Majorization
In mathematics, majorization is a preorder on vectors of real numbers. For two such vectors, x , y ∈ R n {\displaystyle \mathbf {x} ,\ \mathbf {y} \in \mathbb {R} ^{n}} , we say that x {\displaystyle \mathbf {x} } weakly majorizes (or dominates) y {\displaystyle \mathbf {y} } from below, commonly denoted x ≻ w y , {\displaystyle \mathbf {x} \succ _{w}\mathbf {y} ,} when ∑ i = 1 k x i ↓ ≥ ∑ i = 1 k y i ↓ {\displaystyle \sum _{i=1}^{k}x_{i}^{\downarrow }\geq \sum _{i=1}^{k}y_{i}^{\downarrow }} for all k = 1 , … , n {\displaystyle k=1,\,\dots ,\,n} , where x i ↓ {\displaystyle x_{i}^{\downarrow }} denotes the i {\displaystyle i} th largest entry of x {\displaystyle \mathbf {x} } .