For any real vector x=(x1,x2,,xn)n, let x(1)x(2)x(n) denote the components of x in non-increasing order.

For x,yn, we say that x is majorized by y, or y majorizes x, if

i=1mx(i) i=1my(i), for m=1,,n-1, and
i=1nx(i) =i=1ny(i)

A common notation for “x is majorized by y” is xy.


A canonical example is that, if y1, y2,,yn are non-negative real numbers such that their sum is equal to 1, then


In general, xy vaguely means that the components of x is less spread out than are the components of y.


  • 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 majorization
Canonical name Majorization
Date of creation 2013-03-22 14:30:22
Last modified on 2013-03-22 14:30:22
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Definition
Classification msc 26D99
Defines majorize
Defines majorization