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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 that weakly majorizes (or dominates) from below written as iff
Equivalently, we say that is weakly majorized (or dominated) by from below, written as .
If and in addition , we say that majorizes (or dominates) , written as . Equivalently, we say that is majorized (or dominated) by , written as .
Note that the majorization order does 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.
Note that the notation is inconsistent in the mathematical literature: some use the reverse notation, e.g., is replaced with .
A function is said to be Schur convex when implies . Similarly, is Schur concave when implies
The order of the entries does not affect the majorization, e.g., the statement is simply
equivalent to .
(Strong) majorization: . For vectors with n components
(Weak) majorization: . For vectors with n components:
Geometry of majorization
Figure 1. 2D majorization example
For we have
if and only if is in the convex hull of all vectors obtained by permuting the coordinates of .
Figure 1 displays the convex hull in 2D for the vector . Notice that the center of the convex hull, which is an interval in this case, is the vector . This is the "smallest" vector satisfying for this given vector .
Figure 2. 3D Majorization Example
Figure 2 shows the convex hull in 3D. The center of the convex hull, which is a 2D polygon in this case, is the "smallest" vector satisfying for this given vector .
Each of the following statements is true if and only if :
Suppose that for two real vectors, majorizes . Then it can be shown that there exists a set of probabilities and a set of permutations such that . Alternatively it can be shown that there exists a doubly stochastic matrix such that
We say that a Hermitian operator, , majorizes another, , if the set of eigenvalues of majorizes that of .
In recursion theory
Given , then is said to majorize if, for all , . If there is some so that for all , then is said to dominate (or eventually dominate) . Alternatively, the preceding terms are often defined requiring the strict inequality instead of in the foregoing definitions.
Various generalizations of majorization are discussed in chapters 14 and 15 of the reference work Inequalities: Theory of Majorization and Its Applications. Albert W. Marshall, Ingram Olkin, Barry Arnold. Second edition. Springer Series in Statistics. Springer, New York, 2011. ISBN978-0-387-40087-7