Submajorized

Submajorized is a relation between finite real vectors used in majorization theory. It generalizes the classical majorization relation by relaxing the requirement that the total sums are equal. For x, y in R^n, let x↓ and y↓ denote the components of x and y arranged in nonincreasing order. We say that x is submajorized by y, written x ≼ y or x ≺w y, if for every k = 1, 2, ..., n the sum of the k largest components satisfies sum_{i=1}^k x_i↓ ≤ sum_{i=1}^k y_i↓. In particular, this implies sum_{i=1}^n x_i ≤ sum_{i=1}^n y_i.

Submajorization reduces to ordinary majorization when the two vectors have the same total sum, because the inequality at k = n becomes an equality.

Examples help illustrate the concept. Take x = (2, 1, 0) and y = (2, 2, 0). After sorting, x↓ = (2, 1, 0) and y↓ = (2, 2, 0). The partial sums satisfy 2 ≤ 2, 3 ≤ 4, and 3 ≤ 4, so x ≼ y. By contrast, x is not submajorized by y = (1, 3, 0) since the first partial sum would fail.

Properties and use. Submajorization is a partial order on R^n and is transitive and reflexive. It is used in inequalities, matrix and spectral theory, and the study of how spread or concentration of vector components compares under monotone transforms. It is closely related to majorization and to notions like Schur-convexity.

See also majorization, weak majorization, and Schur-convexity.