Majorization

In mathematics, majorization is a preorder on vectors of real numbers. For two such vectors, , we say that weakly majorizes (or dominates) from below, commonly denoted when

for all ,

where denotes th largest entry of . If further satisfy , we say that majorizes (or dominates) , commonly denoted . Majorization is a partial order for vectors whose entries are non-decreasing, but only a preorder for general vectors, since majorization is agnostic to the ordering of the entries in vectors, e.g., the statement is simply equivalent to .

Majorizing also sometimes refers to entrywise ordering, e.g. the real-valued function f majorizes the real-valued function g when for all in the domain, or other technical definitions, such as majorizing measures in probability theory.[1]

  1. ^ Talagrand, Michel (1996-07-01). "Majorizing measures: the generic chaining". The Annals of Probability. 24 (3). doi:10.1214/aop/1065725175. ISSN 0091-1798.