Geodesic
Matrix Links, An Extremization Problem, and the Reduction of a Non-Negative Matrix to One With Prescribed Row and Column Sums
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 225-232

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The word matrix will, in this paper, always connote a matrix with non-negative elements, having in general m rows and n columns. In conformity with the definition in (4), two matrices will be said to have the same pattern if the entry in any row or column is zero or not according as the corresponding entry of the other is zero or not. The symbols ρi and σj will stand for the respective phrases “ith row-sum“ and “jth column-sum” of the matrix under consideration. r1 ... , rm; c1, ... , cn is a set of positive numbers. It will be said to be consistent for the pattern of an m × n matrix, if there exists a matrix of that pattern for which ρi = rj and σj = cj for all i and j. 
Menon, M. V. Matrix Links, An Extremization Problem, and the Reduction of a Non-Negative Matrix to One With Prescribed Row and Column Sums. Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 225-232. doi: 10.4153/CJM-1968-021-9
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[1] 1. Brualdi, Richard A., Parter, Seymour V., and Schneider, Hans, The diagonal equivalence of a non-negative matrix to a stochastic matrix, J. Math. Anal. AppL, 16, 1 (1966), 31–50. Google Scholar

[2] 2. Menon, M. V., Reduction of a matrix with positive elements to a doubly stochastic matrix, Proc. Amer. Math. Soc, 18, 2 (1967), 244–247. Google Scholar

[3] 3. Menon, M. V., Reduction of a matrix with positive elements to a doubly stochastic matrix, Tech. Rept. 658, Math. Res. Center, U.S. Army, Univ. of Wis., Madison, Wis. (1966). Google Scholar

[4] 4. Perfect, H. and Mirsky, L., The distribution of positive elements in doubly stochastic matrices, J. London Math. Soc, 40 (1965), 688–698. Google Scholar

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

[6] 6. Sinkhorn, R., Diagonal equivalence to matrices with prescribed row and column sums, Amer. Math. Monthly, 74, 4 (1967), 402–405. Google Scholar

[7] 7. Sinkhorn, R. and Knopp, P., Concerning non-negative matrices and doubly stochastic matrices, Pac. J. Math., 21, No. 2, 343–348. Google Scholar

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