Difference between revisions of "Majorization"

Revision as of 12:25, 16 January 2023

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 $k=1$ . An interesting consequence of this is that the finite sequence $\{a_i\}$ majorizes $\{b_i\}$ if and only if $\{-a_i\}$ minorizes $\{-b_i\}$ .

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.

@@ Line 9: / Line 9: @@
 == Alternative Criteria ==
-It is also true that <math> \{a_i\}_{i=1}^n </math>[[Image:succ.gif]]<math> \{b_i\}_{i=1}^n </math> if and only if for all <math> 1\le k \le n </math>, <math>\sum_{i=k}^n a_i \le \sum_{i=k}^n b_i</math>, with equality when <math> \displaystyle k=1 </math>.  An interesting consequence of this is that the finite sequence <math> \displaystyle \{a_i\} </math> majorizes <math> \displaystyle \{b_i\} </math> if and only if <math> \displaystyle \{-a_i\} </math> majorizes <math> \displaystyle \{-b_i\} </math>.
+It is also true that <math> \{a_i\}_{i=1}^n </math>[[Image:succ.gif]]<math> \{b_i\}_{i=1}^n </math> if and only if for all <math> 1\le k \le n </math>, <math>\sum_{i=k}^n a_i \le \sum_{i=k}^n b_i</math>, with equality when <math>k=1 </math>.  An interesting consequence of this is that the finite sequence <math>\{a_i\} </math> majorizes <math>\{b_i\} </math> if and only if <math>\{-a_i\} </math> minorizes <math>\{-b_i\} </math>.
 We can also say that this is the case if and only if for all <math> t \in \mathbb{R} </math>,

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Difference between revisions of "Majorization"

Revision as of 12:25, 16 January 2023

Contents

Definition

Minorization

Alternative Criteria

See Also