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Wikipedia / Majorization

In mathematics, majorization is a preorder on vectors of real numbers weakly majorizes from below when

  1. ,

th majorizes.

Both weak majorization and majorization are partial orders for vectors whose entries are non-decreasing, but only a preorder.

.

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

Equivalent conditions

Geometric definition

is in the convex hull for some doubly stochastic matrix. can be written as a convex combination. is in the permutahedron.

.

Other definitions

.

  • .

.

  • and, for every .

  • .

  • .

Examples

.

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.

Schur convexity

is said to be Schur convex is Schur concave

. 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.

Generalizations

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.

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.

See also

Notes

References

  • J. Karamata. "Sur une inegalite relative aux fonctions convexes." Publ. Math. Univ. Belgrade1, 145158, 1932.

  • G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd edition, 1952, Cambridge University Press, London.

  • Inequalities: Theory of Majorization and Its Applications Albert W. Marshall, Ingram Olkin, Barry Arnold, Second edition. Springer Series in Statistics. Springer, New York, 2011.

  • A tribute to Marshall and Olkin's book "Inequalities: Theory of Majorization and its Applications"

  • Matrix Analysis (1996) Rajendra Bhatia, Springer,

  • Topics in Matrix Analysis (1994) Roger A. Horn and Charles R. Johnson, Cambridge University Press,

  • Majorization and Matrix Monotone Functions in Wireless Communications (2007) Eduard Jorswieck and Holger Boche, Now Publishers,

  • The Cauchy Schwarz Master Class (2004) J. Michael Steele, Cambridge University Press,

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