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A Note Concerning Simultaneous Integral Equations
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 855-861

Voir la notice de l'article provenant de la source Cambridge University Press

In (2) Sinkhorn showed that corresponding to each positive n × n matrix A (i.e., every aij > 0) is a unique doubly stochastic matrix of the form D1AD2 , where each Dk is a diagonal matrix with a positive main diagonal. The Dk themselves are unique up to a scalar multiple. In (3) the result was extended to show that D1AD2 could be made to have arbitrarypositive row and column sums (with the reservation, of course, that the sum of the row sums equal the sum of the column sums) where A need no longer be square. 
Knopp, Paul; Sinkhorn, Richard. A Note Concerning Simultaneous Integral Equations. Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 855-861. doi: 10.4153/CJM-1968-082-4
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[1] 1. Hobby, Charles and Pyke, Ronald, Doubly stochastic operators obtained from positive operators, Pacific J. Math. 15 (1965), 153–157. Google Scholar

[2] 2. Sinkhorn, Richard, A relationship between arbitrary positive matrices and doubly stochastic matrices, Ann. Math. Statist. 35 (1964), 876–879. Google Scholar

[3] 3. Sinkhorn, Richard, Diagonal equivalence to matrices with prescribed row and column sums, Amer. Math. Monthly 74 (1967), 402–405. Google Scholar

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