# 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. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 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. 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, , 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 CharacterizationsOfMajorization 2013-03-22 15:26:37 2013-03-22 15:26:37 Mathprof (13753) Mathprof (13753) 7 Mathprof (13753) Theorem msc 26D99 BirkoffVonNeumannTheorem MuirheadsTheorem Pigou-Dalton transfer