Doubly Stochastic Matrices

  • Albert W. Marshall
  • Ingram Olkin
  • Barry C. Arnold
Part of the Springer Series in Statistics book series (SSS)


An important tool in the study of majorization is a theorem due to Hardy, Littlewood, and Pólya (1929) which states that for x, yЄRn,x < y if and only if x = yP for some doubly stochastic matrix P.


Convex Hull Extreme Point Permutation Matrix Convex Polytope Stochastic Matrix 
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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Albert W. Marshall
    • 1
    • 2
  • Ingram Olkin
    • 3
  • Barry C. Arnold
    • 4
  1. 1.Department of StatisticsUniversity of British ColumbiaVancouverCanada
  2. 2.Lummi IslandUSA
  3. 3.Department of StatisticsStanford UniversityStanfordUSA
  4. 4.Department of StatisticsUniversity of CaliforniaRiversideUSA

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