Difference between revisions of "Elementary symmetric sum"
m (→Uses: spelling errors, plus what we sum over in Vieta's formulas.) |
m (The more popular notation is S_n for sums) |
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The <math>k</math>-th '''elmentary 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: | The <math>k</math>-th '''elmentary 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> | + | 1st Symmetric Sum = <math>S_1 = a+b+c+d</math> |
| − | 2nd Symmetric Sum = <math> | + | 2nd Symmetric Sum = <math>S_2 = ab+ac+ad+bc+bd+cd</math> |
| − | 3rd Symmetric Sum = <math> | + | 3rd Symmetric Sum = <math>S_3 = abc+abd+acd+bcd</math> |
| − | 4th Symmetric Sum = <math> | + | 4th Symmetric Sum = <math>S_4 = abcd</math> |
==Notation== | ==Notation== | ||
Revision as of 22:02, 27 June 2013
An elementary symmetric sum is a type of summation.
Contents
Definition
The 
-th elmentary 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 elmentary 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.
