Majorization
In mathematics, majorization is a preorder on vectors of real numbers. For a vector , we denote by the vector with the same components, but sorted in descending order. Given , we say that weakly majorizes (or dominates) from below written as iff Equivalently, we say that is weakly majorized (or dominated) by from below, written as . If and in addition , we say that majorizes (or dominates) , written as . Equivalently, we say that is majorized (or dominated) by , written as . Note that the majorization order does not depend on the order of the components of the vectors or . Majorization is not a partial order, since and do not imply , it only implies that the components of each vector are equal, but not necessarily in the same order. Note that the notation is inconsistent in the mathematical literature: some use the reverse notation, e.g., is replaced with . A function is said to be Schur convex when implies . Similarly, is Schur concave when implies The majorization partial order on finite sets, described here, can be generalized to the , a partial order on distribution functions. For example, a wealth distribution is Lorenz-greater than another iff its Lorenz curve lies below the other. As such, a Lorenz-greater wealth distribution has a higher Gini coefficient, and has more income inequality. (Wikipedia).
