Majorization

In mathematics, majorization is a partial order over vectors of real numbers. Given $$\mathbf{a},\mathbf{b} \in \mathbb{R}^d$$, we say that $$\mathbf{a}$$ majorizes $$\mathbf{b}$$, and we write $$\mathbf{a} \succeq \mathbf{b}$$, if $$\sum_{i=1}^d a_i = \sum_{i=1}^d b_i$$ and for all $$k \in \{1, \ldots, d\}$$,


 * $$\sum_{i=1}^k a_i^{\downarrow} \geq \sum_{i=1}^k b_i^{\downarrow},$$

where $$a^{\downarrow}_i$$ and $$b^{\downarrow}_i$$ are the elements of $$\mathbf{a}$$ and $$\mathbf{b}$$, respectively, sorted in decreasing order. Equivalently, we say that $$\mathbf{a}$$ dominates $$\mathbf{b}$$, or that $$\mathbf{b}$$ is majorized (or dominated) by $$\mathbf{a}$$.

A function $$f:\mathbb{R}^d \to \mathbb{R}$$ is said to be Schur convex when $$\mathbf{a} \succeq \mathbf{b}$$ implies $$f(\mathbf{a}) \geq f(\mathbf{b})$$. Similarly, $$f(\mathbf{a})$$ is Schur concave when $$\mathbf{a} \succeq \mathbf{b}$$ implies $$f(\mathbf{a}) \leq f(\mathbf{b})$$.

The majorization partial order on finite sets can be generalized to the Lorenz ordering, a partial order on distribution functions.

Equivalent conditions
Each of the following statements is true if and only if $$\mathbf{a}\succeq \mathbf{b}$$:


 * $$\mathbf{b} = D\mathbf{a}$$ for some doubly stochastic matrix $$D$$ (see Arnold, Theorem 2.1).
 * From $$\mathbf{a}$$ we can produce $$\mathbf{b}$$ by a finite sequence of "Robin Hood operations" where we replace two elements $$a_i$$ and $$a_j < a_i$$ with $$a_i-\varepsilon$$ and $$a_j+\varepsilon$$, respectively, for some $$\varepsilon \in (0, a_i-a_j)$$ (see Arnold, p. 11).
 * For every convex function $$h:\mathbb{R}\to \mathbb{R}$$, $$\sum_{i=1}^d h(a_i) \geq \sum_{i=1}^d h(b_i)$$ (see Arnold, Theorem 2.9).

In linear algebra
\sum_{i=1}^d p_i=1$$ and a set of permutations $$(P_1,P_2,\ldots,P_d)$$ such that $$v'=\sum_{i=1}^d p_iP_iv$$. Alternatively it can be shown that there exists a doubly stochastic matrix $$D$$ such that $$vD=v'$$
 * Suppose that for two real vectors $$v,v' \in \mathbb{R}^d$$, $$v$$ majorizes $$v'$$. Then it can be shown that there exists a set of probabilities $$(p_1,p_2,\ldots,p_d),


 * We say that a hermitian operator, $$H$$, majorizes another, $$H'$$, if the set of eigenvalues of $$H$$ majorizes that of $$H'$$.

In recursion theory
Given $$f, g : \mathbb{N} \to\mathbb{N}$$, then $$f$$ is said to majorize $$g$$ if, for all $$x$$, $$f(x)\geq g(x)$$. If there is some $$n$$ so that $$f(x)\geq g(x)$$ for all $$x > n$$, then $$f$$ is said to dominate (sometimes written "eventually dominate") $$f$$.