Diagonal Matrix Scaling and Linear Programming

Abstract

A positive semidefinite symmetric matrix either has a nontrivial nonnegative zero or can be scaled by a positive diagonal matrix into a doubly quasi-stochastic matrix. This paper describes a simple path-following Newton algorithm of the complexity $O(\sqrt{n} L)$ iterations to either scale an $n \times n$ matrix or give a nontrivial nonnegative zero. The latter problem is well known to be equivalent to linear programming.