Lesson plan
Majorization
In mathematics, majorization is a preorder on vectors of real numbers. For two such vectors, \mathbf{x},\ \mathbf{y} \in \mathbb{R}^n , we say that \mathbf{x} weakly majorizes (or dominates) \mathbf{y} from below, commonly denoted \mathbf{x} \succ_w \mathbf{y}, when : \sum_{i=1}^k x_i^{\downarrow} \geq \sum_{i=1}^k y_i^{\downarrow} for all k=1,\,\dots,\,n , where x_i^{\downarrow} denotes the i th largest entry of \mathbf{x} . If \mathbf{x}, \mathbf{y} further satisfy \sum_{i=1}^n x_i = \sum_{i=1}^n y_i , we say that \mathbf{x} majorizes (or dominates) \mathbf{y} , commonly denoted \mathbf{x} \succ \mathbf{y} . Both weak majorization and majorization are partial orders for vectors whose entries are non-decreasing, but only a preorder for general vectors, since majorization is agnostic to the ordering of the entries in vectors, e.g., the statement (1,2)\prec (0,3) is simply equivalent to (2,1)\prec (3,0) . Specifically, \mathbf{x} \succ \mathbf{y} \wedge \mathbf{y} \succ \mathbf{x} if and only if \mathbf{x}, \mathbf{y} are permutations of each other. Similarly for \succ_w . Majorizing also sometimes refers to entrywise ordering, e.g. the real-valued function f majorizes the real-valued function g when f(x) \geq g(x) for all x in the domain, or other technical definitions, such as majorizing measures in probability theory.
Article summary
In mathematics, majorization is a preorder on vectors of real numbers. For two such vectors, \mathbf{x},\ \mathbf{y} \in \mathbb{R}^n , we say that \mathbf{x} weakly majorizes (or dominates) \mathbf{y} from below, commonly denoted \mathbf{x} \succ_w \mathbf{y}, when : \sum_{i=1}^k x_i^{\downarrow} \geq \sum_{i=1}^k y_i^{\downarrow} for all k=1,\,\dots,\,n , where x_i^{\downarrow} denotes the i th largest entry of \mathbf{x} . If \mathbf{x}, \mathbf{y} further satisfy \sum_{i=1}^n x_i = \sum_{i=1}^n y_i , we say that \mathbf{x} majorizes (or dominates) \mathbf{y} , commonly denoted \mathbf{x} \succ \mathbf{y} . Both weak majorization and majorization are partial orders for vectors whose entries are non-decreasing, but only a preorder for general vectors, since majorization is agnostic to the ordering of the entries in vectors, e.g., the statement (1,2)\prec (0,3) is simply equivalent to (2,1)\prec (3,0) . Specifically, \mathbf{x} \succ \mathbf{y} \wedge \mathbf{y} \succ \mathbf{x} if and only if \mathbf{x}, \mathbf{y} are permutations of each other. Similarly for \succ_w . Majorizing also sometimes refers to entrywise ordering, e.g. the real-valued function f majorizes the real-valued function g when f(x) \geq g(x) for all x in the domain, or other technical definitions, such as majorizing measures in probability theory.
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.