characterizations of majorization

Let $\mathcal{E}_{n}$ be the set of all $n\times n$ permutation matrices that exchange two components. Such matrices have the form

\begin{bmatrix}\ddots&\\ &0&&1\\ &&\ddots\\ &1&&0\\ &&&&\ddots\end{bmatrix}

A matrix $T$ is called a Pigou-Dalton transfer (PDT) if

T=\alpha I+(1-\alpha)E

for some $\alpha$ between 0 and 1, and $E\in\mathcal{E}_{n}$ .

The following are equivalent

1.

$x$ is majorized (http://planetmath.org/Majorization) by $y$ .
2.

$x=Dy$ for a doubly stochastic matrix $D$ .
3.

$x=T_{1}T_{2}\cdots T_{k}y$ for finitely many PDT $T_{1},\ldots,T_{k}$ .
4.

$\sum_{i=1}^{n}\theta(x_{i})\leq\sum_{i=1}^{n}\theta(y_{i})$ for all convex function $\theta$ .
5.

$x$ lies in the convex hull whose vertex set is

$\big{\{}(y_{\pi(1)},y_{\pi(2)},\ldots,y_{\pi(n)}):\,\pi\text{ is a permutation% of }\{1,\ldots,n\}\big{\}}.$
6.

For any $n$ non-negative real numbers $a_{1},\ldots,a_{n}$ ,

$\sum_{\pi}a_{1}^{x_{\pi(1)}}a_{2}^{x_{\pi(2)}}\cdots a_{n}^{x_{\pi(n)}}\leq% \sum_{\pi}a_{1}^{y_{\pi(1)}}a_{2}^{y_{\pi(2)}}\cdots a_{n}^{y_{\pi(n)}}$

where summation is taken over all permutations of $\{1,\ldots,n\}$ .

The equivalence of the above conditions are due to Hardy, Littlewood, Pólya, Birkhoff, von Neumann and Muirhead.

Reference

•

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

A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, 1979, Acadamic Press, New York.

Title	characterizations of majorization
Canonical name	CharacterizationsOfMajorization
Date of creation	2013-03-22 15:26:37
Last modified on	2013-03-22 15:26:37
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	7
Author	Mathprof (13753)
Entry type	Theorem
Classification	msc 26D99
Related topic	BirkoffVonNeumannTheorem
Related topic	MuirheadsTheorem
Defines	Pigou-Dalton transfer