Scalings of matrices satisfying line-product constraints and generalizations

Abstract

A scaling of a nonnegative matrix A is a matrix having the form A′ = UAV where U and V are square diagonal matrices which have positive diagonal elements. If the matrix A is square and V = U −1, we call A' a symmetric scaling of A. We consider two problems: the first concerns the identification of a scaling of a given nonnegative matrix with prescribed row and column products; the second concerns the problem of finding a symmetric scaling of a given nonnegative square matrix whose row products equal the corresponding column products. For each of these two scaling problems we characterize the solutions in terms of a nonlinear convex optimization problem, we use the characterization to demonstrate that feasibility of either scaling problem is equivalent to the existence of a matrix satisfying the target property and having the same pattern as the given matrix, we establish uniqueness of solutions to either scaling problem whenever it is feasible, and we develop algorithms for computing the desired (unique) scalings in the cases where the problem are feasible.