Difference between revisions of "Majorization"

Latest revision as of 21:48, 5 July 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 $\{a_i\}_{i=1}^n\succ\{b_i\}_{i=1}^n$ 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 $k = n$ . If $\{a_i\}$ and $\{b_i\}$ are not necessarily nonincreasing, then we still write $\{a_i\}\succ\{b_i\}$ 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 $\{b_i\}\prec\{a_i\}$ , if $\{a_i\}\succ\{b_i\}$ .

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 1: / Line 1: @@
-A sequence <math>\displaystyle A=a_1,a_2,\cdots,a_n</math> is said to '''majorize''' a sequence <math>\displaystyle B=b_1,b_2,\cdots,b_n</math> [[if and only if]] all of the following are true:
+== Definition ==
-<math>\displaystyle a_1\geq b_1</math>
+We say a [[nonincreasing]] [[sequence]] of [[real number]]s <math> a_1, \ldots ,a_n</math> '''majorizes''' another nonincreasing sequence <math>b_1,b_2,\ldots,b_n</math>, and write <math>\{a_i\}_{i=1}^n\succ\{b_i\}_{i=1}^n </math> if and only if all for all <math> 1 \le k \le n </math>, <math> \sum_{i=1}^{k}a_i \ge \sum_{i=1}^{k}b_i </math>, with equality when <math>k = n </math>.  If <math>\{a_i\} </math> and <math>\{b_i\} </math> are not necessarily nonincreasing, then we still write <math>\{a_i\}\succ\{b_i\} </math> if this is true after the sequences have been sorted in nonincreasing order.
-<math>\displaystyle a_1+a_2\geq b_1+b_2</math>
+=== Minorization ===
-<math>\displaystyle \vdots</math>
+We will occasionally say that <math> b_1, \ldots, b_n </math> ''minorizes'' <math> a_1, \ldots, a_n </math>, and write <math>\{b_i\}\prec\{a_i\} </math>, if <math>\{a_i\}\succ\{b_i\} </math>.
-<math>\displaystyle a_1+a_2+\cdots+a_{n-1}\geq b_1+b_2+\cdots+b_{n-1}</math>
+== Alternative Criteria ==
-<math>\displaystyle a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n</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>,
+<center>
+<math>
+\sum_{i=1}^{n}|t-a_i| \ge \sum_{i=1}^{n}|t-b_i|
+</math>.
+</center>
+Both of these conditions are equivalent to our original definition.
+== See Also ==
+* [[Inequalities]]
+* [[Karamata's Inequality]]
+* [[Convex function]]
 {{stub}}

Page

Toolbox

Search

Difference between revisions of "Majorization"

Latest revision as of 21:48, 5 July 2023

Contents

Definition

Minorization

Alternative Criteria

See Also