Difference between revisions of "Elementary symmetric sum"
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| − | + | An '''elementary symmetric sum''' is a type of [[summation]]. | |
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== Definition == | == Definition == | ||
| + | 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>e_1 = a+b+c+d</math> | |
| − | + | 2nd Symmetric Sum = <math>e_2 = ab+ac+ad+bc+bd+cd</math> | |
| − | + | 3rd Symmetric Sum = <math>e_3 = abc+abd+acd+bcd</math> | |
| − | + | 4th Symmetric Sum = <math>e_4 = abcd</math> | |
| − | + | ==Notation== | |
| + | The first elmentary symmetric sum of <math>f(x)</math> is often written <math>\sum_{sym}f(x)</math>. The <math>n</math>th can be written <math>\sum_{sym}^{n}f(x)</math> | ||
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== Uses == | == Uses == | ||
| + | Any symmetric sum can be written as a [[polynomial]] of the elmentary symmetric sum functions. For example, <math>x^3 + y^3 + z^3 = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - xz) + 3xyz = e_1^3 - 3e_1e_2 + 3e_3</math>. This is often used to solve systems of equations involving [[power sum]]s, combined with Vieta's. | ||
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| + | Elmentary symmetric sums show up in [[Vieta's formulas]]. In a monic polynomial, the coefficient of the <math>x^1</math> term is <math>e_1</math>, and the coefficient of the <math>x^k</math> term is <math>e_k</math>. | ||
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==See Also== | ==See Also== | ||
| + | *[[Symmetric sum]] | ||
*[[Cyclic sum]] | *[[Cyclic sum]] | ||
[[Category:Algebra]] | [[Category:Algebra]] | ||
[[Category:Definition]] | [[Category:Definition]] | ||
Revision as of 19:23, 23 November 2007
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 elmentary symmetric sum functions. For example, 
. This is often used to solve systems of equations involving power sums, combined with Vieta's.
Elmentary 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 
.
