Majorisation of Vectors

Let $\bm{x}, \bm{y}~\in~\mathbb{R}^{n}$ with components $(x_1, x_2, \ldots, x_n)$ and $(y_1, y_2, \ldots, y_n)$ respectively.

If $\bm{x}$ majorises $\bm{y}$, then $\bm{y}$ can be considered to be more mixed or closer to the maximally mixed vector, $I = (1/n, 1/n, \ldots, 1/n)$, then $\bm{x}$.

If $\bm{x}$ majorises $\bm{y}$ then it is typically denoted by $\bm{x} \succ \bm{y}$.

Majorisation Conditions¶

$\bm{x}$ majorises $\bm{y}$:

• If and only if,

where $x^{\downarrow}_{i}$ and $y^{\downarrow}_{i}$ are the components of $\bm{x}$ and $\bm{y}$ respectively, ordered in non-increasing order, such that $x_i \geq x_{i+1}~\forall~i$.

• If and only if there exists a stochastic matrix $A$ satisfying

(where the second condition ensures $A$ needs to be doubly stochastic).

Schur-convex Functions¶

A function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ is called Schur-convex if and only if

A function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ is called Schur-concave if and only if