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Difference between revisions of "Majorization"

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== Definition ==
 
== Definition ==
  
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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</math>[[Image:succ.gif]]<math>\{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> \displaystyle k = n </math>.  If <math> \displaystyle \{a_i\} </math> and <math> \displaystyle \{b_i\} </math> are not necessarily nonincreasing, then we still write <math> \displaystyle \{a_i\} </math>[[Image:succ.gif]]<math> \displaystyle \{b_i\} </math> if this is true after the sequences have been sorted in nonincreasing order.
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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.
  
 
=== Minorization ===
 
=== Minorization ===
  
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We will occasionally say that <math> b_1, \ldots, b_n </math> ''minorizes'' <math> a_1, \ldots, a_n </math>, and write <math> \displaystyle \{b_i\} </math>[[Image:prec.gif]]<math> \displaystyle \{a_i\} </math>, if <math> \displaystyle \{a_i\} </math>[[Image:succ.gif]]<math> \displaystyle \{b_i\} </math>.
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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>.
  
 
== Alternative Criteria ==
 
== Alternative Criteria ==
  
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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>.
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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>,
 
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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* [[Inequalities]]
 
* [[Inequalities]]
 
* [[Karamata's Inequality]]
 
* [[Karamata's Inequality]]
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* [[Convexity]]
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* [[Convex function]]
  
 
{{stub}}
 
{{stub}}

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 $\{a_i\}_{i=1}^n$Succ.gif$\{b_i\}_{i=1}^n$ 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.

See Also

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