Continuous dependence on $A$ in the $D_{1}AD_{2}$ theorems

Abstract:

It has been shown by Sinkhorn and Knopp and others that if A is a nonnegative square matrix such that there exists a doubly stochastic matrix B with the same zero pattern as A, then there exists a unique doubly stochastic matrix of the form ${D_1}A{D_2}$ where ${D_1}$ and ${D_2}$ are diagonal matrices with positive main diagonals. Sinkhorn and Knopp have also shown that if A has at least one positive diagonal, then the sequence of matrices obtained by alternately normalizing the row and column sums of A will converge to a doubly stochastic limit. It is the intent of this paper to show that ${D_1}A{D_2}$ and/or the limit of this iteration, when either exists, is continuously dependent upon the matrix A.