submajorization

Submajorization is a partial order on vectors used in inequality theory and matrix analysis. Let x and y be vectors in R^n. Denote by x↓ and y↓ the components of x and y arranged in nonincreasing order. We say x is submajorized by y, written x ≼ y, if for every k = 1, ..., n the sum of the k largest components satisfies sum_{i=1}^k x↓_i ≤ sum_{i=1}^k y↓_i. There is no requirement that the total sums are equal. If equality holds for k = n, then the relation becomes the standard majorization order, with y majorizing x. In many texts, submajorization is referred to as weak majorization.

Example: Take x = (1, 1, 0) and y = (2, 0, 0). Then x↓ = (1, 1, 0) and y↓ = (2, 0, 0). The partial sums are 1 ≤ 2, 2 ≤ 2, and 2 ≤ 2, so x ≼ y.

Submajorization is a natural counterpart to majorization: it compares how much a vector can be spread out in its largest components without enforcing equality of total mass. It is a transitive relation and plays a role in the study of dispersion inequalities and matrix theory. It commonly arises in the context of eigenvalue and singular value inequalities, and in the analysis of Schur-convex and Schur-concave functions through related comparisons of coordinate dispersion.

See also majorization and related notions such as weak or submajorization, Schur-convexity, and their applications in matrix analysis and inequalities.