Elementary symmetric sum: Difference between revisions

Revision as of 19:23, 23 November 2007

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 = $e_1 = a+b+c+d$

2nd Symmetric Sum = $e_2 = ab+ac+ad+bc+bd+cd$

3rd Symmetric Sum = $e_3 = abc+abd+acd+bcd$

4th Symmetric Sum = $e_4 = abcd$

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 elmentary 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.

Elmentary 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 $e_k$ .

@@ Line 1: / Line 1: @@
-A '''symmetric sum''' is a type of [[summation]].
+An '''elementary symmetric sum''' is a type of [[summation]].
 == 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:
-The <math>k</math>-th '''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:
+st Symmetric Sum = <math>e_1 = a+b+c+d</math>
-st Symmetric Sum = <math>a+b+c+d</math>
+nd Symmetric Sum = <math>e_2 = ab+ac+ad+bc+bd+cd</math>
-nd Symmetric Sum = <math>ab+ac+ad+bc+bd+cd</math>
+rd Symmetric Sum = <math>e_3 = abc+abd+acd+bcd</math>
-rd Symmetric Sum = <math>abc+abd+acd+bcd</math>
+th Symmetric Sum = <math>e_4 = abcd</math>
-th Symmetric Sum = <math>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>
-==Notation==
-The first 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 ==
+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.
+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>.
-Symmetric sums show up in [[Vieta's formulas]]
 ==See Also==
+*[[Symmetric sum]]
 *[[Cyclic sum]]
 [[Category:Algebra]]
 [[Category:Definition]]

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Elementary symmetric sum: Difference between revisions

Revision as of 19:23, 23 November 2007

Contents

Definition

Notation

Uses

See Also