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Continuous dependence on $ A$ in the $ D\sb{1}AD\sb{2}$ theorems


Author: Richard Sinkhorn
Journal: Proc. Amer. Math. Soc. 32 (1972), 395-398
MSC: Primary 15A48
DOI: https://doi.org/10.1090/S0002-9939-1972-0297792-4
MathSciNet review: 0297792
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0297792-4
Keywords: Doubly stochastic matrix, permanent, doubly stochastic pattern, doubly stochastic subpattern
Article copyright: © Copyright 1972 American Mathematical Society