Majorization

Definition

Let $n \ge 1$ and let $x, y \in \R^n$ . Write $x^{\downarrow}$ for the vector with the entries of $x$ in decreasing order, so $x^{\downarrow}_1 \ge x^{\downarrow}_2 \ge \cdots \ge x^{\downarrow}_n$ . The vector $x$ majorizes $y$ , written $y \prec x$ , if:

$\sum_{i=1}^{k} y^{\downarrow}_i \le \sum_{i=1}^{k} x^{\downarrow}_i$ for each $k = 1, \ldots, n - 1$ .
$\sum_{i=1}^{n} y_i = \sum_{i=1}^{n} x_i$ .

Remarks

Example: $y = (\frac{1}{2}, \frac{1}{2}, 0) \prec x = (1, 0, 0)$ . Check: $\frac{1}{2} \le 1$ for $k = 1$ , $1 \le 1$ for $k = 2$ , and both sums equal $1$ . Non-example: $(1, 0) \not\prec (2, 0)$ , because clause 1 holds but the sums $1$ and $2$ differ, so clause 2 fails. The relation is a preorder: $y \prec x$ and $x \prec y$ hold together exactly when $y$ is a permutation of $x$ . By a theorem of Hardy, Littlewood and Pólya, $y \prec x$ if and only if $y = Dx$ for some $n \times n$ doubly stochastic matrix $D$ .

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Doubly stochastic matrix concept
Majorization definition

Tag	`WYHKN0TE3Z`
URI	`m2://WYHKN0TE3Z`
Kind	definition
Writing	W0 stub
MSC	26D15, 15B51
Defines	Majorization
Uses	Doubly stochastic matrix
Changed	2 November 2026

Majorization

Definition

Remarks

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