Applications of an inequality in information theory to matrices

Abstract

If x and y are nonnegative vectors of order n, and if Σn_i=₁x_i=Σn_i=₁y_i, then a well-known inequality asserts that ∏n_i=₁xx_ii≥∏n_i=₁yx_ii , with equality if and only if x = y. In this paper various situations are considered where this inequality can be applied to obtain inequalities concerning nonnegative matrices. In particular, inequalities are proved concerning nonnegative matrices which are diagonally equivalent, permanents and functions more general than the permanent, and diagonal products and circuit products of nonnegative matrices.