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majorization (Definition)

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 non-increasing order.

For $ x, y \in \mathbb{R}^n$, we say that $ x$ is majorized by $ y$, or $ y$ majorizes $ x$, if

$\displaystyle \sum_{i=1}^m x_{(i)}$ $\displaystyle \leq \sum_{i=1}^m y_{(i)},$    for $ m=1,\ldots, n-1$, 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 non-negative real numbers such that their sum is equal to 1, then

$\displaystyle \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.



"majorization" is owned by rspuzio. [ full author list (2) | owner history (1) ]
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Also defines:  majorize, majorization

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characterizations of majorization (Theorem) by Mathprof
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Cross-references: sum, canonical, order, components, vector, real
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This is version 5 of majorization, born on 2004-07-28, modified 2006-11-11.
Object id is 6043, canonical name is Majorization.
Accessed 4664 times total.

Classification:
AMS MSC: 26D99 (Real functions :: Inequalities :: Miscellaneous)
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