# Difference between revisions of "Cauchy-Schwarz Inequality"

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The '''Cauchy-Schwarz Inequality''' (which is known by other names, including Cauchy's Inequality) states that, for two sets of real numbers <math>a_1,a_2,\ldots,a_n</math> and <math>b_1,b_2,\ldots,b_n</math>, the following inequality is always true: | The '''Cauchy-Schwarz Inequality''' (which is known by other names, including Cauchy's Inequality) states that, for two sets of real numbers <math>a_1,a_2,\ldots,a_n</math> and <math>b_1,b_2,\ldots,b_n</math>, the following inequality is always true: | ||

− | <math>\displaystyle({a_1}^2+{a_2}^2+...+{a_n}^2)({b_1}^2+{b_2}^2+...+{b_n}^2)\geq(a_1b_1+a_2b_2+...+a_nb_n)^2</math> | + | <math> \displaystyle ({a_1}^2+{a_2}^2+...+{a_n}^2)({b_1}^2+{b_2}^2+...+{b_n}^2)\geq(a_1b_1+a_2b_2+...+a_nb_n)^2</math> |

Equality holds if and only if <math>\frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n}</math>. | Equality holds if and only if <math>\frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n}</math>. | ||

There are many ways to prove this; one of the more well-known is to consider the equation | There are many ways to prove this; one of the more well-known is to consider the equation | ||

− | <math>(a_1x+b_1)^2+(a_2x+b_2)^2+...+(a_nx+b_n)^2=0</math>. Expanding, we find the equation to be of the form <math>Ax^2+Bx+C</math>, where <math>A=\sum_{i=1}^n a_i^2</math>, <math>B=2\sum_{j=1}^n a_jb_j</math>, and <math>C=\sum_{k=1}^n b_k^2.</math>. By the [[Trivial inequality | Trivial Inequality]], we know that the left-hand-side of the original equation is always at least 0, so either both roots are [[Complex Numbers]], or there is a double root at <math>x=0</math>. Either way, the [[discriminant]] of the equation is nonpositive. Taking the [[discriminant]], <math>B^2-4AC \leq 0</math> and substituting the above values of A, B, and C leaves us with the '''Cauchy-Schwarz Inequality''', <math>(a_1b_1+a_2b_2+...+a_nb_n)^2 \leq (a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2)</math>, | + | |

+ | <math> \displaystyle (a_1x+b_1)^2+(a_2x+b_2)^2+...+(a_nx+b_n)^2=0</math>. | ||

+ | |||

+ | Expanding, we find the equation to be of the form <math>Ax^2+Bx+C</math>, where <math>A=\sum_{i=1}^n a_i^2</math>, <math>B=2\sum_{j=1}^n a_jb_j</math>, and <math>C=\sum_{k=1}^n b_k^2.</math>. By the [[Trivial inequality | Trivial Inequality]], we know that the left-hand-side of the original equation is always at least 0, so either both roots are [[Complex Numbers]], or there is a double root at <math>x=0</math>. Either way, the [[discriminant]] of the equation is nonpositive. Taking the [[discriminant]], <math>B^2-4AC \leq 0</math> and substituting the above values of A, B, and C leaves us with the '''Cauchy-Schwarz Inequality''', <math>(a_1b_1+a_2b_2+...+a_nb_n)^2 \leq (a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2)</math>, | ||

or, in the more compact [[sigma notation]], | or, in the more compact [[sigma notation]], | ||

<math>\left(\sum a_ib_i\right)</math> <math>\leq \left(\sum a_i^2\right)\left(\sum b_i^2\right)</math> | <math>\left(\sum a_ib_i\right)</math> <math>\leq \left(\sum a_i^2\right)\left(\sum b_i^2\right)</math> |

## Revision as of 01:29, 18 June 2006

The **Cauchy-Schwarz Inequality** (which is known by other names, including Cauchy's Inequality) states that, for two sets of real numbers and , the following inequality is always true:

Equality holds if and only if .

There are many ways to prove this; one of the more well-known is to consider the equation

.

Expanding, we find the equation to be of the form , where , , and . By the Trivial Inequality, we know that the left-hand-side of the original equation is always at least 0, so either both roots are Complex Numbers, or there is a double root at . Either way, the discriminant of the equation is nonpositive. Taking the discriminant, and substituting the above values of A, B, and C leaves us with the **Cauchy-Schwarz Inequality**, ,
or, in the more compact sigma notation,

Note that this also gives us the equality case; equality holds if and only if the discriminant is equal to 0, which is true if and only if the equation has 0 as a double root, which is true if and only if .

This inequality is used very frequently to solve Olympiad-level Inequality problems, such as those on the USAMO and IMO.