# Majorization

## Definition

We say a nonincreasing sequence of real numbers $a_1, \ldots ,a_n$ majorizes another nonincreasing sequence $b_1,b_2,\ldots,b_n$, and write $\{a_i\}_{i=1}^n$  $\{b_i\}_{i=1}^n$ if and only if all for all $1 \le k \le n$, $\sum_{i=1}^{k}a_i \ge \sum_{i=1}^{k}b_i$, with equality when $\displaystyle k = n$. If $\displaystyle \{a_i\}$ and $\displaystyle \{b_i\}$ are not necessarily nonincreasing, then we still write $\displaystyle \{a_i\}$  $\displaystyle \{b_i\}$ if this is true after the sequences have been sorted in nonincreasing order.

### Minorization

We will occasionally say that $b_1, \ldots, b_n$ minorizes $a_1, \ldots, a_n$, and write $\displaystyle \{b_i\}$  $\displaystyle \{a_i\}$, if $\displaystyle \{a_i\}$  $\displaystyle \{b_i\}$.

## Alternative Criteria

It is also true that $\{a_i\}_{i=1}^n$  $\{b_i\}_{i=1}^n$ if and only if for all $1\le k \le n$, $\sum_{i=k}^n a_i \le \sum_{i=k}^n b_i$, with equality when $\displaystyle k=1$. An interesting consequence of this is that the finite sequence $\displaystyle \{a_i\}$ majorizes $\displaystyle \{b_i\}$ if and only if $\displaystyle \{-a_i\}$ majorizes $\displaystyle \{-b_i\}$.

We can also say that this is the case if and only if for all $t \in \mathbb{R}$, $\sum_{i=1}^{n}|t-a_i| \ge \sum_{i=1}^{n}|t-b_i|$.

Both of these conditions are equivalent to our original definition.