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A Note Concerning Simultaneous Integral Equations

Published online by Cambridge University Press:  20 November 2018

Paul Knopp and
Richard Sinkhorn
Paul Knopp
Affiliation:
University of Houston, Houston, Texas
Richard Sinkhorn
Affiliation:
University of Houston, Houston, Texas
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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.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968, pp. 855 - 861
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

1.

This author was partially supported by NASA grant NGR-44-005-037.

2.

This author was partially supported by NASA grants NGR-44-005-037 and NGR-44-005-021.

References

1. Hobby, Charles and Pyke, Ronald, Doubly stochastic operators obtained from positive operators, Pacific J. Math. 15 (1965), 153–157.Google Scholar
2. Sinkhorn, Richard, A relationship between arbitrary positive matrices and doubly stochastic matrices, Ann. Math. Statist. 35 (1964), 876–879.Google Scholar
3. Sinkhorn, Richard, Diagonal equivalence to matrices with prescribed row and column sums, Amer. Math. Monthly 74 (1967), 402–405.Google Scholar