majorization
For any real vector $x=({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})\in {\mathbb{R}}^{n}$, let ${x}_{(1)}\ge {x}_{(2)}\ge \mathrm{\cdots}\ge {x}_{(n)}$ denote the components of $x$ in nonincreasing order.
For $x,y\in {\mathbb{R}}^{n}$, we say that $x$ is majorized by $y$, or $y$ majorizes $x$, if
$\sum _{i=1}^{m}}{x}_{(i)$  $\le {\displaystyle \sum _{i=1}^{m}}{y}_{(i)},\text{for}m=1,\mathrm{\dots},n1\text{, and}$  
$\sum _{i=1}^{n}}{x}_{(i)$  $={\displaystyle \sum _{i=1}^{n}}{y}_{(i)}$ 
A common notation for “$x$ is majorized by $y$” is $x\prec y$.
Remark:
A canonical example is that, if ${y}_{1}$, ${y}_{2},\mathrm{\dots},{y}_{n}$ are nonnegative real numbers such that their sum is equal to 1, then
$$(\frac{1}{n},\mathrm{\dots},\frac{1}{n})\prec ({y}_{1},\mathrm{\dots},{y}_{n}).$$ 
In general, $x\prec y$ vaguely means that the components of $x$ is less spread out than are the components of $y$.
Reference

•
G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities^{}, 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  20130322 14:30:22 
Last modified on  20130322 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 