Submajorization

Submajorization is a partial order on vectors that compares the sizes of their largest components. For x and y in R^n, let x* and y* denote their nonincreasing rearrangements. We say that x is submajorized by y, written x ≼ y (also called weak submajorization), if for every k = 1, ..., n the partial sums satisfy

sum_{i=1}^k x_i* ≤ sum_{i=1}^k y_i*.

If, in addition, the total sums are equal, i.e., sum_{i=1}^n x_i = sum_{i=1}^n y_i, then x is majorized by y (x ≺ y).

Submajorization thus compares the distributions of mass in a vector by looking at the accumulative top-k values. It becomes majorization when the total mass is fixed to be equal.

Basic properties include that the relation is a partial order: it is reflexive, transitive, and antisymmetric up to permutation. If x is a permutation of y, the inequalities hold with equality for all k, so x ≼ y and y ≼ x.

Example: let x = (4, 3, 3) and y = (5, 3, 2). After sorting, x* = (4,3,3) and y* = (5,3,2). The partial sums are 4 ≤ 5, 7 ≤ 8, 10 ≤ 10, so x ≼ y, and since total sums are equal, x ≺ y as well.

Applications appear in matrix analysis, probability and statistics, and the study of Schur-convex and Schur-concave functions, where submajorization provides a framework for comparing distributions and bounding norms or eigenvalue-like quantities. It is closely related to, but weaker than, the classical majorization order.