Majorization

Majorization is a mathematical term from set theory .

Definition

Let be $X=\{x_{1},x_{2},\ldots ,x_{n}\},Y=\{y_{1},y_{2},\ldots ,y_{n}\}$ ${\ displaystyle X = \ {x_ {1}, x_ {2}, \ ldots, x_ {n} \}, Y = \ {y_ {1}, y_ {2}, \ ldots, y_ {n} \} }$ where $x_{1}\geqslant x_{2}\geqslant \ldots \geqslant x_{n},y_{1}\geqslant y_{2}\geqslant \ldots \geqslant y_{n}$ ${\ displaystyle x_ {1} \ geqslant x_ {2} \ geqslant \ ldots \ geqslant x_ {n}, y_ {1} \ geqslant y_ {2} \ geqslant \ ldots \ geqslant y_ {n}}$ .

They say that many $X$ ${\ displaystyle X}$ majorizes a lot $Y$ ${\ displaystyle Y}$ (indicated by $X\succ Y$ ${\ displaystyle X \ succ Y}$ ) if the following is true:

for anyone

k=1\ldots n-1

{\ displaystyle k = 1 \ ldots n-1}

,

\sum _{i=1}^{k}x_{i}\geqslant \sum _{i=1}^{k}y_{i}

{\ displaystyle \ sum _ {i = 1} ^ {k} x_ {i} \ geqslant \ sum _ {i = 1} ^ {k} y_ {i}}

; and

\sum _{i=1}^{n}x_{i}=\sum _{i=1}^{n}y_{i}

{\ displaystyle \ sum _ {i = 1} ^ {n} x_ {i} = \ sum _ {i = 1} ^ {n} y_ {i}}

If the last equality is replaced by a less strong condition $\sum _{i=1}^{n}x_{i}\geqslant \sum _{i=1}^{n}y_{i}$ ${\ displaystyle \ sum _ {i = 1} ^ {n} x_ {i} \ geqslant \ sum _ {i = 1} ^ {n} y_ {i}}$ then $X$ ${\ displaystyle X}$ lax majorizes $Y$ ${\ displaystyle Y}$ .

Majorization can be generalized to the case of disordered sets of numbers. A bunch of $X$ ${\ displaystyle X}$ majorizes a lot $Y$ ${\ displaystyle Y}$ if nonincreasing permutation $X$ ${\ displaystyle X}$ majorizes non-increasing permutation $Y$ ${\ displaystyle Y}$ .

Examples

$\{8,7,1\}\succ \{6,5,5\}$ ${\ displaystyle \ {8,7,1 \} \ succ \ {6,5,5 \}}$ , as $8\geqslant 6,\ 8+7\geqslant 6+5,\ 8+7+1=6+5+5$ ${\ displaystyle 8 \ geqslant 6, \ 8 + 7 \ geqslant 6 + 5, \ 8 + 7 + 1 = 6 + 5 + 5}$

$\{3,2\}\succ \{3,2\}$ ${\ displaystyle \ {3,2 \} \ succ \ {3,2 \}}$ , as $3\geqslant 3,\ 3+2=3+2$ ${\ displaystyle 3 \ geqslant 3, \ 3 + 2 = 3 + 2}$

Generally, for any $x_{1}\geqslant x_{2}\geqslant \ldots \geqslant x_{n}$ ${\ displaystyle x_ {1} \ geqslant x_ {2} \ geqslant \ ldots \ geqslant x_ {n}}$ the following is true:

$\{x_{1}+x_{2}+\ldots +x_{n},\underbrace {0,0,\ldots ,0} _{n-1}\}\succ \{x_{1},x_{2},\ldots ,x_{n}\}\succ \{\underbrace {{\frac {x_{1}+\ldots +x_{n}}{n}},{\frac {x_{1}+\ldots +x_{n}}{n}},\ldots ;{\frac {x_{1}+\ldots +x_{n}}{n}}} _{n}\}$ ${\ displaystyle \ {x_ {1} + x_ {2} + \ ldots + x_ {n}, \ underbrace {0,0, \ ldots, 0} _ {n-1} \} \ succ \ {x_ {1 }, x_ {2}, \ ldots, x_ {n} \} \ succ \ {\ underbrace {{\ frac {x_ {1} + \ ldots + x_ {n}} {n}}, {\ frac {x_ {1} + \ ldots + x_ {n}} {n}}, \ ldots; {\ frac {x_ {1} + \ ldots + x_ {n}} {n}}} _ {n} \}}$

Muirhead Inequality

Let be $F_{1}$ ${\ displaystyle F_ {1}}$ - monomial symmetrization $x_{1}^{\alpha _{1}}\ldots x_{n}^{\alpha _{n}}$ ${\ displaystyle x_ {1} ^ {\ alpha _ {1}} \ ldots x_ {n} ^ {\ alpha _ {n}}}$ , $F_{2}$ ${\ displaystyle F_ {2}}$ - monomial symmetrization $x_{1}^{\beta _{1}}\ldots x_{n}^{\beta _{n}}$ ${\ displaystyle x_ {1} ^ {\ beta _ {1}} \ ldots x_ {n} ^ {\ beta _ {n}}}$ . If $\{\alpha _{1};\ldots ;\alpha _{n}\}\succ \{\beta _{1};\ldots ;\beta _{n}\}$ ${\ displaystyle \ {\ alpha _ {1}; \ ldots; \ alpha _ {n} \} \ succ \ {\ beta _ {1}; \ ldots; \ beta _ {n} \}}$ then for all non-negative $x_{1},\dots ,x_{n}$ ${\ displaystyle x_ {1}, \ dots, x_ {n}}$ inequality holds $F_{1}(\alpha _{1},\ldots \alpha _{n})\geqslant F_{2}(\beta _{1},\ldots \beta _{n})$ ${\ displaystyle F_ {1} (\ alpha _ {1}, \ ldots \ alpha _ {n}) \ geqslant F_ {2} (\ beta _ {1}, \ ldots \ beta _ {n})}$ .

Majorization

Definition

Examples

Muirhead Inequality

More articles: