The Diagonal Equivalence of a Nonnegative Matrix to a Stochastic Matrix

In [l] R. Sinkhorn proved the following theorem: Let A be a positive square matrix. Then there exist two diagonal matrices D, , D, whose diagonal elements are positive such that D,AD, is doubly stochastic. Moreover, these matrices are uniquely determkd up to scalar factors. In addition, Sinkhorn gave some examples which show that the theorem fails for some nonnegative matrices A. Marcus and Newman [2] and Maxfield and Mint [3] also studied this problem. Recently M. V. Menon [4] gave a simplified proof of Sinkhorn’s theorem based on the Brouwer fixed-point theorem. Perfect and Mirsky [5] have shown that given a fully indecomposable matrix B, there exists a doubly stochastic nonnegative matrix with the same zero pattern. The operator T defined by Menon in his proof of Sinkhorn’s theorem is a homogeneous positive nonlinear operator. Morishima [6] and Thompson [7] have studied such operators in extending the theorems of Perron and Frobenius. We define the operator T in the case when A is an irreducible matrix with a positive main diagonal. Using the Wielandt approach to the PerronFrobenius theory, we show that T has some but not all of the properties of of an irreducible nonnegative matrix. Thus T has a unique eigenvector in the interior of the positive cone, but there may also exist eigenvectors on the boundary. We then deduce Sinkhorn’s theorem when A is a nonnegative fully indecomposable matrix. It is an easy matter to establish that this condition is essentially necessary (see [5] or 6.2). After completing our work we learned that Sinkhorn and Knopp [8] have also obtained a proof of the D,AD, theorem under the full indecomposability assumption. Their method of proof is quite different from ours, and we feel