalize M -matrices), which play an important role in a wide range of interesting applications [see, 18, and the references therein]. In contrast with the problem of obtaining bounds on the minimum H-eigenvalue of M -tensors that has received significant attention in the literature [14,18,24,43]; here, we use our characterization of H + -tensors to compute H-eigenvalues of M -tensors by solving a power cone optimization problem (see Corollary 21). A comparison of the H-eigenvalues obtained in this way with bounds proposed in the literature is provided in Table 1.…”