Proof of a conjecture concerning the Hadamard powers of inverse M-matrices

The main result of this paper is the following: if A = (a(ij)) is an inverse M-matrix, A((r)) = (a(ij)(r)) denotes the rth Hadamard power of A, then A((r)) is again an inverse M-matrix for any real number r > 1. This settles a conjecture proposed by Wang et al. [B.Y. Wang, X.P. Zhang, F.Z. Zhang, On the Hadamard product of inverse M-matrices, Linear Algebra Appl. 305 (2000) 23-31] affirmatively. Naturally, it shows that the conjecture of Neumann [M. Neumann, A conjecture concerning the Hadamard product of inverses of M-matrices, Linear Algebra Appl. 285 (1998) 277-290] is also valid. (c) 2007 Elsevier Inc. All rights reserved.