Scaling of symmetric matrices by positive diagonal congruence
Charles R. Johnson, Robert Reams · Linear and Multilinear Algebra · 2008
We consider the problem of characterizing n-by-n real symmetric matrices A for which there is an n-by-n diagonal matrix D, with positive diagonal entries, so that DAD has row (and column) sums 1. Under certain conditions we provide necessary and sufficient conditions for the existence of a scaling for A, based upon both the positive definiteness of A on a cone lying in the nonnegative orthant and the semipositivity of A. This generalizes known results for strictly copositive matrices. Also given are (1) a condition sufficient for a unique scaling; (2) a characterization of those positive semidefinite matrices that are scalable; and (3) a new condition equivalent to strict copositivity, which we call total scalability. When A has positive entries, a simple iterative algorithm (different from Sinkhorn's) is given to calculate the unique scaling.