Majorization

（重定向自Majorise）

In mathematics, majorization is a preorder on vectors of real numbers. For a vector $\mathbf{a}\in\mathbb{R}^d$ , we denote by $\mathbf{a}^{\downarrow}\in\mathbb{R}^d$ the vector with the same components, but sorted in descending order. Given $\mathbf{a},\mathbf{b} \in \mathbb{R}^d$ , we say that $\mathbf{a}$ weakly majorizes (or dominates) $\mathbf{b}$ from below written as $\mathbf{a} \succ_w \mathbf{b}$ iff

where $a^{\downarrow}_i$ and $b^{\downarrow}_i$ are the elements of $\mathbf{a}$ and $\mathbf{b}$ , respectively, sorted in decreasing order. Equivalently, we say that $\mathbf{b}$ is weakly majorized (or dominated) by $\mathbf{a}$ from below, denoted as $\mathbf{b} \prec_w \mathbf{a}$ .

单词	Majorise
释义	Majorise 英语百科 Majorization （重定向自Majorise） In mathematics, majorization is a preorder on vectors of real numbers. For a vector $\mathbf{a}\in\mathbb{R}^d$ , we denote by $\mathbf{a}^{\downarrow}\in\mathbb{R}^d$ the vector with the same components, but sorted in descending order. Given $\mathbf{a},\mathbf{b} \in \mathbb{R}^d$ , we say that $\mathbf{a}$ weakly majorizes (or dominates) $\mathbf{b}$ from below written as $\mathbf{a} \succ_w \mathbf{b}$ iff where $a^{\downarrow}_i$ and $b^{\downarrow}_i$ are the elements of $\mathbf{a}$ and $\mathbf{b}$ , respectively, sorted in decreasing order. Equivalently, we say that $\mathbf{b}$ is weakly majorized (or dominated) by $\mathbf{a}$ from below, denoted as $\mathbf{b} \prec_w \mathbf{a}$ .
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