Some New Inequalities for Eigenvalues of the Hadamard Product and the Fan Product of Matrices

Abstract

Let A and B be nonnegative matrices. A new upper bound on the spectral radius ρ(A◦B) is obtained. Meanwhile, a new lower bound on the smallest eigenvalue q(A B) for the Fan product, and a new lower bound on the minimum eigenvalue q(B ◦A−1) for the Hadamard product of B and A−1 of two nonsingular M -matrices A and B are given. Some results of comparison are also given in theory. To illustrate our results, numerical examples are considered. Keywords—Hadamard product, Fan product; nonnegative matrix, M -matrix, Spectral radius, Minimum eigenvalue, 1-path cover.