Generalized ultrametric matrices — a class of inverse M-matrices Original Research Article

Recently, Marti´nez, Michon, and San Marti´n introduced the new class of (symmetric)strictly ultrametric matrices. They proved that the inverse of a strictly ultrametric matrix is a strictly row and strictly column diagonally dominant Stieltjes matrix. Here, we generalize their result by introducing a class of nonsymmetric matrices, calledgeneralized ultrametric matrices. We give a necessary and sufficient condition for the regularity of these matrices and prove that the inverse of a nonsingular generalized ultrametric matrix is a row and column diagonally dominant M-matrix. We establish that a nonnegative matrix is a generalized ultrametric matrix if and only if the matrix is a certain sum of at most rank-two matrices. Moreover, we give a characterization of generalized ultrametric matrices, based on weighted trees. The entries of generalized ultrametric matrices then arise as certain “distances” between the leaves and the root of the tree.