Notes on inverse M-matrices

Abstract We show that a nonsingular p -by- p matrix A is an inverse M -matrix if and only if Q T AQ + D is an n -by- n inverse M -matrix whenever Q is a p -by- n nonnegative matrix with exactly one positive entry in each column and D is a positive diagonal matrix. This includes several facts about inverse M -matrices as special cases. We also show that a nonnegative n -by- n matrix A is in the closure of the inverse M -matrices if and only if A + D is nonsingular and ( A + D ) −1 ⩽ D −1 for each positive diagonal matrix D . Related results allow characterization of nilpotent matrices on the boundary of the inverse M -matrices.