### What does **majorization** mean?

# Definitions for majorization

ma·joriza·tion

#### This dictionary definitions page includes all the possible meanings, example usage and translations of the word **majorization**.

### Wiktionary

majorizationnoun

A partial order over vectors of real numbers

### Wikipedia

Majorization

In mathematics, majorization is a preorder on vectors of real numbers. Let x ( i ) , i = 1 , … , n {\displaystyle {x}_{(i)}^{},\ i=1,\,\ldots ,\,n} denote the i {\displaystyle i} -th largest element of the vector x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} . Given x , y ∈ R n {\displaystyle \mathbf {x} ,\ \mathbf {y} \in \mathbb {R} ^{n}} , we say that x {\displaystyle \mathbf {x} } weakly majorizes (or dominates) y {\displaystyle \mathbf {y} } from below (or equivalently, we say that y {\displaystyle \mathbf {y} } is weakly majorized (or dominated) by x {\displaystyle \mathbf {x} } from below) denoted as x ≻ w y {\displaystyle \mathbf {x} \succ _{w}\mathbf {y} } if ∑ i = 1 k x ( i ) ≥ ∑ i = 1 k y ( i ) {\displaystyle \sum _{i=1}^{k}x_{(i)}^{}\geq \sum _{i=1}^{k}y_{(i)}^{}} for all k = 1 , … , d {\displaystyle k=1,\,\dots ,\,d} . If in addition ∑ i = 1 d x i = ∑ i = 1 d y i {\displaystyle \sum _{i=1}^{d}x_{i}^{}=\sum _{i=1}^{d}y_{i}^{}} , we say that x {\displaystyle \mathbf {x} } majorizes (or dominates) y {\displaystyle \mathbf {y} } , written as x ≻ y {\displaystyle \mathbf {x} \succ \mathbf {y} } , or equivalently, we say that y {\displaystyle \mathbf {y} } is majorized (or dominated) by x {\displaystyle \mathbf {x} } . The order of the entries of the vectors x {\displaystyle \mathbf {x} } or y {\displaystyle \mathbf {y} } does not affect the majorization, e.g., the statement ( 1 , 2 ) ≺ ( 0 , 3 ) {\displaystyle (1,2)\prec (0,3)} is simply equivalent to ( 2 , 1 ) ≺ ( 3 , 0 ) {\displaystyle (2,1)\prec (3,0)} . As a consequence, majorization is not a partial order, since x ≻ y {\displaystyle \mathbf {x} \succ \mathbf {y} } and y ≻ x {\displaystyle \mathbf {y} \succ \mathbf {x} } do not imply x = y {\displaystyle \mathbf {x} =\mathbf {y} } , it only implies that the components of each vector are equal, but not necessarily in the same order. The majorization partial order on finite dimensional vectors, described here, can be generalized to the Lorenz ordering, a partial order on distribution functions. For example, a wealth distribution is Lorenz-greater than another if its Lorenz curve lies below the other. As such, a Lorenz-greater wealth distribution has a higher Gini coefficient, and has more income disparity. Various other generalizations of majorization are discussed in chapters 14 and 15 of.

### Wikidata

Majorization

In mathematics, majorization is a preorder on vectors of real numbers. For a vector, we denote by the vector with the same components, but sorted in descending order. Given, we say that weakly majorizes from below written as iff where and are the elements of and, respectively, sorted in decreasing order. Equivalently, we say that is weakly majorized by from below, denoted as . Similarly, we say that weakly majorizes from above written as iff Equivalently, we say that is weakly majorized by from above, denoted as . If and in addition we say that majorizes written as . Equivalently, we say that is majorized by, denoted as . It is easy to see that if and only if and . Note that the majorization order do not depend on the order of the components of the vectors or . Majorization is not a partial order, since and do not imply, it only implies that the components of each vector are equal, but not necessarily in the same order. Regrettably, to confuse the matter, some literature sources use the reverse notation, e.g., is replaced with, most notably, in Horn and Johnson, Matrix analysis, Definition 4.3.24, while the same authors switch to the traditional notation, introduced here, later in their Topics in Matrix Analysis.

### Numerology

Chaldean Numerology

The numerical value of majorization in Chaldean Numerology is:

**5**Pythagorean Numerology

The numerical value of majorization in Pythagorean Numerology is:

**7**

## Translations for **majorization**

### From our Multilingual Translation Dictionary

### Get even more translations for majorization »

### Translation

#### Find a translation for the **majorization** definition in other languages:

Select another language:

- - Select -
- 简体中文 (Chinese - Simplified)
- 繁體中文 (Chinese - Traditional)
- Español (Spanish)
- Esperanto (Esperanto)
- 日本語 (Japanese)
- Português (Portuguese)
- Deutsch (German)
- العربية (Arabic)
- Français (French)
- Русский (Russian)
- ಕನ್ನಡ (Kannada)
- 한국어 (Korean)
- עברית (Hebrew)
- Gaeilge (Irish)
- Українська (Ukrainian)
- اردو (Urdu)
- Magyar (Hungarian)
- मानक हिन्दी (Hindi)
- Indonesia (Indonesian)
- Italiano (Italian)
- தமிழ் (Tamil)
- Türkçe (Turkish)
- తెలుగు (Telugu)
- ภาษาไทย (Thai)
- Tiếng Việt (Vietnamese)
- Čeština (Czech)
- Polski (Polish)
- Bahasa Indonesia (Indonesian)
- Românește (Romanian)
- Nederlands (Dutch)
- Ελληνικά (Greek)
- Latinum (Latin)
- Svenska (Swedish)
- Dansk (Danish)
- Suomi (Finnish)
- فارسی (Persian)
- ייִדיש (Yiddish)
- հայերեն (Armenian)
- Norsk (Norwegian)
- English (English)

### Word of the Day

#### Would you like us to send you a **FREE** new word definition delivered to your inbox daily?

## Citation

#### Use the citation below to add this definition to your bibliography:

**Style:**MLAChicagoAPA

"majorization." *Definitions.net.* STANDS4 LLC, 2024. Web. 2 Nov. 2024. <https://www.definitions.net/definition/majorization>.

## Discuss these majorization definitions with the community:

## Report Comment

We're doing our best to make sure our content is useful, accurate and safe.

If by any chance you spot an inappropriate comment while navigating through our website please use this form to let us know, and we'll take care of it shortly.

## Attachment

## You need to be logged in to favorite.

## Log In