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>e_1 = a+b+c+d</math>
+
1st Symmetric Sum = <math>S_1 = a+b+c+d</math>
  
2nd Symmetric Sum = <math>e_2 = ab+ac+ad+bc+bd+cd</math>
+
2nd Symmetric Sum = <math>S_2 = ab+ac+ad+bc+bd+cd</math>
  
3rd Symmetric Sum = <math>e_3 = abc+abd+acd+bcd</math>
+
3rd Symmetric Sum = <math>S_3 = abc+abd+acd+bcd</math>
  
4th Symmetric Sum = <math>e_4 = abcd</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.

Definition

The $k$-th elmentary symmetric sum of a set of $n$ numbers is the sum of all products of $k$ of those numbers ($1 \leq k \leq n$). For example, if $n = 4$, and our set of numbers is $\{a, b, c, d\}$, then:

1st Symmetric Sum = $S_1 = a+b+c+d$

2nd Symmetric Sum = $S_2 = ab+ac+ad+bc+bd+cd$

3rd Symmetric Sum = $S_3 = abc+abd+acd+bcd$

4th Symmetric Sum = $S_4 = abcd$

Notation

The first elmentary symmetric sum of $f(x)$ is often written $\sum_{sym}f(x)$. The $n$th can be written $\sum_{sym}^{n}f(x)$

Uses

Any symmetric sum can be written as a polynomial of the elementary symmetric sum functions. For example, $x^3 + y^3 + z^3 = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - xz) + 3xyz = e_1^3 - 3e_1e_2 + 3e_3$. 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 $x^1$ term is $e_1$, and the coefficient of the $x^k$ term is $e_k$, where the symmetric sums are taken over the roots of the polynomial.

See Also

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