Concerning nonnegative matrices and doubly stochastic matrices
Richard Sinkhorn, Paul J. Knopp · Pacific Journal of Mathematics · 1967
This paper is concerned with the condition for the convergence to a doubly stochastic limit of a sequence of matrices obtained from a nonnegative matrix A by alternately scaling the rows and columns of A and with the condition for the existence of diagonal matrices A and D 2 with positive main diagonals such that Ώ γ AΏ 2 is doubly stochastic.The result is the following.The sequence of matrices converges to a doubly stochastic limit if and only if the matrix A contains at least one positive diagonal.A necessary and sufficient condition that there exist diagonal matrices A and D 2 with positive main diagonals such that D 1 AD 2 is both doubly stochastic and the limit of the iteration is that AφO and each positive entry of A is contained in a positive diagonal.The form DιAD 2 is unique, and A and D 2 are unique up to a positive scalar multiple if and only if A is fully indecomposable.Sinkhorn [6] has shown that corresponding to each positive square matrix A there is a unique doubly stochastic matrix of the form