Majorization

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

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