Difference between revisions of "Elementary symmetric sum"
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== Definition == | == Definition == | ||
| − | The <math>k</math>-th ''' | + | The <math>k</math>-th '''elementary symmetric sum''' of a [[set]] of <math>n</math> numbers is the sum of all products of <math>k</math> of those numbers (<math>1 \leq k \leq n</math>). For example, if <math>n = 4</math>, and our set of numbers is <math>\{a, b, c, d\}</math>, then: |
1st Symmetric Sum = <math>S_1 = a+b+c+d</math> | 1st Symmetric Sum = <math>S_1 = a+b+c+d</math> | ||
Revision as of 00:45, 28 November 2014
An elementary symmetric sum is a type of summation.
Contents
Definition
The 
-th elementary symmetric sum of a set of 
numbers is the sum of all products of of those numbers (
). For example, if 
, and our set of numbers is 
, then:
1st Symmetric Sum = 
2nd Symmetric Sum = 
3rd Symmetric Sum = 
4th Symmetric Sum = 
Notation
The first elementary symmetric sum of 
is often written 
. The th can be written 
Uses
Any symmetric sum can be written as a polynomial of the elementary symmetric sum functions. For example, 
. This is often used to solve systems of equations involving power sums, combined with Vieta's formulas.
Elementary symmetric sums show up in Vieta's formulas. In a monic polynomial, the coefficient of the 
term is 
, and the coefficient of the 
term is 
, where the symmetric sums are taken over the roots of the polynomial.
