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Concerning nonnegative matrices and doubly stochastic matrices
Open accessAbstract
This paper is concerned with the condition for the convergence to a doubly stochastic limit of a sequence of matrices obtained from a nonnegative matrix A by alternately scaling the rows and columns of A and with the condition for the existence of diagonal matrices A and D 2 with positive main diagonals such that Ώ γ AΏ 2 is doubly stochastic.The result is the following.The sequence of matrices converges to a doubly stochastic limit if and only if the matrix A contains at least one positive diagonal.A necessary and sufficient condition that there exist diagonal matrices A and D 2 with positive main diagonals such that D 1 AD 2 is both doubly stochastic and the limit of the iteration is that AφO and each positive entry of A is contained in a positive diagonal.The form DιAD 2 is unique, and A and D 2 are unique up to a positive scalar multiple if and only if A is fully indecomposable.Sinkhorn [6] has shown that corresponding to each positive square matrix A there is a unique doubly stochastic matrix of the form
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Metadata
- Authors
- Sinkhorn, R., & Knopp, P.
- Venue
- Pacific Journal of Mathematics
- Publisher
- Mathematical Sciences Publishers
- Published
- 1967-05-01
- Volume
- 21
- Issue
- 2
- Pages
- 343-348
- ISSN
- 0030-8730 (ISSN-L)
- OpenAlex
- W2003447360
- Cited by
- 887
- References
- 0
- FWCI
- 10.62
- Type
- journal article
- Language
- en
- Open access
- bronze
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