Concerning nonnegative matrices and doubly stochastic matrices

Abstract

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 D2 with positive main diagonals such that ΏγAΏ2 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 D2 with positive main diagonals such that D1AD2 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 DιAD2 is unique, and A and D2 are unique up to a positive scalar multiple if and only if A is fully indecomposable.