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A Relationship between Arbitrary Positive Matrices and Stochastic Matrices

Published online by Cambridge University Press:  20 November 2018

Richard Sinkhorn
Richard Sinkhorn*
Affiliation:
University of Houston
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The author (2) has shown that corresponding to each positive square matrix A (i.e. every aij > 0) is a unique doubly stochastic matrix of the form D1AD2, where the Di are diagonal matrices with positive diagonals. This doubly stochastic matrix can be obtained as the limit of the iteration defined by alternately normalizing the rows and columns of A.

In this paper, it is shown that with a sacrifice of one diagonal D it is still possible to obtain a stochastic matrix. Of course, it is necessary to modify the iteration somewhat. More precisely, it is shown that corresponding to each positive square matrix A is a unique stochastic matrix of the form DAD where D is a diagonal matrix with a positive diagonal. It is shown further how this stochastic matrix can be obtained as a limit to an iteration on A.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 303 - 306
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Marcus, M. and Newman, M., The permanent of a symmetric matrix, Amer. Math. Soc. Not., 8 (1961), 595.Google Scholar
2. Sinkhorn, R., A relationship between arbitrary positive matrices and doubly stochastic matrices, Ann. Math. Statist., 35 (1964), 876–879.Google Scholar