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- Majorization abstract "In mathematics, majorization is a preorder on vectors of real numbers. For a vector , we denote by the vector with the same components, but sorted in descending order.Given , we say thatweakly majorizes (or dominates) from below written as iff where and are the elements of and , respectively, sorted in decreasing order.Equivalently, we say that is weakly majorized (or dominated) by from below, denoted as .Similarly, we say thatweakly majorizes from above written as iff Equivalently, we say that is weakly majorized by from above, denoted as .If and in addition we say thatmajorizes (or dominates) written as .Equivalently, we say that is majorized (or dominated) by , denoted as .It is easy to see that if and only if and .Note that the majorization order do not depend on the order of the components of the vectors or . Majorization is not a partial order, since and do not imply , it only implies that the components of each vector are equal, but not necessarily in the same order.Regrettably, to confuse the matter, some literature sources use the reverse notation, e.g., is replaced with , most notably, in Horn and Johnson, Matrix analysis (Cambridge Univ. Press, 1985), Definition 4.3.24, while the same authors switch to the traditional notation, introduced here, later in their Topics in Matrix Analysis (1994).A function is said to be Schur convex when implies . Similarly, is Schur concave when implies The majorization partial order on finite sets, described here, can be generalized to the Lorenz ordering, a partial order on distribution functions.".
- Majorization thumbnail 2D_Majorization_Example.png?width=300.
- Majorization wikiPageExternalLink 0801.4221v1.
- Majorization wikiPageExternalLink Majorization.html.
- Majorization wikiPageExternalLink Majorization.html.
- Majorization wikiPageExternalLink 26962-majorization-check.
- Majorization wikiPageID "2864495".
- Majorization wikiPageRevisionID "605059984".
- Majorization hasPhotoCollection Majorization.
- Majorization subject Category:Linear_algebra.
- Majorization subject Category:Order_theory.
- Majorization comment "In mathematics, majorization is a preorder on vectors of real numbers.".
- Majorization label "Majorisation".
- Majorization label "Majorization".
- Majorization label "Мажоризация".
- Majorization sameAs Majorisation.
- Majorization sameAs m.087p3_.
- Majorization sameAs Q1994695.
- Majorization sameAs Q1994695.
- Majorization wasDerivedFrom Majorization?oldid=605059984.
- Majorization depiction 2D_Majorization_Example.png.
- Majorization isPrimaryTopicOf Majorization.
