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

Let A be an n×n matrix, q(A)=min{λ:λset membership, variantσ(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(Aring operatorA−1) is 2/n. In this paper the following results are presented: if q(Akring operatorAk−1) tends to the infimum 2/n for n×n (n>2) M-matrices Ak, k=1,2,…, then the spectral radius ρ(Jk) of the Jacobi iterative matrix of Ak tends to 1. That is, if q(Aring operatorA−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,imagewhere ρ(J) is the spectral radius of the Jacobi iterative matrix of A. Furthermore, if A is an H-matrix, then q(Aring operatorA−1)greater-or-equal, slanted(1−ρ(Jm(A))2)/(1+ρ(Jm(A))2), where ρ(Jm(A)) is the spectral radius of the Jacobi iterative matrix of the comparison matrix m(A).