Characterizations of optimal scalings of matrices

A scaling of a non-negative, square matrix A .. 0 is a matrix of the form DAD-I, where Dis a non~negatjve. non-singular, diagonal, square matrix. For a non.-negative, rectangular matrix B .. 0 we define a scaling to be a matrix CBEI where C and E are non-negative, non-singular, diagonal, square matrices of the corresponding dimension. (For square matrices the latter definition allows more scalings.) A measure of the goodness of a scaling X is the maximal ratio of non-zero elements of X. We characterize the minimal value of this measure over the set of all scalings of a given matrix. This is obtained in terms of cyclic products .associated with a graph corresponding to the matrix. Our analysis is based on converting the scaling problem into a linear program. We then characterize the extreme points of the polytope which occurs in the linear program.