Difference between revisions of "Elementary symmetric sum"
m (The more popular notation is S_n for sums) |
m (Corrected invalid wikipedia link to Sums of Powers) |
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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> | ||
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==Notation== | ==Notation== | ||
− | The first | + | The first elementary 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> |
== Uses == | == Uses == | ||
− | Any symmetric sum can be written as a [[polynomial]] of the elementary 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 = | + | Any [[symmetric sum]] can be written as a [[polynomial]] of the elementary 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 = S_1^3 - 3S_1S_2 + 3S_3</math>. This is often used to solve systems of equations involving [https://en.wikipedia.org/wiki/Sums_of_powers sums of powers], combined with Vieta's formulas. |
Elementary 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>, where the symmetric sums are taken over the roots of the polynomial. | Elementary 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>, where the symmetric sums are taken over the roots of the polynomial. |
Revision as of 21:53, 25 October 2018
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 sums of powers, 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.