# Difference between revisions of "Majorization"

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− | + | == Definition == | |

− | <math>\displaystyle | + | 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. |

− | + | === Minorization === | |

− | <math>\displaystyle \ | + | 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>. |

− | + | == Alternative Criteria == | |

− | <math>\displaystyle | + | 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>. |

+ | |||

+ | 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}} | {{stub}} |

## Latest revision as of 13:47, 12 September 2008

## Definition

We say a nonincreasing sequence of real numbers **majorizes** another nonincreasing sequence , and write if and only if all for all , , with equality when . If and are not necessarily nonincreasing, then we still write if this is true after the sequences have been sorted in nonincreasing order.

### Minorization

We will occasionally say that *minorizes* , and write , if .

## Alternative Criteria

It is also true that if and only if for all , , with equality when . An interesting consequence of this is that the finite sequence majorizes if and only if majorizes .

We can also say that this is the case if and only if for all ,

.

Both of these conditions are equivalent to our original definition.

## See Also

*This article is a stub. Help us out by expanding it.*