Maggiorizzazione

In mathematics, majorization is a preorder on vectors of real numbers. For two such vectors, \mathbf{x},\ \mathbf{y} \in \mathbb{R}^n, we say that \mathbf{x} weakly majorizes (or dominates) \mathbf{y} from below, commonly denoted \mathbf{x} \succ_w \mathbf{y}, when \sum_{i=1}^k x_i^{\downarrow} \geq \sum_{i=1}^k y_i^{\downarrow} for all k=1,\,\dots,\,n, where x_i^{\downarrow} denotes the ith largest entry of \mathbf{x}. If \mathbf{x}, \mathbf{y} further satisfy \sum_{i=1}^n x_i = \sum_{i=1}^n y_i, we say that \mathbf{x} majorizes (or dominates) \mathbf{y} , commonly denoted \mathbf{x} \succ \