Matrix scaling: A geometric proof of Sinkhornʹs theorem

Matrix scaling problems have been extensively studied since Sinkhorn established in 1964 the following result: Any positive square matrix of order n is diagonally equivalent to a unique doubly stochastic matrix of order n, and the diagonal matrices which take part in the equivalence are unique up to scalar factors. We present a new elementary proof of the existence part of Sinkhornʹs theorem which is based on well-known geometrical interpretations of doubly stochastic matrices and left and right multiplication by diagonal matrices.