Majorization
In mathematics, majorization is a preorder on vectors of real numbers.
weakly majorizes (or dominates)
from below, commonly denoted
majorizes (or dominates)
, commonly denoted
Both weak majorization and majorization are partial orders 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
Majorizing also sometimes refers to entrywise ordering, e.g. the real-valued function f majorizes the real-valued function g when
in the domain, or other technical definitions, such as majorizing measures in probability theory.
is in the convex hull of all vectors obtained by permuting the coordinates of
for some doubly stochastic matrix
can be written as a convex combination 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
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
Among non-negative vectors with three components,
and permutations of it majorize all other vectors
This behavior extends to general-length probability vectors: the singleton vector majorizes all other probability vectors, and the uniform distribution is majorized by all probability vectors.
Hence, Schur-convex functions translate the ordering of vectors to a standard ordering in
Schur convex functions are necessarily symmetric that the entries of it argument can be switched without modifying the value of the function.
Therefore, linear functions, which are convex, are not Schur-convex unless they are symmetric.
If a function is symmetric and convex, then it is Schur-convex.
Majorization can be generalized to the Lorenz ordering, a partial order on distribution functions.
For example, a wealth distribution is Lorenz-greater than another if its Lorenz curve lies below the other.
As such, a Lorenz-greater wealth distribution has a higher Gini coefficient, and has more income disparity.
[6] The majorization preorder can be naturally extended to density matrices in the context of quantum information.
denotes the state's spectrum).
Similarly, one can say a Hermitian operator,
, if the set of eigenvalues of
