HADAMARD FUNCTIONS OF INVERSE M-MATRICES

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 alpha >= 1, in the sense of Hadamard functions, of an inverse M- matrix is also inverse M- matrix. This was conjectured for alpha = 2 by Neumann in [Linear Algebra Appl., 285 (1998), pp. 277-290], and solved for integer alpha >= 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.