characterizations of majorization

Let n be the set of all n×n permutation matricesMathworldPlanetmath that exchange two componentsPlanetmathPlanetmathPlanetmath. Such matrices have the form


A matrix T is called a Pigou-Dalton transfer (PDT) if


for some α between 0 and 1, and En.

The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath

  1. 1.

    x is majorized ( by y.

  2. 2.

    x=Dy for a doubly stochastic matrix D.

  3. 3.

    x=T1T2Tky for finitely many PDT T1,,Tk.

  4. 4.

    i=1nθ(xi)i=1nθ(yi) for all convex function θ.

  5. 5.

    x lies in the convex hull whose vertex set is

    {(yπ(1),yπ(2),,yπ(n)):π is a permutation of {1,,n}}.
  6. 6.

    For any n non-negative real numbers a1,,an,


    where summation is taken over all permutationsMathworldPlanetmath of {1,,n}.

The equivalence of the above conditions are due to Hardy, Littlewood, Pólya, Birkhoff, von Neumann and Muirhead.


  • G. H. Hardy, J. E. Littlewood and G. Pólya, InequalitiesMathworldPlanetmath, 2nd edition, 1952, Cambridge University Press, London.

  • A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, 1979, Acadamic Press, New York.

Title characterizations of majorization
Canonical name CharacterizationsOfMajorization
Date of creation 2013-03-22 15:26:37
Last modified on 2013-03-22 15:26:37
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 7
Author Mathprof (13753)
Entry type Theorem
Classification msc 26D99
Related topic BirkoffVonNeumannTheorem
Related topic MuirheadsTheorem
Defines Pigou-Dalton transfer