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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
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 $$ \left(\frac{1}{n},\ldots,\frac{1}{n} \right) \prec (y_1,\ldots,y_n). $$ 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.
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