Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Continuous dependence on $ A$ in the $ D\sb{1}AD\sb{2}$ theorems


Author: Richard Sinkhorn
Journal: Proc. Amer. Math. Soc. 32 (1972), 395-398
MSC: Primary 15A48
DOI: https://doi.org/10.1090/S0002-9939-1972-0297792-4
MathSciNet review: 0297792
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It has been shown by Sinkhorn and Knopp and others that if A is a nonnegative square matrix such that there exists a doubly stochastic matrix B with the same zero pattern as A, then there exists a unique doubly stochastic matrix of the form $ {D_1}A{D_2}$ where $ {D_1}$ and $ {D_2}$ are diagonal matrices with positive main diagonals. Sinkhorn and Knopp have also shown that if A has at least one positive diagonal, then the sequence of matrices obtained by alternately normalizing the row and column sums of A will converge to a doubly stochastic limit. It is the intent of this paper to show that $ {D_1}A{D_2}$ and/or the limit of this iteration, when either exists, is continuously dependent upon the matrix A.


References [Enhancements On Off] (What's this?)

  • [1] Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Boston, Mass., 1964. MR 29 #112. MR 0162808 (29:112)
  • [2] Richard Sinkhorn and Paul Knopp, Concerning nonnegative matrices and doubly stochastic matrices, Pacific J. Math. 21 (1967), 343-348. MR 35 #1617. MR 0210731 (35:1617)
  • [3] -, Problems involving diagonal products in nonnegative matrices, Trans. Amer. Math. Soc. 136 (1969), 67-75. MR 38 #2151. MR 0233830 (38:2151)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 15A48

Retrieve articles in all journals with MSC: 15A48


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0297792-4
Keywords: Doubly stochastic matrix, permanent, doubly stochastic pattern, doubly stochastic subpattern
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society