In mathematics, majorization is a preorder on vectors of real numbers. For two such vectors, 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, commonly denoted x ≻ w y , {\displaystyle \mathbf {x} \succ _{w}\mathbf {y} ,} when
∑ i = 1 k x i ↓ ≥ ∑ i = 1 k y i ↓ {\displaystyle \sum _{i=1}^{k}x_{i}^{\downarrow }\geq \sum _{i=1}^{k}y_{i}^{\downarrow }} for all k = 1 , … , n {\displaystyle k=1,\,\dots ,\,n} , where x i ↓ {\displaystyle x_{i}^{\downarrow }} denotes the i {\displaystyle i} th largest entry of x {\displaystyle \mathbf {x} } . If x , y {\displaystyle \mathbf {x} ,\mathbf {y} } further satisfy ∑ i = 1 n x i = ∑ i = 1 n y i {\displaystyle \sum _{i=1}^{n}x_{i}=\sum _{i=1}^{n}y_{i}} , we say that x {\displaystyle \mathbf {x} } majorizes (or dominates) y {\displaystyle \mathbf {y} } , commonly denoted x ≻ y {\displaystyle \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 ) ≺ ( 0 , 3 ) {\displaystyle (1,2)\prec (0,3)} is simply equivalent to ( 2 , 1 ) ≺ ( 3 , 0 ) {\displaystyle (2,1)\prec (3,0)} . Specifically, x ≻ y ∧ y ≻ x {\displaystyle \mathbf {x} \succ \mathbf {y} \wedge \mathbf {y} \succ \mathbf {x} } if and only if x , y {\displaystyle \mathbf {x} ,\mathbf {y} } are permutations of each other. Similarly for ≻ w {\displaystyle \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 ) ≥ g ( x ) {\displaystyle f(x)\geq g(x)} for all x {\displaystyle x} in the domain, or other technical definitions, such as majorizing measures in probability theory.
Equivalent conditions
Geometric definition
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