Difference between revisions of "Majorization"
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== Definition == | == Definition == | ||
| − | 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 | + | 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\succ\{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>k = n </math>. If <math>\{a_i\} </math> and <math>\{b_i\} </math> are not necessarily nonincreasing, then we still write <math>\{a_i\}\succ\{b_i\} </math> if this is true after the sequences have been sorted in nonincreasing order. |
=== Minorization === | === Minorization === | ||
| − | We will occasionally say that <math> b_1, \ldots, b_n </math> ''minorizes'' <math> a_1, \ldots, a_n </math>, and write <math> | + | We will occasionally say that <math> b_1, \ldots, b_n </math> ''minorizes'' <math> a_1, \ldots, a_n </math>, and write <math>\{b_i\}\prec\{a_i\} </math>, if <math>\{a_i\}\succ\{b_i\} </math>. |
== Alternative Criteria == | == Alternative Criteria == | ||
Latest revision as of 21:48, 5 July 2023
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 
minorizes 
.
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
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