# 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 | + | The '''Cauchy-Schwarz Inequality''' (which is known by other names, including '''Cauchy's Inequality''', '''Schwarz's Inequality''', and the '''Cauchy-Bunyakovsky-Schwarz Inequality''') is a well-known [[inequality]] with many elegant applications. |

− | + | == Elementary Form == | |

− | + | For any real numbers <math> a_1, \ldots, a_n </math> and <math> b_1, \ldots, b_n </math>, | |

+ | <center> | ||

+ | <math> | ||

+ | \left( \sum_{i=1}^{n}a_ib_i \right)^2 \le \sum_{i=1}^{n}a_i^2 \sum_{i=1}^{n}b_i^2 | ||

+ | </math>, | ||

+ | </center> | ||

+ | with equality when there exist constants <math> \displaystyle \mu, \lambda </math> not both zero such that for all <math> 1 \le i \le n </math>, <math> \displaystyle \mu a_i = \lambda b_i </math>. | ||

− | == Proof == | + | === Proof === |

− | |||

− | + | There are several proofs; we will present an elegant one that does not generalize. | |

− | + | Consider the vectors <math> \mathbf{a} = \langle a_1, \ldots a_n \rangle </math> and <math> {} \mathbf{b} = \langle b_1, \ldots b_n \rangle </math>. If <math> \displaystyle \theta </math> is the [[angle]] formed by <math> \mathbf{a} </math> and <math> \mathbf{b} </math>, then the left-hand side of the inequality is equal to the square of the [[dot product]] of <math> \mathbf{a} </math> and <math> \mathbf{b} </math>, or <math> \left( ||\mathbf{a}|| \cdot ||\mathbf{b}|| \cos\theta \right)^2 </math>. The right hand side of the inequality is equal to <math> \left( ||\mathbf{a}|| \cdot ||\mathbf{b}|| \right)^2 </math>. The inequality then follows from <math> |\cos\theta | \le 1 </math>, with equality when one of <math> \mathbf{a,b} </math> is a multiple of the other, as desired. | |

− | + | === Complex Form === | |

− | + | The inequality sometimes appears in the following form. | |

− | <math> \ | + | Let <math> a_1, \ldots, a_n </math> and <math> b_1, \ldots, b_n </math> be [[complex numbers]]. Then |

+ | <center> | ||

+ | <math> | ||

+ | \left| \sum_{i=1}^na_ib_i \right|^2 \le \sum_{i=1}^{n}|a_i^2| \sum_{i=1}^n |b_i^2| | ||

+ | </math>. | ||

+ | </center> | ||

+ | This appears to be more powerful, but it follows immediately from | ||

+ | <center> | ||

+ | <math> | ||

+ | \left| \sum_{i=1}^n a_ib_i \right| ^2 \le \left( \sum_{i=1}^n |a_i| \cdot |b_i| \right)^2 \le \sum_{i=1}^n |a_i^2| \sum_{i=1}^n |b_i^2| | ||

+ | </math>. | ||

+ | </center> | ||

− | + | == General Form == | |

− | <math>{\ | + | Let <math> \displaystyle V </math> be a [[vector space]], and let <math> \langle \cdot, \cdot \rangle : V \times V \mapsto \mathbb{R} </math> be an [[inner product]]. Then for any <math> \mathbf{a,b} \in V </math>, |

− | + | <center> | |

− | + | <math> | |

+ | \langle \mathbf{a,b} \rangle^2 \le \langle \mathbf{a,a} \rangle \langle \mathbf{b,b} \rangle | ||

+ | </math>, | ||

+ | </center> | ||

+ | with equality if and only if there exist constants <math> \displaystyle \mu, \lambda </math> not both zero such that <math> \mu\mathbf{a} = \lambda\mathbf{b} </math>. | ||

− | == | + | === Proof 1 === |

− | This | + | |

+ | Consider the polynomial of <math> \displaystyle t </math> | ||

+ | <center> | ||

+ | <math> | ||

+ | \langle t\mathbf{a + b}, t\mathbf{a + b} \rangle = t^2\langle \mathbf{a,a} \rangle + 2t\langle \mathbf{a,b} \rangle + \langle \mathbf{b,b} \rangle | ||

+ | </math>. | ||

+ | </center> | ||

+ | This must always be greater than or equal to zero, so it must have a non-positive discriminant, i.e., <math> \langle \mathbf{a,b} \rangle^2 </math> must be less than or equal to <math> \langle \mathbf{a,a} \rangle \langle \mathbf{b,b} \rangle </math>, with equality when <math> \mathbf{a = 0} </math> or when there exists some scalar <math> \displaystyle -t </math> such that <math> -t\mathbf{a} = \mathbf{b} </math>, as desired. | ||

+ | |||

+ | === Proof 2 === | ||

+ | |||

+ | We consider | ||

+ | <center> | ||

+ | <math> | ||

+ | \langle \mathbf{a-b, a-b} \rangle = \langle \mathbf{a,a} \rangle + \langle \mathbf{b,b} \rangle - 2 \langle \mathbf{a,b} \rangle | ||

+ | </math>. | ||

+ | </center> | ||

+ | Since this is always greater than or equal to zero, we have | ||

+ | <center> | ||

+ | <math> | ||

+ | \langle \mathbf{a,b} \rangle \le \frac{1}{2} \langle \mathbf{a,a} \rangle + \frac{1}{2} \langle \mathbf{b,b} \rangle | ||

+ | </math>. | ||

+ | </center> | ||

+ | Now, if either <math> \mathbf{a} </math> or <math> \mathbf{b} </math> is equal to <math> \mathbf{0} </math>, then <math> \langle \mathbf{a,b} \rangle^2 = \langle \mathbf{a,a} \rangle \langle \mathbf{b,b} \rangle = 0 </math>. Otherwise, we may [[normalize]] so that <math> \langle \mathbf {a,a} \rangle = \langle \mathbf{b,b} \rangle = 1 </math>, and we have | ||

+ | <center> | ||

+ | <math> | ||

+ | \langle \mathbf{a,b} \rangle \le 1 = \langle \mathbf{a,a} \rangle^{1/2} \langle \mathbf{b,b} \rangle^{1/2} | ||

+ | </math>, | ||

+ | </center> | ||

+ | with equality when <math>\mathbf{a} </math> and <math> \mathbf{b} </math> may be scaled to each other, as desired. | ||

+ | |||

+ | === Examples === | ||

+ | |||

+ | The elementary form of the Cauchy-Schwarz inequality is a special case of the general form, as is the '''Cauchy-Schwarz Inequality for Integrals''': for integrable functions <math> f,g : [a,b] \mapsto \mathbb{R} </math>, | ||

+ | <center> | ||

+ | <math> | ||

+ | \left( \int_{a}^b f(x)g(x)dx \right)^2 \le \int_{a}^b [f(x)]^2dx \cdot \int_a^b [g(x)]^2 dx | ||

+ | </math>, | ||

+ | </center> | ||

+ | with equality when there exist constants <math> \displaystyle \mu, \lambda </math> not both equal to zero such that for <math> t \in [a,b] </math>, | ||

+ | <center> | ||

+ | <math> | ||

+ | \mu \int_a^t f(x)dx = \lambda \int_a^t g(x)dx | ||

+ | </math>. | ||

+ | </center> | ||

== Other Resources == | == Other Resources == | ||

− | * [http://en.wikipedia.org/wiki/Cauchy- | + | * [http://en.wikipedia.org/wiki/Cauchy-Schwarz_inequality Wikipedia entry] |

===Books=== | ===Books=== | ||

* [http://www.amazon.com/exec/obidos/ASIN/052154677X/artofproblems-20 The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities] by J. Michael Steele. | * [http://www.amazon.com/exec/obidos/ASIN/052154677X/artofproblems-20 The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities] by J. Michael Steele. | ||

* [http://www.amazon.com/exec/obidos/ASIN/0387982191/artofproblems-20 Problem Solving Strategies] by Arthur Engel contains significant material on inequalities. | * [http://www.amazon.com/exec/obidos/ASIN/0387982191/artofproblems-20 Problem Solving Strategies] by Arthur Engel contains significant material on inequalities. |

## Revision as of 16:21, 15 April 2007

The **Cauchy-Schwarz Inequality** (which is known by other names, including **Cauchy's Inequality**, **Schwarz's Inequality**, and the **Cauchy-Bunyakovsky-Schwarz Inequality**) is a well-known inequality with many elegant applications.

## Contents

## Elementary Form

For any real numbers and ,

,

with equality when there exist constants not both zero such that for all , .

### Proof

There are several proofs; we will present an elegant one that does not generalize.

Consider the vectors and . If is the angle formed by and , then the left-hand side of the inequality is equal to the square of the dot product of and , or . The right hand side of the inequality is equal to . The inequality then follows from , with equality when one of is a multiple of the other, as desired.

### Complex Form

The inequality sometimes appears in the following form.

Let and be complex numbers. Then

.

This appears to be more powerful, but it follows immediately from

.

## General Form

Let be a vector space, and let be an inner product. Then for any ,

,

with equality if and only if there exist constants not both zero such that .

### Proof 1

Consider the polynomial of

.

This must always be greater than or equal to zero, so it must have a non-positive discriminant, i.e., must be less than or equal to , with equality when or when there exists some scalar such that , as desired.

### Proof 2

We consider

.

Since this is always greater than or equal to zero, we have

.

Now, if either or is equal to , then . Otherwise, we may normalize so that , and we have

,

with equality when and may be scaled to each other, as desired.

### Examples

The elementary form of the Cauchy-Schwarz inequality is a special case of the general form, as is the **Cauchy-Schwarz Inequality for Integrals**: for integrable functions ,

,

with equality when there exist constants not both equal to zero such that for ,

.

## Other Resources

### Books

- The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele.
- Problem Solving Strategies by Arthur Engel contains significant material on inequalities.