Inverse M-Matrix Inequalities and Generalized Ultrametric Matrices

Abstract We use weighted directed graphs to introduce a class of nonnegative matrices which, under a simple condition, are inverse M-matrices. We call our class the generalized ultrametic matrices, since it contains the class of (symmetric) ultrametric matrices and some unsymmetric matrices. We show that a generalized ultrametric matrix is the inverse of a row and column diagonally dominant M-matrix if and only if it contains no zero row and no two of its rows are identical. This theorem generalizes the known result that a (symmetric) strictly ultrametric matrix is the inverse of a strictly diagonally dominant M-matrix. We also present inequalities and conditions for equality among the entries of the inverse of a row diagonally dominant M-matrix. Some of these inequalities and conditions for equality generalize results of Stieltjes on inverses of symmetric diagonally dominant M-matrices.

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