What is "majorization"

n. (context mathematics English) A partial order over vectors of real numbers

In mathematics, majorization is a preorder on vectors of real numbers. For a vector a ∈ R, we denote by a ∈ R the vector with the same components, but sorted in descending order. Given a, b ∈ R, we say that a weakly majorizes (or dominates) b from below written as a ≻ b iff

∑a ≥ ∑b for k = 1, …, d, 

where a and b are the elements of a and b, respectively, sorted in decreasing order. Equivalently, we say that b is weakly majorized (or dominated) by a from below, denoted as b ≺ a.

Similarly, we say that a weakly majorizes b from above written as a ≻ b iff

∑a ≤ ∑b for k = 1, …, d, 

Equivalently, we say that b is weakly majorized by a from above, denoted as b ≺ a.

If a ≻ b and in addition ∑a = ∑b we say that a majorizes (or dominates) b written as a ≻ b. Equivalently, we say that b is majorized (or dominated) by a, denoted as b ≺ a.

It is easy to see that a ≻ b if and only if a ≻ b and a ≻ b.

Note that the majorization order do not depend on the order of the components of the vectors a or b. Majorization is not a partial order, since a ≻ b and b ≻ a do not imply a = b, it only implies that the components of each vector are equal, but not necessarily in the same order.

Regrettably, to confuse the matter, some literature sources use the reverse notation, e.g.,  ≻  is replaced with  ≺ , most notably, in Horn and Johnson, Matrix analysis (Cambridge Univ. Press, 1985), Definition 4.3.24, while the same authors switch to the traditional notation, introduced here, later in their Topics in Matrix Analysis (1994), and in the second edition of Matrix analysis (2013).

A function f : R → R is said to be Schur convex when a ≻ b implies f(a) ≥ f(b). Similarly, f(a) is Schur concave when a ≻ b implies f(a) ≤ f(b).

The majorization partial order on finite sets, described here, can be generalized to the Lorenz ordering, a partial order on distribution functions.