Hadamard functions of inverse M-matricese

Abstract

We prove that the class of generalized ultrametric matrices (GUM) is the largest class of bipotential matrices stable under Hadamard increasing functions. We also show that any power α ≥ 1, in the sense of Hadamard functions, of an inverse M-matrix is also inverse M-matrix. This was conjectured for α = 2 by Neumann in [Linear Algebra Appl., 285 (1998), pp. 277–290], and solved for integer α ≥ 1 by Chen in [Linear Algebra Appl., 381 (2004), pp. 53–60]. We study the class of filtered matrices, which include naturally the GUM matrices, and present some sufficient conditions for a filtered matrix to be a bipotential.