Abstract

Motivated by a proof due to Fiedler of an inequality on the determinants of M-matrices and a recent paper by the authors, we now obtain various inequalities on permanents and determinants of nonsingular M-matrices. This is done by extending the multilinear considerations of Fiedler and, subsequently, of the authors, to fractional multilinear functionals on pairs of nonnegative matrices. Two examples of our results: For an n× n nonsingular M-matrix M (i) we give a sharp upper bound for det( M)+per( M), when M is a nonsingular M-matrix, (ii) we determine an upper bound on the relative error |per( M+ E)−per( M)|/|per( M)|, when M+ E is a certain componentwise perturbation of M.

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