On an inequality for the Hadamard product of an M-matrix or an H-matrix and its inverse

Abstract

Let A be an n× n matrix, q( A)=min{| λ|: λ∈ σ( A)} and σ( A) denote the spectrum of A. From Fiedler and Markham [Linear Algebra Appl. 101 (1988) 1], Song [Linear Algebra Appl. 305 (2000) 99] and Yong [Linear Algebra Appl. 320 (2000) 167], for the Hadamard products of n× n M-matrices and their inverses, the infimum of q( A∘ A −1) is 2/ n. In this paper the following results are presented: if q( A k ∘ A k −1) tends to the infimum 2/ n for n× n ( n>2) M-matrices A k , k=1,2,…, then the spectral radius ρ( J k ) of the Jacobi iterative matrix of A k tends to 1. That is, if q( A∘ A −1) is close to 2/ n, then ρ( J) is close to 1; and another lower bound is given for A being an n× n M-matrix, q(A∘A −1)⩾ max 1−ρ(J) 2, 1+ρ(J) 1 n+2 1+(n−1)ρ(J) 1 n+2 , where ρ( J) is the spectral radius of the Jacobi iterative matrix of A. Furthermore, if A is an H-matrix, then q( A∘ A −1)⩾(1− ρ( J m( A) ) 2)/(1+ ρ( J m( A) ) 2), where ρ( J m( A) ) is the spectral radius of the Jacobi iterative matrix of the comparison matrix m( A).