You are here
Homemajorization
Primary tabs
majorization
For any real vector $x=(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}$, let $x_{{(1)}}\geq x_{{(2)}}\geq\cdots\geq x_{{(n)}}$ 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
$\displaystyle\sum_{{i=1}}^{m}x_{{(i)}}$  $\displaystyle\leq\sum_{{i=1}}^{m}y_{{(i)}},\quad\text{ for $m=1,\ldots,n1$, and}$  
$\displaystyle\sum_{{i=1}}^{n}x_{{(i)}}$  $\displaystyle=\sum_{{i=1}}^{n}y_{{(i)}}$ 
A common notation for “$x$ is majorized by $y$” is $x\prec y$.
Remark:
A canonical example is that, if $y_{1}$, $y_{2},\ldots,y_{n}$ are nonnegative real numbers such that their sum is equal to 1, then
$\left(\frac{1}{n},\ldots,\frac{1}{n}\right)\prec(y_{1},\ldots,y_{n}).$ 
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.
Mathematics Subject Classification
26D99 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new image: informationtheoreticdistributedmeasurement4.2 by rspuzio
new image: informationtheoreticdistributedmeasurement4.1 by rspuzio
new image: informationtheoreticdistributedmeasurement3.2 by rspuzio
new image: informationtheoreticdistributedmeasurement3.1 by rspuzio
new image: informationtheoreticdistributedmeasurement2.1 by rspuzio
Apr 19
new collection: On the InformationTheoretic Structure of Distributed Measurements by rspuzio
Apr 15
new question: Prove a formula is part of the Gentzen System by LadyAnne
Mar 30
new question: A problem about Euler's totient function by mbhatia
new problem: Problem: Show that phi(a^n1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia
new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz