Majorization
![Figure 1. 2D Majorization Example](/Images/godic/202501/25/2D_Majorization_Example4123.png")
![Figure 2. 3D Majorization Example](/Images/godic/202501/25/3D_Majorization_Example4123.png")
In mathematics, majorization is a preorder on vectors of real numbers. For a vector ![\mathbf{a}\in\mathbb{R}^d](/Images/godic/202501/25/e471f49076925d2d827b1071b91ab85a4123.png")
, we denote by ![\mathbf{a}^{\downarrow}\in\mathbb{R}^d](/Images/godic/202501/25/9e633fbcff558c7b43af7823aa9f3c924123.png")
the vector with the same components, but sorted in descending order.
Given ![\mathbf{a},\mathbf{b} \in \mathbb{R}^d](/Images/godic/202501/25/ba134adfeeaf3c10cc9926c906dba3b04123.png")
, we say that
![\mathbf{a}](/Images/godic/202501/25/3c47f830945ee6b24984ab0ba188e10e4123.png")
weakly majorizes (or dominates) ![\mathbf{b}](/Images/godic/202501/25/51e355f73e82b295cc3b04e2a2bf3d2c4123.png")
from below written as ![\mathbf{a} \succ_w \mathbf{b}](/Images/godic/202501/25/85755e91424c08e3c72f2d933c4b347f4123.png")
iff
where ![a^{\downarrow}_i](/Images/godic/202501/25/8cd1bc0d7393d1284570a5e88216ca744123.png")
and ![b^{\downarrow}_i](/Images/godic/202501/25/7bca41f10942046149425ff7c9e9cabf4124.png")
are the elements of and , respectively, sorted in decreasing order.
Equivalently, we say that is weakly majorized (or dominated) by from below, denoted as ![\mathbf{b} \prec_w \mathbf{a}](/Images/godic/202501/25/f06922ded870de627d42a814bc7247554124.png")
.