Diagonal equivalence to matrices with prescribed row and column sums. II

TLDR: In this article, it was shown that if A is a nonnegative fully indecomposable matrix, i.e. A contains no s x (n s) zero submatrix, then there exists a doubly stochastic matrix of the form D 1AD2 where DI and D2 are diagonal matrices with positive main diagonals.

Abstract: Let A be a nonnegative m x n matrix and let r= (rl, ** I rm) and c = (c1' * c ) be positive vectors such that ?m r. = zn. c1. It is well known that if there exists a nonnegative z =1 z ]-=1 m x n matrix B with the same zero pattern as A having the ith row sum ri and jth column sum c,, there exist diagonal matrices D1 and D with positive main diagonals such that D1AD2 has ith row sum r. and jth column sum cjHowever the known proofs are at best cumbersome. It is shown here that this result can be obtained by considering the minimum of a certain real-valued function of n positive variables. It has been shown originally by Sinkhorn and Knopp [81 and Brualdi, Parter, and Schneider [31 that if A is a nonnegative fully indecomposable matrix, i.e. A contains no s x (n s) zero submatrix, then there exists a doubly stochastic matrix of the form D 1AD2 where DI and D2 are diagonal matrices with positive main diagonals. Later Djokovic? [41, and independently, London [5], proved the same theorem by considering the minimum of