Cauchy-Schwarz Inequality

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

$\displaystyle ({a_1}^2+{a_2}^2+ \cdots +{a_n}^2)({b_1}^2+{b_2}^2+ \cdots +{b_n}^2) \geq (a_1b_1+a_2b_2+ \cdots +a_nb_n)^2$

Equality holds if and only if $\frac{a_1}{b_1}=\frac{a_2}{b_2}= \cdots =\frac{a_n}{b_n}$ .

Proof

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

$\displaystyle (a_1x+b_1)^2+(a_2x+b_2)^2+ \cdots +(a_nx+b_n)^2=0$ .

Expanding, we find the equation to be of the form

$\displaystyle Ax^2 + Bx + C,$

where $A=\sum_{i=1}^n a_i^2$ , $B=2\sum_{j=1}^n a_jb_j$ , and $C=\sum_{k=1}^n b_k^2.$ . 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 $x=0$ . Either way, the discriminant of the equation is nonpositive. Taking the discriminant, $B^2-4AC \leq 0$ and substituting the above values of A, B, and C leaves us with the Cauchy-Schwarz Inequality,

$\displaystyle (a_1b_1+a_2b_2+ \cdots +a_nb_n)^2 \leq (a_1^2+a_2^2+ \cdots +a_n^2)(b_1^2+b_2^2+ \cdots +b_n^2),$

or, in the more compact sigma notation,

$\left(\sum a_ib_i\right) \leq \left(\sum a_i^2\right)\left(\sum b_i^2\right).$

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 $\frac{a_1}{b_1}=\frac{a_2}{b_2}= \cdots =\frac{a_n}{b_n}$ .

Contest Problem Solving

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

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Cauchy-Schwarz Inequality

Proof

Contest Problem Solving

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