Majorization

Majorization is a partial order on finite vectors that compares how concentrated or spread out their entries are. It is defined by inequalities between sums of sorted entries, together with equality of total sums.

Majorization order: A partial order in which one vector majorizes another when its sorted partial sums dominate and their totals agree. This formal order is defined by the inequalities that give majorization its meaning.
Permutation: A rearrangement of a finite set of objects or positions. Majorization ignores the original order of entries by sorting or permuting vectors.
Lorenz curve: A graph showing the cumulative share of a total held by the least wealthy fractions of a population. Its cumulative comparisons visualize the same concentration ordering as majorization.
Weak majorization: A variant of majorization that compares partial sums without requiring equality of total sums. It relaxes the equal-total condition central to ordinary majorization.
Doubly stochastic matrix: A square matrix with nonnegative entries whose rows and columns each sum to one. Multiplication by such a matrix transforms a vector into one it is majorized by.
Decreasing rearrangement: The sequence formed by arranging a vector's entries from largest to smallest. The defining inequalities compare the decreasing rearrangements of both vectors.
Gini coefficient: A statistic summarizing inequality in a distribution, often represented by the area between its Lorenz curve and equality line. Majorization orders distributions more finely than this single inequality summary.
Log-majorization: An order comparing positive vectors through the partial sums of their logarithms. It replaces additive comparisons with multiplicative ones.
Birkhoff–von Neumann theorem: The theorem that every doubly stochastic matrix is a convex combination of permutation matrices. It explains why doubly stochastic transformations amount to averaging and rearranging entries.
Partial sum: The sum of an initial segment of a sequence. Majorization compares every initial partial sum after entries are sorted.

Linked from 7 pages