Majorization

From AoPSWiki

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 $\displaystyle \{-b_i\}$ .