Lower Bound Estimation of the Minimum Eigenvalue of Hadamard Product of an M-Matrix and its Inverse

In this paper, we propose a lower bound sequence for τ(A ◦ A -1 ), the minimum eigenvalue of Hadamard product of an M -matrix and its inverse by constructing a vector x and a constant k such that A◦A -1 x ≥ kx. We prove the convergence of the lower bound sequence, which is an improvement on some of the existing results. By introducing a parameter and modifying constantly x and k, a more precise lower bound sequence is obtained. An example is given to show that the truth value of the minimum eigenvalue can be obtained by applying the new theorem to some kind of cyclic matrix. And several numerical experiments are given to demonstrate that the new bounds are sharper than some existing ones in most cases.