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 D1 and
D2 with positive main diagonals such that D1AD2 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
D1 and D2 with positive main diagonals such that D1AD2 is both doubly
stochastic and the limit of the iteration is that A≠0 and each positive entry of A
is contained in a positive diagonal. The form D1AD2 is unique, and D1
and D2 are unique up to a positive scalar multiple if and only if A is fully
indecomposable.
