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

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.